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Foundations of Software Testing 2E

Part I

Chapter 1 Chapter 2

Part II

Chapter 3 Chapter 4 Chapter 5 Chapter 6

Part III

Chapter 7 Chapter 8

Part IV

Chapter 9 Chapter 10 Chapter 11

Updated: July 21, 2013
Foundations of Software Testing 2E

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd

ADITYA P. MATHUR

Contents
Author: Aditya P. Mathur

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Chapter 1:

Updated: July 17, 2013

Foundations of Software Testing 2E

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Preliminaries: Software Testing

Contents
Author: Aditya P. Mathur

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Learning Objectives
Errors, Testing, debugging, test process, CFG, correctness, reliability,

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n 

oracles.

n 

Finite state machines

n 

Testing techniques

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1.1 Humans, errors and testing

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Errors
Errors are a part of our daily life.

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Humans make errors in their thoughts, actions, and in the products that
might result from their

actions.

Errors occur wherever humans are involved in taking actions and making
decisions.

These fundamental facts of human existence
make testing an essential activity.

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Errors: Examples

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Error, faults, failures

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1.2 Software Quality

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Software quality

Static quality attributes: structured, maintainable, testable code as well as
the availability of correct and complete documentation.

Dynamic quality attributes: software reliability, correctness,
completeness, consistency, usability, and performance

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Software quality (contd.)

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Completeness refers to the availability of all features listed in the requirements,
or in the user manual. An incomplete software is one that does not fully
implement all features required.

Consistency refers to adherence to a common set of conventions and
assumptions. For example, all buttons in the user interface might follow a
common color coding convention. An example of inconsistency would be when
a database application displays the date of birth of a person in the database.

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Software quality (contd.)
Usability refers to the ease with which an application can be used. This is an
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area in itself and there exist techniques for usability testing. Psychology plays
an important role in the design of techniques for usability testing.

Performance refers to the time the application takes to perform a requested
task. It is considered as a non-functional requirement. It is specified in terms
such as ``This task must be performed at the rate of X units of activity in one
second on a machine running at speed Y, having Z gigabytes of memory."

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1.3 Requirements, behavior, and correctness

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Requirements, behavior, correctness

Requirements leading to two different programs:

Requirement 1: It is required to write a

program that inputs two integers and outputs the maximum of these.

Requirement 2: It is required to write a

program that inputs a sequence of integers and outputs the sorted version of
this sequence.

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Requirements: Incompleteness

of max when the input integers are 13 and 19 can be easily determined to be 19.

Suppose now that the tester wants to know if the two integers are to be input to the
program on one line followed by a carriage return, or on two separate lines with a
carriage return typed in after each number. The requirement as stated above fails to
provide an answer to this question.

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Suppose that program max is developed to satisfy Requirement 1. The expected output

Ltd Requirements: Ambiguity . Contents Foundations of Software Testing 2E Author: Aditya P. The behavior of sort program. It is not clear whether the input sequence is to sorted in ascending or in descending order. will depend on the decision taken by the programmer while writing sort. written to satisfy this requirement. Mathur 15 Copyright © 2013 Dorling Kindersley (India) Pvt.Requirement 2 is ambiguous.

The set of all possible inputs to a program P is known as the input domain or input space.767. Mathur 16 Copyright © 2013 Dorling Kindersley (India) Pvt.768 till 32. Ltd Input domain (Input space) . of P. Contents Foundations of Software Testing 2E Author: Aditya P. Using Requirement 1 above we find the input domain of max to be the set of all pairs of integers where each element in the pair integers is in the range -32. Using Requirement 2 it is not possible to find the input domain for the sort program.

While providing input to the program.Input domain (Continued) Modified Requirement 2: Copyright © 2013 Dorling Kindersley (India) Pvt. the sequence is terminated with a period. the request character is input first followed by the sequence of integers to be sorted. Ltd It is required to write a program that inputs a sequence of integers and outputs the integers in this sequence sorted in either ascending or descending order. The order of the output sequence is determined by an input request character which should be ``A'' when an ascending sequence is desired. and ``D'' otherwise. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 17 .

The second element of the pair is a sequence of zero or more integers ending with a period. The first element of the pair is a character. Ltd Input domain (Continued) . the input domain for sort is a set of pairs. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 18 Copyright © 2013 Dorling Kindersley (India) Pvt.Based on the above modified requirement.

Valid/Invalid Inputs The modified requirement for sort mentions that the Copyright © 2013 Dorling Kindersley (India) Pvt. The requirement for sort does not specify what action it should take when an invalid input is encountered. but fails to answer the question ``What if the user types a different character ?’’ When using sort it is certainly possible for the user to type a character other than ``A'' and ``D''. Contents Foundations of Software Testing 2E Author: Aditya P. Any character other than ``A'’ and ``D'' is considered as invalid input to sort. Mathur 19 . Ltd request characters can be ``A'' and ``D''.

Ltd 1.Copyright © 2013 Dorling Kindersley (India) Pvt.4 Correctness versus reliability Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 20 .

Thus. it is almost .Correctness never the objective of testing. Mathur 21 Copyright © 2013 Dorling Kindersley (India) Pvt. this is impossible to accomplish. To establish correctness via testing would imply testing a program on all elements in the input domain. In most cases that are encountered in practice. correctness is established via mathematical proofs of programs. Ltd Though correctness of a program is desirable. Contents Foundations of Software Testing 2E Author: Aditya P.

Correctness and Testing Copyright © 2013 Dorling Kindersley (India) Pvt. completeness of testing does not necessarily demonstrate that a program is error free. and the error removal processes together increase our confidence in the correct functioning of the program under test. testing attempts to find if there are any errors in it. debugging. Contents Foundations of Software Testing 2E Author: Aditya P. Testing. Mathur 22 . Ltd While correctness attempts to establish that the program is error free. Thus.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Software reliability: two definitions Software reliability [ANSI/IEEE Std 729-1983]: is the probability of failure free operation of software over a given time interval and under given conditions. Software reliability is the probability of failure free operation of software in its intended environment. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 23 .

Contents Foundations of Software Testing 2E Author: Aditya P.Operational profile Copyright © 2013 Dorling Kindersley (India) Pvt. Consider a sort program which. allows any one of two types of input sequences. on any given execution. Mathur 24 . Sample operational profiles for sort follow. Ltd An operational profile is a numerical description of how a program is used.

Ltd Operational profile Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 25 .

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Operational profile Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 26 .

Mathur 27 .Copyright © 2013 Dorling Kindersley (India) Pvt.5 Testing and debugging Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 1.

Mathur 28 Copyright © 2013 Dorling Kindersley (India) Pvt. When testing reveals an error. Ltd Testing and debugging .Testing is the process of determining if a program has any errors. is known as debugging. Contents Foundations of Software Testing 2E Author: Aditya P. the process used to determine the cause of this error and to remove it.

Ltd A test/debug cycle Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 29 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P.Test plan Example: The sort program is to be tested to meet the requirements given earlier. Mathur 30 Copyright © 2013 Dorling Kindersley (India) Pvt. one with ``A'' and the other with ``D'' as request characters. . the following needs to be done. •  Execute sort on at least two input sequences. Specifically. Ltd A test cycle is often guided by a test plan.

•  Execute the program on an empty input sequence. •  Test the program for robustness against erroneous inputs such as ``R'' Copyright © 2013 Dorling Kindersley (India) Pvt. •  All failures of the test program should be recorded in a suitable file using the Company Failure Report Form. Mathur 31 .) typed in as the request character. Ltd Test plan (contd. Contents Foundations of Software Testing 2E Author: Aditya P.

Sample test case for sort: Test data: <''A'’ 12 -29 32 > Expected output: -29 12 32 Contents Foundations of Software Testing 2E Author: Aditya P. The test data is a set of values.Test case/data A test case is a pair consisting of test data to be input to the program and the Copyright © 2013 Dorling Kindersley (India) Pvt. A test set is a collection of zero or more test cases. Ltd expected output. one for each input variable. Mathur 32 .

A state diagram specifies program states and how the program changes its state on an input sequence. Contents Foundations of Software Testing 2E Author: Aditya P. inputs. a state diagram. Ltd Program behavior Can be specified in several ways: plain natural language. formal mathematical specification. etc.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 33 .

Contents Foundations of Software Testing 2E Author: Aditya P.Program behavior: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 34 . Ltd Consider a menu driven application.

Mathur 35 .) Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Program behavior: Example (contd.Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 36 . In the second step one analyzes the observed behavior to check if it is correct or not. Both these steps could be quite complex for large commercial programs. The entity that performs the task of checking the correctness of the observed behavior is known as an oracle. Ltd In the first step one observes the behavior.Behavior: observation and analysis Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Oracle: Example Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 37 .

Mathur 38 . For example. Ltd Oracles can also be programs designed to check the behavior of other programs. one might use a matrix multiplication program to check if a matrix inversion program has produced the correct output. the matrix inversion program inverts a given matrix A and generates B as the output matrix. Contents Foundations of Software Testing 2E Author: Aditya P.Oracle: Programs Copyright © 2013 Dorling Kindersley (India) Pvt. In this case.

Contents Foundations of Software Testing 2E Author: Aditya P. such as the one to check a matrix multiplication program or a sort program. Mathur 39 . the construction of automated oracles is a complex undertaking. Ltd Oracle: Construction Construction of automated oracles. requires the determination of inputoutput relationship.Copyright © 2013 Dorling Kindersley (India) Pvt. In general.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 40 . Ltd Oracle construction: Example Contents Foundations of Software Testing 2E Author: Aditya P.

In practice. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 41 Copyright © 2013 Dorling Kindersley (India) Pvt. program verification is often avoided.Program verification aims at proving the correctness of programs by showing that it contains no errors. This is very different from testing that aims at uncovering errors in a program. Ltd Testing and verification . and the focus is on testing. Program verification and testing are best considered as complementary techniques.

there might be an incorrect assumption on the input conditions. and so on. Verified and published programs have been shown to be incorrect. However. Mathur 42 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Testing is not a perfect technique in that a program might contain errors .) despite the success of a set of tests. Contents Foundations of Software Testing 2E Author: Aditya P. Verification promises to verify that a program is free from errors.Testing and verification (contd. incorrect assumptions might be made regarding the components that interface with the program. the person/tool who verified a program might have made a mistake in the verification process.

Test generation strategies Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 1.Copyright © 2013 Dorling Kindersley (India) Pvt.10. Mathur 43 .

the process is a bit more formal. In more advanced test processes. the source document resides in the mind of the tester who generates tests based on a knowledge of the requirements. The tests are generated using a mix of formal and informal methods either directly from the requirements document serving as the source.Test generation Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 44 . requirements serve as a source for the development of formal models. In several commercial environments. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Any form of test generation uses a source document. In the most informal of test methods.

Ltd Model based: require that a subset of the requirements be modeled using a formal notation (usually graphical). Petri net. Mathur 45 . Specification based: require that a subset of the requirements be modeled using a formal mathematical notation.Test generation strategies Copyright © 2013 Dorling Kindersley (India) Pvt. Z. Code based: generate tests directly from the code. etc. Examples: B. Models: Finite State Machines. Contents Foundations of Software Testing 2E Author: Aditya P. and Larch. Timed automata.

Ltd Test generation strategies (Summary) Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 46 .Copyright © 2013 Dorling Kindersley (India) Pvt.

13 Types of software testing Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 1.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 47 .

Mathur 48 . Ltd One possible classification is based on the following four classifiers: C1: Source of test generation. C2: Lifecycle phase in which testing takes place C3: Goal of a specific testing activity C4: Characteristics of the artifact under test Contents Foundations of Software Testing 2E Author: Aditya P.Types of testing Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd C1: Source of test generation Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 49 .

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd C2: Lifecycle phase Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 50 .

Ltd C3: Goal of specific testing activity Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 51 .

Mathur 52 . Ltd C4: Artifact under test Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 53 . Exercises at the end of Chapter 1 are to help you sharpen your understanding. Ltd Summary We have dealt with some of the most basic concepts in software testing.Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 54 . 2013 Foundations of Software Testing 2E Contents Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Chapter 2: Preliminaries: Mathematical Updated: July 12.

Ltd 2.Copyright © 2013 Dorling Kindersley (India) Pvt.1 Predicates and Boolean expressions Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 55 .

” V 4.Copyright © 2013 Dorling Kindersley (India) Pvt. pp 133-157. Here is an example from Paradkar. 1997. Annals of Software Engineering. A boiler needs to be to be shut down when the following conditions hold: Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 56 . “Specification based testing using cause-effect graphs. Ltd Where do predicates arise? Predicates arise from requirements in a variety of applications. and Vouk. Tai.

We combine these five conditions to form a compound condition (predicate) for boiler shutdown. Mathur 57 Copyright © 2013 Dorling Kindersley (India) Pvt.  Steam meter has failed.  A water pump has failed. (d) Boiler in degraded mode when either is true.Boiler shutdown conditions 2. (c) 4.  The water level in the boiler is below X lbs. (a) . (b) 3. 5. (e) The boiler is to be shut down when a or b is true or the boiler is in degraded mode and the steam meter fails. Ltd 1.  The water level in the boiler is above Y lbs.  A pump monitor has failed. Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd Denoting the five conditions above as a through e.Boiler shutdown conditions Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 58 .” The goal of predicate-based test generation is to generate tests from a predicate p that guarantee the detection of any error that belongs to a class of errors in the coding of p. we obtain the following Boolean expression E that when true must force a boiler shutdown: E=a+b+(c+d)e where the + sign indicates “OR” and a multiplication indicates “AND.

The following predicate represents the condition part of the statement. .Another example For example. also known as a Boolean expression. consider the requirement ``if the printer is ON and has paper then send document to printer. pr: (printer_status=ON) ∧ (printer_tray!= empty) Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 59 Copyright © 2013 Dorling Kindersley (India) Pvt.” This statement consists of a condition part and an action part. Ltd A condition is represented formally as a predicate.

Ltd Test generation from predicates . Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 60 Copyright © 2013 Dorling Kindersley (India) Pvt. named BOR and BRO for generating tests that are guaranteed to detect certain faults in the coding of conditions. Conditions guard actions. For example.We will now examine two techniques. The conditions from which tests are generated might arise from requirements or might be embedded in the program to be tested. if condition then action is a typical format of many functional requirements.

≤.∨. >.g.Predicates Relational operators (relop): {<. (e.} Copyright © 2013 Dorling Kindersley (India) Pvt. =. Relational expression: e1 relop e2.∧. Simple predicate: A Boolean variable or a relational expression. (x<0) Compound predicate: Join one or more simple predicates using bop. ≥. AND. Boolean operators (bop): {!. XOR}. OR. Ltd = and == are equivalent. a+b<c) e1 and e2 are expressions whose values can be compared using relop. xor} also known as {not. ≠. Mathur 61 . (gender==“female”∧age>65) Contents Foundations of Software Testing 2E Author: Aditya P.

(a∧b∨!c) a. Mathur 62 Copyright © 2013 Dorling Kindersley (India) Pvt. e. Negation is also denoted by placing a bar over a Boolean expression such as in (a ∧ b) We also write ab for a∧b and a+b for a∨b when there is no confusion. and c are also known as literals. b. Ltd Boolean expressions . in (a∧b∨!c) Contents Foundations of Software Testing 2E Author: Aditya P.. Singular Boolean expression: When each literal appears only once.g.Boolean expression: one or more Boolean variables joined by bop.

e. (p q) +(rs) + (a c). Conjunctive normal form (CNF): Product of sums: e. CNF: (p+!r)(p+s)(q+!r)(q+s) is equivalent to DNF: (pq+!rs) Contents Foundations of Software Testing 2E Author: Aditya P.: (p+q)(r+s)(a+c) Any Boolean expression in DNF can be converted to an equivalent CNF and vice versa.g.g. Ltd Disjunctive normal form (DNF): Sum of product terms: e.g..Boolean expressions (contd. Mathur 63 .) Copyright © 2013 Dorling Kindersley (India) Pvt.

) .. Ltd Boolean expressions (contd.. Contents Foundations of Software Testing 2E Author: Aditya P.Mutually singular: Boolean expressions e1 and e2 are mutually singular when they do not share any literal. e2. then ei is considered singular only if it is non-singular and mutually singular with the remaining elements of E. If expression E contains components e1. Mathur 64 Copyright © 2013 Dorling Kindersley (India) Pvt.

Copyright © 2013 Dorling Kindersley (India) Pvt.Abstract syntax tree (AST) for: (a+b)<c ∧!p. Ltd Boolean expressions: Syntax tree representation Root node (AND-node) Notice that internal nodes are labeled by ∧ Boolean and relational operators Root node: OR-node is labeled as ∨. Mathur 65 . < (a+b) ! c p Leaf nodes Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd 2. Mathur 66 .Copyright © 2013 Dorling Kindersley (India) Pvt.2 Program representation: Control flow graphs Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 67 Copyright © 2013 Dorling Kindersley (India) Pvt. There is no possibility of exit or a halt at any point inside the basic block except at its exit point. Ltd Program representation: Basic blocks . The entry and exit points of a basic block coincide when the block contains only one statement. Contents Foundations of Software Testing 2E Author: Aditya P.A basic block in program P is a sequence of consecutive statements with a single entry and a single exit point. a block has unique entry and exit points. Control always enters a basic block at its entry point and exits from its exit point. Thus.

Ltd Example: Computing x raised to y Contents Foundations of Software Testing 2E Author: Aditya P.Basic blocks: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 68 .

Ltd Basic blocks Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 69 .) Copyright © 2013 Dorling Kindersley (India) Pvt.Basic blocks: Example (contd.

We often write G= (N. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 70 Copyright © 2013 Dorling Kindersley (India) Pvt. An edge (i.A control flow graph (or flow graph) G is defined as a finite set N of nodes and a finite set E of edges. j) in E connects two nodes ni and nj in N. E) to denote a flow graph G with nodes given by N and edges by E. Ltd Control Flow Graph (CFG) .

that has no outgoing edge. also in N. Blocks and nodes are labeled such that block bi corresponds to node ni.Control Flow Graph (CFG) indicate the flow of control between blocks. each basic block becomes a node and edges are used to . and another node labeled End. Mathur 71 Copyright © 2013 Dorling Kindersley (India) Pvt. We also assume that there is a node labeled Start in N that has no incoming edge. j) connecting basic blocks bi and bj implies that control can go from block bi to block bj. Contents Foundations of Software Testing 2E Author: Aditya P. An edge (i. Ltd In a flow graph of a program.

1. 6. 3). 7. 7). (4. End} E={(Start. (7. (1. (2.1). 5). 8. 9). Mathur 72 . 2). (5. 9. (3. (6. (9. 5). 3. 6).Copyright © 2013 Dorling Kindersley (India) Pvt. (7. 5. 8). 4). Ltd CFG Example N={Start. (1.4). 4. End)} Contents Foundations of Software Testing 2E Author: Aditya P. (5. 2.

CFG Example Copyright © 2013 Dorling Kindersley (India) Pvt. End} E={(Start.4). 8). (3. 1. 6. 5. (1. Ltd Same CFG with statements removed. (7. 3. (1. (6. 4. (5. (9. 8. 5). 2. 9.1). 6). 2). 3). (5. 4). 7. 9). 7). 5). (2. Mathur 73 . (4. N={Start. (7. End)} Contents Foundations of Software Testing 2E Author: Aditya P.

k>0. E). Mathur 74 .Paths Copyright © 2013 Dorling Kindersley (India) Pvt. denotes a path of length k through the flow graph if the following sequence condition holds. nq. and ns are nodes belonging to N. if ei = (np. ns) then nq = nr. A sequence of k edges. Given that np. … e_k) . e_2. } Contents Foundations of Software Testing 2E Author: Aditya P. and 0< i<k. (e_1. nr. Ltd Consider a flow graph G= (N. nq) and ei+1 = (nr.

6. (6. Contents Foundations of Software Testing 2E Author: Aditya P. (5. 2. (9. Dashed edges: subpath. 5. 3. 1). 9. (2. (1. 7. 1. 5. 4. (7. 5. 7). 4. 5. End)) Bold edges: complete path. 5). (5. 4). 9). (4. End) p2= (Start. 2). 7. 5). Ltd Two feasible and complete paths: p1= ( Start. 1. 6. 6). 9. End) Specified unambiguously using edges: p1= ( (Start.Paths: sample paths through the exponentiation flow graph Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 75 .

Mathur 76 . 6. 4. 5. 4.Paths: infeasible Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd A path p through a flow graph for program P is considered feasible if there exists at least one test case which when input to P causes p to be traversed. 1. . 3. 5. 5. 7. 2. 9. End) Contents Foundations of Software Testing 2E Author: Aditya P. 8. End) p2= (Start. 1. 9. 1. 7. p1= ( Start.

Number of paths condition contains exactly one path that begins at node Start and terminates at node End. Each additional condition in the program can increases the number of distinct paths by at least one. A program with no . conditions can have a multiplicative effect on the number of paths. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd There can be many distinct paths through a program. Mathur 77 Copyright © 2013 Dorling Kindersley (India) Pvt. Depending on their location.

and regular expressions Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. languages.6 Strings. Ltd 2. Mathur 78 .

Ltd Strings . Examples: 1011. Mathur 79 Copyright © 2013 Dorling Kindersley (India) Pvt. “Hello world”. AaBc. a set of all strings consisting of zeros and ones is the language of binary numbers.Strings play an important role in testing. Contents Foundations of Software Testing 2E Author: Aditya P. A string serves as a test input. In this section we provide a brief introduction to strings and languages. A collection of strings also forms a language. For example.

We use an upper case letter such as X and Y to denote alphabets. Mathur 80 . Another alphabet is Y={dog. 1} is an alphabet consisting of two symbols 0 and 1. For example. and ``lion". Though alphabets can be infinite. Ltd A collection of symbols is known as an alphabet. ``cat". we are concerned only with finite alphabets. X={0. cat. horse. lion}that consists of four symbols ``dog". Contents Foundations of Software Testing 2E Author: Aditya P. ``horse".Alphabet Copyright © 2013 Dorling Kindersley (India) Pvt.

Foundations of Software Testing 2E Contents Author: Aditya P. 1}. Thus. |1011|=4 and |dog cat dog|=3. also known as an empty string. is denoted by ε. q. Mathur 81 . 0110 is a string over the alphabet {0. Note that ε denotes an empty string and also stands for “element of” when used with sets. horse. we denote its length by |s|. Ltd A string over an alphabet X is any sequence of zero or more symbols that belong to X. cat. r to denote strings. dog cat dog dog lion is a string over the alphabet {dog.Strings over an Alphabet Copyright © 2013 Dorling Kindersley (India) Pvt. The length of a string is the number of symbols in that string. Also. A string of length 0. For example. Given a string s. We will use lower case letters such as p. lion}.

s2 to denote the concatenation of strings s1 and s2. 1}. ε =s and ε. for any string s.s2|=|s1|+|s2|. Also. we obtain 011. For example.s=s. Contents Foundations of Software Testing 2E Author: Aditya P. It is easy to see that |s1. we have s. We write s1. given the alphabet X={0.Let s1 and s2 be two strings over alphabet X.101=011101. and two strings 011 and 101 over X. Mathur 82 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd String concatenation .

11.Languages Copyright © 2013 Dorling Kindersley (India) Pvt. The following sets are finite languages over the binary alphabet {0. Mathur 83 . 0101}: A language containing three strings Contents Foundations of Software Testing 2E Author: Aditya P. Ltd A set L of strings over an alphabet X is known as a language. 1}: ∅: The empty set {ε}: A language consisting only of one string of length zero {00. A language can be finite or infinite.

Ltd Regular expressions . Contents Foundations of Software Testing 2E Author: Aditya P.r2 is a regular expression that denotes the set L1. respectively. Let r1 and r2 be two regular expressions over the alphabet X that denote.L2. Mathur 84 Copyright © 2013 Dorling Kindersley (India) Pvt.Given a finite alphabet X. then a is a regular expression that denotes the set {a}. Then r1. the following are regular expressions over X: If a belongs to X. sets L1 and L2.

Contents Foundations of Software Testing 2E Author: Aditya P.) . r* known as the Kleene closure of r. then r1r2 is also a regular expression that denotes the set L1 ∪ L2. Ltd Regular expressions (contd. Mathur 85 Copyright © 2013 Dorling Kindersley (India) Pvt. respectively. If r denotes the set L then r* denotes the set {ε}∪ L+.If r is a regular expression that denotes the set L then r+ is a regular expression that denotes the set obtained by concatenating L with itself one or more times also written as L+ Also. sets L1 and L2. If r1 and r2 are regular expressions that denote. is a regular expression.

Mathur 86 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Summary We have introduced mathematical preliminaries an understanding of which will be useful while you go through the remaining parts of this book. Contents Foundations of Software Testing 2E Author: Aditya P. Exercises at the end of Chapter 2 will help you sharpen your understanding.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 87 . 2013 Foundations of Software Testing 2E Contents Author: Aditya P. Ltd Chapter 3 Domain Partitioning Updated: July 12.

Mathur 88 . Ltd Learning Objectives Essential black-box techniques for generating tests for functional testing. Cause effect graphing has been omitted from these slides. Contents Foundations of Software Testing 2E Author: Aditya P.§  Equivalence class partitioning §  Boundary value analysis Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 89 . unit. system.Applications of test generation techniques Copyright © 2013 Dorling Kindersley (India) Pvt. or subsystem. correctly implements the functionality as per the given requirements Contents Foundations of Software Testing 2E Author: Aditya P. These techniques are useful during functional testing where the objective is to test whether or not an application. Ltd Test generation techniques described in this chapter belong to the black-box testing category.

Test Procedure Test incident Test log Test item transmittal report report Test summary Test generation techniques report Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 90 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Functional Testing: Test Documents (IEEE829 Standard) . Spec.Requirements Test Plan Model Reference: Lee Copland. A Practitioners Guide to software Test Design Test Design Test Case Spec.

Functional Testing: Documents Test Plan: Describe scope. Each test case has a unique ID for reference in other documents. items to be Copyright © 2013 Dorling Kindersley (India) Pvt. Test case spec: Lists inputs. Ltd tested. Mathur 91 . resources. test schedule. Dependencies with other test cases are specified here. expected outputs. responsibilities. deliverables. approach. Could be used at the system test level or at lower levels. setting of environment variables and test procedures. features to be tested by this test case. Foundations of Software Testing 2E Contents Author: Aditya P. and any other special requirements e. Test design spec: Identifies a subset of features to be tested and identifies the test cases to test the features in this subset. approvals needed.g.

Mathur 92 . Contents Foundations of Software Testing 2E Author: Aditya P. Test log: A log observations during the execution of a test. Test incident report: Document any special event that is recommended for further investigation.Functional Testing: Documents (contd) Copyright © 2013 Dorling Kindersley (India) Pvt. Test summary: Summarize the results of testing activities and provide an evaluation. a database. Ltd Test procedure spec: Describe the procedure for executing a test case. Test transmittal report: Identifies the test items being provided for testing. e.g.

Ltd Test generation techniques in this chapter Three techniques are considered: equivalence partitioning. Mathur 93 .Copyright © 2013 Dorling Kindersley (India) Pvt. boundary value analysis. Each of these test generation techniques is black-box and useful for generating test cases during functional testing. Contents Foundations of Software Testing 2E Author: Aditya P. and category partitioning.

Ltd 3.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 94 .2 The test selection problem Contents Foundations of Software Testing 2E Author: Aditya P.

and statecharts in UML.Requirements and test generation Copyright © 2013 Dorling Kindersley (India) Pvt. Rigorously specified requirements are often transformed into formal requirements using requirements specification languages such as Z. Contents Foundations of Software Testing 2E Author: Aditya P. sequence diagrams. Ltd Requirements serve as the starting point for the generation of tests. S. more aptly ideas. are then specified rigorously using modeling elements such as use cases. These requirements. During the initial phases of development. requirements may exist only in the minds of one or more people. and RSML. Mathur 95 .

Mathur 96 . Ltd Test generation techniques Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

In general there does not exist any algorithm to construct such a test set. there are heuristics and model based methods that can be used to generate tests that will reveal certain type of faults. Ltd Test selection problem Let D denote the input domain of a program P. Contents Foundations of Software Testing 2E Author: Aditya P. The test selection problem is to select a subset T of tests such that execution of P against each element of T will reveal all errors in P.Copyright © 2013 Dorling Kindersley (India) Pvt. However. Mathur 97 .

) The challenge is to construct a test set T⊆D that will reveal as many errors in P as possible. Ltd Test selection problem (contd. The problem of test selection is difficult due primarily to the size and complexity of the input domain of P. Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 98 .

By ``exhaustive" testing we mean testing the given program against every element in its input domain. Contents Foundations of Software Testing 2E Author: Aditya P. The complexity makes it harder to select individual tests.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Exhaustive testing The large size of the input domain prevents a tester from exhaustively testing the program under test against all possible inputs. Mathur 99 .

Ltd Large input domain . say Nmax>1.Consider program P that is required to sort a sequence of integers into ascending order. Mathur 100 Contents Copyright © 2013 Dorling Kindersley (India) Pvt. If the size of the input sequence is limited to. Assuming that P will be executed on a machine in which integers range from -32768 to 32767. Calculate the size of the input domain. 32767]. the input domain of P consists of all possible sequences of integers in the range [-32768. Foundations of Software Testing 2E Author: Aditya P. then the input domain of P is infinitely large and P can never be tested exhaustively. then the size of the input domain depends on the value of N. If there is no limit on the size of the sequence that can be input.

For simplicity. Mathur 101 Contents . Foundations of Software Testing 2E Author: Aditya P.Complex input domain Copyright © 2013 Dorling Kindersley (India) Pvt. assume that the employee record consists of the following items with their respective types and constraints: Calculate the size of the input domain. Ltd Consider a procedure P in a payroll processing system that takes an employee record as input and computes the weekly salary.

3 Equivalence partitioning Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 102 . Ltd 3.Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 103 . the sub-domains by definition are disjoint.Copyright © 2013 Dorling Kindersley (India) Pvt. as shown (next slide (a)). In strict mathematical terms. say N>1. Ltd Equivalence partitioning Test selection using equivalence partitioning allows a tester to subdivide the input domain into a relatively small number of sub-domains. The four subsets shown in (a) constitute a partition of the input domain while the subsets in (b) are not. Each subset is known as an equivalence class.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Subdomains Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 104 .

Ltd Program behavior and equivalence classes The equivalence classes are created assuming that the program under test exhibits the same behavior on all elements. This assumption allow the tester to select exactly one test from each equivalence class resulting in a test suite of exactly N tests. i. Mathur 105 .Copyright © 2013 Dorling Kindersley (India) Pvt. within a class.e. tests. Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd Faults targeted . can be further subdivided into subsets on which the application is required to behave differently (e. U2). E1. or illegal. and U1. Each of the two subsets. or legal.The entire set of inputs to any application can be divided into at least two subsets: one containing all the expected.g. Mathur 106 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. inputs (U). E2. E3. inputs (E) and the other containing all unexpected.

Mathur 107 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Faults targeted (contd.) .Equivalence class partitioning selects tests that target any faults in the application that cause it to behave incorrectly when the input is in either of the two classes or their subsets.

All integers Other integers [1. The set of input values is now divided into a set E containing all integers in the range [1.120]..Example 1 the only legal values of age are in the range [1..120] and a set U containing the remaining integers. Mathur 108 Copyright © 2013 Dorling Kindersley (India) Pvt..120] Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Consider an application A that takes an integer denoted by age as input. Let us suppose that .

) .. Similarly.. This leads to a subdivision of U into two categories. assume that the application is required to process all values in the range [1. Ltd Example 1 (contd. E is further subdivided into two regions depending on the expected behavior. Thus. Mathur 109 Copyright © 2013 Dorling Kindersley (India) Pvt.Further.61] in accordance with requirement R1 and those in the range [62. it is expected that all invalid inputs less than or equal to 1 are to be treated in one way while all greater than 120 are to be treated differently.120] according to requirement R2. Contents Foundations of Software Testing 2E Author: Aditya P.

All integers <1 [62-120] Copyright © 2013 Dorling Kindersley (India) Pvt..) >120 [1.61] Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 110 . Ltd Example 1 (contd.

. Contents Foundations of Software Testing 2E Author: Aditya P.) . A similar expectation applies to the two regions containing the unexpected inputs. Ltd Example 1 (contd. i. any test selected from the region [62. Mathur 111 Copyright © 2013 Dorling Kindersley (India) Pvt.120] will reveal any fault with respect to R2. It is expected that any single test selected from the range [1..61] will reveal any fault with respect to R1.e.. Similarly.Tests selected using the equivalence partitioning technique aim at targeting faults in the application under test with respect to inputs in any of the four regions. two regions containing expected inputs and two regions containing the unexpected inputs.

is judged by the ratio of the number of faults these tests are able to expose to the total faults lurking in A. The effectiveness can be improved through an unambiguous and complete specification of the requirements and carefully selected tests using the equivalence partitioning technique described in the following sections. Contents Foundations of Software Testing 2E Author: Aditya P. the effectiveness of tests selected using equivalence partitioning is less than 1 for most practical applications. Mathur 112 Copyright © 2013 Dorling Kindersley (India) Pvt.The effectiveness of tests generated using equivalence partitioning for testing application A. As is the case with any test selection technique in software testing. Ltd Effectiveness .

Consider that wordCount method takes a word w and a filename f as input and returns the number of occurrences of w in the text contained in the file named f. Mathur 113 .Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Example 2 This example shows a few ways to define equivalence classes based on the knowledge of requirements and the program text. An exception is raised if there is no file with name f.

Example 2 (contd. we obtain the equivalence end classes (next slide). f Input w. Ltd begin .f)). return(0). examples above. if(empty(f))return(0). Mathur 114 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. f if (not exists(f) {raise exception.) String w. Using the partitioning method described in the return(getCount(w.} if(length(w)==0)return(0).

Mathur 115 . not empty E2 non-null does not exist E3 non-null exists. empty Copyright © 2013 Dorling Kindersley (India) Pvt.Example 2 (contd. not empty E5 null does not exist E6 null exists. Ltd Equivalence class Contents Foundations of Software Testing 2E Author: Aditya P. empty E4 null exists.) w f E1 non-null exists.

whereas the number of equivalence classes derived with the knowledge of partial code is 6. even before the code is available Contents Foundations of Software Testing 2E Author: Aditya P.Example 2 (contd. an experienced tester will likely derive the six equivalence classes given above. Mathur 116 .) Copyright © 2013 Dorling Kindersley (India) Pvt. Of course. and perhaps more. Ltd Note that the number of equivalence classes without any knowledge of the program code is 2.

It is worth asking: ``Does the program ever generate a 0? What are the maximum and minimum possible values of the output?" These two questions lead to two the following equivalence classes based on outputs: Contents Foundations of Software Testing 2E Author: Aditya P. For example. Mathur 117 . suppose that a program outputs an integer. Ltd Equivalence classes based on program output In some cases the equivalence classes are based on the output generated by the program.Copyright © 2013 Dorling Kindersley (India) Pvt.

) E1: Output value v is 0. E4: All other output values. each of the four classes given above might lead to one equivalence class consisting of inputs. Based on the output equivalence classes one may now derive equivalence classes for the inputs. E2: Output value v is the maximum possible. Contents Foundations of Software Testing 2E Author: Aditya P. E3: Output value v is the minimum possible. Mathur 118 . Ltd Equivalence classes based on program output (contd.Copyright © 2013 Dorling Kindersley (India) Pvt. Thus.

{56}..0}.90] {50}. {75}. area: float area≥0.Equivalence classes for variables: range Example Constraints Classes Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 119 . {15. {132}} letter:bool {{J}. Classes One class with values speed ∈[60. inside the range and {92} two with values outside the range.0 {{-1.52}} age: int {{-1}. {3}} Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Eq.

Mathur 120 . Contents Foundations of Software Testing 2E Author: Aditya P. {Sue}. Ltd Equivalence classes for variables: strings Example Constraints Classes firstname: string {{ε}.Equivalence Classes At least one containing all legal strings and one all Copyright © 2013 Dorling Kindersley (India) Pvt. {Loooong Name}} illegal strings based on any constraints.

green} {{red.} {blue}. Mathur 121 . {false}} Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Equivalence classes for variables: enumeration Example Classes autocolor:{red. {green}} up:boolean {{true}. blue.Equivalence Classes Constraints Each value in a separate class Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. one Copyright © 2013 Dorling Kindersley (India) Pvt. {[ ]}. {[-9. Ltd Equivalence classes for variables: arrays Example Classes int [ ] aName: new int[3]. 0.Equivalence Classes Constraints One class containing all legal arrays. 12. 20]}. 15]} containing the empty array. {[-10. Mathur 122 . and one containing a larger than expected array.

in C++. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 123 . The next example illustrates the derivation of equivalence classes for an input variable that has a compound type. Such input types may arise while testing components of an application such as a function or an object. are compound types. While generating equivalence classes for such inputs. or structures.Copyright © 2013 Dorling Kindersley (India) Pvt. one must consider legal and illegal values for each component of the structure. Ltd Equivalence classes for variables: compound data type Arrays in Java and records.

// Course titles. // Letter grades corresponding to course titles. } In-class exercise: Derive equivalence classes for each component of R and combine them! Contents Foundations of Software Testing 2E Author: Aditya P.Equivalence classes for variables: compound data type: Example Copyright © 2013 Dorling Kindersley (India) Pvt. char grades [200]. string cTitle [200]. // First name. Ltd struct transcript { string fName. string lName. // Last name. Mathur 124 .

This type of partitioning is used commonly. We refer to this style of partitioning as uni-dimensional equivalence partitioning or simply uni-dimensional partitioning. each input variable leads to a partition of the input domain.One way to partition the input domain is to consider one input variable at a time. Thus. Mathur 125 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd uni-dimensional partitioning . Contents Foundations of Software Testing 2E Author: Aditya P.

Many classes so created might be infeasible. equivalence classes so created offer an increased variety of tests as is illustrated in the next section. This procedure creates one partition consisting of several equivalence classes. Nevertheless.Multidimensional partitioning variables and define a relation on I. Contents Foundations of Software Testing 2E Author: Aditya P. We refer to this method as multidimensional equivalence partitioning or simply multidimensional partitioning. Ltd Another way is to consider the input domain I as the set product of the input . Multidimensional partitioning leads to a large number of equivalence classes that are difficult to manage manually. Mathur 126 Copyright © 2013 Dorling Kindersley (India) Pvt.

This leads to the following six equivalence classes. For uni-dimensional partitioning we apply the partitioning guidelines to x and y individually. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Partitioning Example Consider an application that requires two integer inputs x and y.Copyright © 2013 Dorling Kindersley (India) Pvt. Each of these inputs is expected to lie in the following ranges: 3≤ x≤7 and 5≤y≤9. Mathur 127 .

x ignored.) y ignored.E1: x<3 E2: 3≤x≤7 E3: x>7 E4: y<5 E5: 5≤y≤9 E6: y>9 Copyright © 2013 Dorling Kindersley (India) Pvt. For multidimensional partitioning we consider the input domain to be the set product X x Y. Ltd Partitioning Example (contd. Contents Foundations of Software Testing 2E Author: Aditya P. This leads to 9 equivalence classes. Mathur 128 .

y<5 E8: x>7. y>9 Copyright © 2013 Dorling Kindersley (India) Pvt. y<5 E2: x<3. y<5 E5: 3≤x≤7. Ltd Partitioning Example (contd. Mathur 129 . 5≤y≤9 E6: 3≤x≤7. y>9 E7: >7. 5≤y≤9 E9: x>7.E1: x<3.) Contents Foundations of Software Testing 2E Author: Aditya P. y>9 E4: 3≤x≤7. 5≤y≤9 E3: x<3.

Partitioning Example (contd. y<5 E3: x<3. y>9 E7: >7.) Copyright © 2013 Dorling Kindersley (India) Pvt. y>9 9 equivalence classes: Contents Foundations of Software Testing 2E Author: Aditya P. 5≤y≤9 E9: x>7. 5≤y≤9 E4: 3≤x≤7. Mathur 130 . y<5 E5: 3≤x≤7. Ltd 6 equivalence classes: E1: x<3. y<5 E8: x>7. 5≤y≤9 E6: 3≤x≤7. y>9 E2: x<3.

an approximation to the input domain is the product of these sets. their types. Contents Foundations of Software Testing 2E Author: Aditya P. also serve as input variables. Windows. and any conditions associated with their use. Environment variables. Given the set of values each variable can assume. Identify the input domain: Read the requirements carefully and identify all input and output variables. Mathur 131 Copyright © 2013 Dorling Kindersley (India) Pvt. such as class variables used in the method under test and environment variables in Unix. Ltd Systematic procedure for equivalence partitioning . and other operating systems.1.

Each subset is an equivalence class. partitioning the input domain using values of one variable. is done based on the the expected behavior of the program. Equivalence classing: Partition the set of values of each variable into disjoint subsets.2. Contents Foundations of Software Testing 2E Author: Aditya P.) . the equivalence classes based on an input variable partition the input domain. Together. Values for which the program is expected to behave in the ``same way" are grouped together. Mathur 132 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Systematic procedure for equivalence partitioning (contd. Note that ``same way" needs to be defined by the tester.

Ltd Systematic procedure for equivalence partitioning (contd. Mathur 133 Copyright © 2013 Dorling Kindersley (India) Pvt. by not combining the equivalence classes. Contents Foundations of Software Testing 2E Author: Aditya P.3. Combine equivalence classes: This step is usually omitted and the equivalence classes defined for each variable are directly used to select test cases. The equivalence classes are combined using the multidimensional partitioning approach described earlier. one misses the opportunity to generate useful tests. However.) .

Ltd Systematic procedure for equivalence partitioning (contd. There might also be constraints in the requirements that render certain equivalence infeasible. i. For example.e. Contents Foundations of Software Testing 2E Author: Aditya P. suppose that an application is tested via its GUI. Such an equivalence class might arise due to several reasons.Copyright © 2013 Dorling Kindersley (India) Pvt.) 4. Identify infeasible equivalence classes: An infeasible equivalence class is one that contains a combination of input data that cannot be generated during test. The GUI might disallow invalid inputs by offering a palette of valid inputs only. Mathur 134 . data is input using commands available in the GUI.

One .. Command temp causes CS to ask the operator to enter the amount by which the temperature is to be changed (tempch). Values of tempch are in the range -10.10 in increments of 5 degrees Fahrenheit. and cancel the request (cancel). shut down the boiler (shut).Boiler control example (BCS) of the options. Mathur Contents 135 Copyright © 2013 Dorling Kindersley (India) Pvt. is used by a human operator to give one of four commands (cmd): change the boiler temperature (temp). Foundations of Software Testing 2E Author: Aditya P. Ltd The control software of BCS. abbreviated as CS. is required to offer several options. C (for control). An temperature change of 0 is not an option.

BCS obtains the command from a command file. the operator is asked to enter one of the three commands via a GUI. Contents Foundations of Software Testing 2E Author: Aditya P. together with the value of the temperature to be changed if the command is temp. The command file may contain any one of the three commands. However. If V is set to GUI.) . if V is set to file. Mathur 136 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd BCS: example (contd.Selection of option C forces the BCS to examine variable V. The file name is obtained from variable F.

cancel) tempch GUI cmd Control Software (CS) tempch: desired temperature change (-10. shut. Mathur 137 . F: Environment variables (temp. Ltd BCS: example (contd. file} F: file name if V is set to “file..” Contents Foundations of Software Testing 2E Author: Aditya P.) V.V F cmd: command Copyright © 2013 Dorling Kindersley (India) Pvt.10) datafile V ∈{GUI.

Contents Foundations of Software Testing 2E Author: Aditya P.Values of V and F can be altered by a different module in BCS. Mathur 138 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd BCS: example (contd. the control software is required to generate appropriate signals to be sent to the boiler heating system.) . In response to temp and shut commands.

5. The GUI forces the tester to select from a limited set of values as specified in the requirements. For example. We refer to these four values of tempch as tvalid while all other values as tinvalid. Ltd BCS: example (contd. and 10.We assume that the control software is to be tested in a simulated environment. -5.) . Contents Foundations of Software Testing 2E Author: Aditya P. The tester takes on the role of an operator and interacts with the CS via a GUI. Mathur 139 Copyright © 2013 Dorling Kindersley (India) Pvt. the only options available for the value of tempch are -10.

These are listed in the following table. Identify input domain . First we examine the requirements. their types. Ltd BCS: 1. identify input variables. Contents Foundations of Software Testing 2E Author: Aditya P. Recall that the domain identified in this step will likely be a superset of the complete input domain of the control software. Mathur 140 Copyright © 2013 Dorling Kindersley (India) Pvt.The first step in generating equivalence partitions is to identify the (approximate) input domain. and values.

Mathur 141 . cancel. GUI F Environment String A file name cmd Input via GUI/File Enumerated {temp. values Contents Foundations of Software Testing 2E Author: Aditya P. -5. shut} tempch Input via GUI/File Enumerated {-10. 10} Copyright © 2013 Dorling Kindersley (India) Pvt. types. 5. Ltd BCS: Variables.Variable Kind Type Value(s) V Environment Enumerated File.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 142 . --) (file. Ltd BCS: Input domain Input domain⊆S=V×F×cmd×tempch Sample values in the input domain (--: don’t care): (GUI. (file. --). 0) Does this belong to the input domain? Contents Foundations of Software Testing 2E Author: Aditya P. shut. --. temp. cmdfile. shut. cmdfile.

Mathur 143 . Equivalence classing Contents Foundations of Software Testing 2E Author: Aditya P. {undefined}} F {{fvalid}. {finvalid}} cmd {{temp}. Ltd BCS: 2. {file}.Variable Partition V {{GUI}. {cinvalid}} tempch {{tvalid}. {tinvalid}} Copyright © 2013 Dorling Kindersley (India) Pvt. {shut}. {cancel}.

Ltd BCS: 3. {(GUI}}. -10)} Note that each of the classes listed above represents an infinite number of input values Copyright © 2013 Dorling Kindersley (India) Pvt. Sample equivalence class: {(GUI. Contents Foundations of Software Testing 2E Author: Aditya P. tvalid. Combine equivalence classes (contd. There are a total of 3×4×2×5=120 equivalence classes.Note that tinvalid. For example. and fvalid denote sets of values.) for the control software. “undefined” denotes one value. -10)} denotes an infinite set of values obtained by replacing fvalid by a string that corresponds to the name of an existing file. Mathur 144 . fvalid. temp. fvalid. Each value is a potential input to the BCS. temp. finvalid.

Thus. all equivalence classes that match the following template are infeasible. Ltd Note that the GUI requests for the amount by which the boiler temperature is to be changed only when the operator selects temp for cmd. tvalid∪ tinvalid)} This parent-child relationship between cmd and tempch renders infeasible a total of 3×2×3×5=90 equivalence classes. {(V. F. shut. Exercise: How many additional equivalence classes are infeasible? Contents Foundations of Software Testing 2E Author: Aditya P. cinvalid}. Mathur 145 . Discard infeasible equivalence classes Copyright © 2013 Dorling Kindersley (India) Pvt. {cancel.BCS: 4.

Ltd BCS: 4. we are left with a total of 18 testable (or feasible) equivalence classes. Contents Foundations of Software Testing 2E Author: Aditya P.) .After having discarded all infeasible equivalence classes. Mathur 146 Copyright © 2013 Dorling Kindersley (India) Pvt. Discard infeasible equivalence classes (contd.

complications could arise in the presence of infeasible data and don't care values. Ltd Given a set of equivalence classes that form a partition of the input domain. it is relatively straightforward to select tests. In the most general case. Exercise: Generate sample tests for BCS from the remaining feasible equivalence classes.Selecting test data Copyright © 2013 Dorling Kindersley (India) Pvt. However. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 147 . a tester simply selects one test that serves as a representative of each equivalence class.

However.. Contents Foundations of Software Testing 2E Author: Aditya P.While designing equivalence classes for programs that obtain input exclusively from a keyboard. Ltd GUI design and equivalence classes . For example. Mathur 148 Copyright © 2013 Dorling Kindersley (India) Pvt.4. one must account for the possibility of errors in data entry. the requirement for an application. testing must account for the possibility that a user may inadvertently enter a value for X that is out of range. The application places a constraint on an input variable X such that it can assume integral values in the range 0.

Hence only the correct values of X will be input. Mathur 149 Copyright © 2013 Dorling Kindersley (India) Pvt. In such a situation it is impossible to test the application with a value of X that is out of range.) .Suppose that all data entry to the application is via a GUI front end. Contents Foundations of Software Testing 2E Author: Aditya P. Suppose also that the GUI offers exactly five correct choices to the user for X. See figure on the next slide. Ltd GUI design and equivalence classes (contd.

Mathur 150 .Copyright © 2013 Dorling Kindersley (India) Pvt.) Contents Foundations of Software Testing 2E Author: Aditya P. Ltd GUI design and equivalence classes (contd.

4 Boundary value analysis Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 151 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 3.

when M is tested against x=0 but not if the input test set is. suppose that method M is required to compute a function f1 when x≤ 0 is true and function f2 otherwise. lies at the boundary of the equivalence classes x≤0 and x>0. In this example. though not necessarily. the value x=0. 7} derived using equivalence partitioning. For example. Obviously. this fault is revealed. Mathur 152 Copyright © 2013 Dorling Kindersley (India) Pvt. for example. However. Foundations of Software Testing 2E Contents Author: Aditya P. {-4. M has an error due to which it computes f1 for x<0 and f2 otherwise.Errors at the boundaries boundaries of equivalence classes. Ltd Experience indicates that programmers make mistakes in processing values at and near the .

boundary value analysis focuses on tests at and near the boundaries of equivalence classes. Mathur 153 Copyright © 2013 Dorling Kindersley (India) Pvt. Certainly. tests derived using either of the two techniques may overlap. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Boundary value analysis (BVA) .Boundary value analysis is a test selection technique that targets faults in applications at the boundaries of equivalence classes. While equivalence partitioning selects tests from within equivalence classes.

3  Select test data such that each boundary value occurs in at least one test input. Boundaries may also be identified using special relationships amongst the inputs. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 1  . We will generate several sub-domains in this step. Mathur 154 Copyright © 2013 Dorling Kindersley (India) Pvt. This leads to as many partitions as there are input variables. Alternately.BVA: Procedure Partition the input domain using uni-dimensional partitioning. 2  Identify the boundaries for each partition. a single partition of an input domain can be created using multidimensional partitioning.

. E6: Values greater than 100. Create equivalence classes Copyright © 2013 Dorling Kindersley (India) Pvt.999 and quantity in the range 1. Mathur 155 .. E2: Values in the range. Ltd Assuming that an item code must be in the range 99. Contents Foundations of Software Testing 2E Author: Aditya P. Equivalence classes for code: E1: Values less than 99. Equivalence classes for qty: E4: Values less than 1. E3: Values greater than 999. E5: Values in the range.100.BVA: Example: 1.

Foundations of Software Testing 2E Author: Aditya P. Mathur Contents 156 .BVA: Example: 2. Boundaries are indicated with an x. Identify boundaries * x 99 E1 100 998 * * E2 0 * E4 x 1 2 99 * * E5 1000 x * 999 E3 Copyright © 2013 Dorling Kindersley (India) Pvt. Points near the boundary are marked *. Ltd 98 101 x * 100 E6 Equivalence classes and boundaries for findPrice.

t4: (code=998. Consider the . qty=2). qty=0). qty=100). Ltd must include. Illegal values of code and t3: (code=100. t5: (code=999.BVA: Example: 3. qty=101) } Contents Foundations of Software Testing 2E Author: Aditya P. t2: (code=99. Construct test set Test selection based on the boundary value analysis technique requires that tests following test set: T={ t1: (code=98. qty=99). qty included. qty=1). Mathur 157 Copyright © 2013 Dorling Kindersley (India) Pvt. values at and around the boundary. for each variable. t6: (code=1000.

11. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 158 Copyright © 2013 Dorling Kindersley (India) Pvt. Is there an advantage of separating the invalid values of code and age into different test cases? Answer: Refer to Example 3. Ltd Is T the best possible test set for findPrice? Answer this question based on T’s .12.BVA: In-class exercise ability to detect missing code for checking the validity of age. Highly recommended: Go through Example 3.

Ltd Relationships amongst the input variables must be examined carefully while identifying boundaries along the input domain. Mathur 159 . This examination may lead to boundaries that are not evident from equivalence classes obtained from the input and output variables.BVA: Recommendations Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Additional tests may be obtained when using a partition of the input domain obtained by taking the product of equivalence classes created using individual variables.

4. Mathur 160 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 4. Tests using predicate syntax Contents Foundations of Software Testing 2E Author: Aditya P.

1997. and Vouk. Ltd Where do predicates arise? Predicates arise from requirements in a variety of applications. Tai. A boiler needs to be shut down when the following conditions hold: Contents Foundations of Software Testing 2E Author: Aditya P. Here is an example from Paradkar.” V 4. “Specification based testing using cause-effect graphs.” Annals of Software Engineering. pp 133-157.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 161 .

Contents Foundations of Software Testing 2E Author: Aditya P. 5. (a) 2. (e) The boiler is to be shut down when a or b is true or the boiler is in degraded mode and the steam meter fails.  The water level in the boiler is above Y lbs. (b) 3.Boiler shutdown conditions Copyright © 2013 Dorling Kindersley (India) Pvt. (d) Boiler in degraded mode when either is true. Ltd 1. Mathur 162 . (c) 4. We combine these five conditions to form a compound condition (predicate) for boiler shutdown.  Steam meter has failed.  A pump monitor has failed.  A water pump has failed.  The water level in the boiler is below X lbs.

” The goal of predicate-based test generation is to generate tests from a predicate p that guarantee the detection of any error that belongs to a class of errors in the coding of p.Boiler shutdown conditions Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 163 . Ltd Denoting the five conditions above as a through e. we obtain the following Boolean expression E that when true must force a boiler shutdown: E=a+b+(c+d)e where the + sign indicates “OR” and a multiplication indicates “AND. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 164 Copyright © 2013 Dorling Kindersley (India) Pvt. . The following predicate represents the condition part of the statement. Ltd A condition is represented formally as a predicate.” This statement consists of a condition part and an action part. pr: (printer_status=ON) ∧ (printer_tray!= empty) Contents Foundations of Software Testing 2E Author: Aditya P. also known as a Boolean expression. consider the requirement ``if the printer is ON and has paper then send document to printer.Another example For example.

one can apply these techniques to generate tests for f. Given a function f to be tested in an application. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 165 . Ltd Summary Equivalence partitioning and boundary value analysis are the most commonly used methods for test generation while doing functional testing.Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 166 . 2013 Foundations of Software Testing 2E Contents Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Chapter 4 Predicate Analysis Updated: July 12.

Mathur 167 . Ltd Learning Objectives Contents Foundations of Software Testing 2E Author: Aditya P.§  Domain testing §  Cause-effect graphing §  Test generation from predicates Copyright © 2013 Dorling Kindersley (India) Pvt.

4.4 Tests using predicate syntax 4. Mathur 168 .Copyright © 2013 Dorling Kindersley (India) Pvt.1: A fault model Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 4.

Fault model for predicate testing Copyright © 2013 Dorling Kindersley (India) Pvt. c. and d are integer variables and e is a Boolean variable. Here a. Mathur 169 . Ltd What faults are we targeting when testing for the correct implementation of predicates? Boolean operator fault: Suppose that the specification of a software module requires that an action be performed when the condition (a<b) ∨ (c>d) ∧e is true. Contents Foundations of Software Testing 2E Author: Aditya P. b.

Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Boolean operator faults Correct predicate: (a<b) ∨ (c>d) ∧e (a<b) ∧ (c>d) ∧e Incorrect Boolean operator (a<b) ∨ ! (c>d) ∧e Incorrect negation operator (a<b) ∧(c>d) ∨ e Incorrect Boolean operators (a<b) ∨ (e>d) ∧c Incorrect Boolean variable.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 170 .

Mathur 171 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Relational operator faults Correct predicate: (a<b) ∨ (c>d) ∧e (a==b) ∨ (c>d) ∧e Incorrect relational operator (a==b) ∨ (c≤d) ∧e Two relational operator faults (a==b) ∨ (c>d) ∨ e Incorrect Boolean operators Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd Correct predicate: Ec: e1 relop1 e2. Incorrect predicate: Ei: : e3 relop2 e4. Ei has an off-by-ε* fault if |e3-e4|≥ ε for any test case for which e1=e2. Ei has an off-by-ε fault if |e3-e4|= ε for any test case for which e1=e2. Ei has an off-by-ε+ fault if |e3-e4|> ε for any test case for which e1=e2.Arithmetic expression faults Copyright © 2013 Dorling Kindersley (India) Pvt. Assume that Ec and Ei use the same set of variables. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 172 .

b=2. c=2> Ei: a<b-1. Ei: a<b+1. Ei: a<b. Ltd Arithmetic expression faults: Examples . Ei has an off-by-1+ fault as |a-(b-1)|>1 for any test case for which a=b+c. Contents Foundations of Software Testing 2E Author: Aditya P. e. b=1. Given c>0. Ei has an off-by-1* fault as |a-(b+1)|≥ 1 for any test case for which a=b+c.Correct predicate: Ec: a<(b+c). b=2. <a=4. Mathur 173 Copyright © 2013 Dorling Kindersley (India) Pvt. <a=3. c=1>. c=1>.g. <a=2. Given c=1. Assume ε=1. Ei has an off-by-1 fault as |a-b|= 1 for a test case for which a=b+c. Given c=2.

Assume ε=1. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Arithmetic expression faults: In class exercise Given the correct predicate: Ec: 2*X+Y>2.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 174 . Find an incorrect version of Ec that has off-by-1 fault. Find an incorrect version of Ec that has off-by-1+ fault. Find an incorrect version of Ec that has off-by-1* fault.

Such a test set is said to guarantee the detection of any fault of the kind in the fault model introduced above. evaluate to different truth values. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 175 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Goal of predicate testing .Given a correct predicate pc. the goal of predicate testing is to generate a test set T such that there is at least one test case t∈ T for which pc and its faulty version pi.

t2} where t1: <a=0. Consider a test set T={t1. b=1.) . the fault is revealed by t2 as pc evaluates to true and pi to false when evaluated against t2. However. Mathur 176 Copyright © 2013 Dorling Kindersley (India) Pvt. suppose that pc: a<b+c and pi: a>b+c. c=1>. The fault in pi is not revealed by t1 as both pc and pi evaluate to false when evaluated against t1. c=0> and t2: <a=0. Ltd Goal of predicate testing (contd. Contents Foundations of Software Testing 2E Author: Aditya P.As an example. b=0.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 177 . Ltd Missing or extra Boolean variable faults Correct predicate: a ∨ b Missing Boolean variable fault: a Extra Boolean variable fault: a ∨ b∧c Contents Foundations of Software Testing 2E Author: Aditya P.

4 Tests using predicate syntax 4. Ltd 4. Mathur 178 .1: Predicate constraints Contents Foundations of Software Testing 2E Author: Aditya P.4.Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Consider the following Boolean-Relational set of BR-symbols: BR={t. >. Contents Foundations of Software Testing 2E Author: Aditya P. consider the predicate E: a<b and the constraint “>” . <. A test case that satisfies this constraint for E must cause E to evaluate to false.Predicate constraints: BR symbols Copyright © 2013 Dorling Kindersley (India) Pvt. -ε} A BR symbol is a constraint on a Boolean variable or a relational expression. f. Mathur 179 . +ε. For example. =.

For example. Mathur 180 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.A constraint C is considered infeasible for predicate pr if there exists no input values for the variables in pr that satisfy c. Ltd Infeasible constraints . the constraint t is infeasible for the predicate a>b∧ b>d if it is known that d>a.

if each component of pr satisfies the corresponding constraint in C when evaluated against t.. A predicate constraint C for predicate pr is a sequence of (n+1) BR symbols. Contents Foundations of Software Testing 2E Author: Aditya P. it offers hints on what the values of the variables should be for pr to satisfy C. When clear from context. Test case t satisfies C for predicate pr.Predicate constraints Copyright © 2013 Dorling Kindersley (India) Pvt. n>0. ∨ and ∧ operators. Ltd Let pr denote a predicate with n.e. Constraint C for predicate pr guides the development of a test for pr. i. one for each Boolean variable or relational expression in pr. Mathur 181 . we refer to ``predicate constraint" as simply constraint.

A set of constraints S is partitioned into subsets St and Sf. Mathur 182 .True and false constraints Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd pr(C) denotes the value of predicate pr evaluated using a test case that satisfies C. Contents Foundations of Software Testing 2E Author: Aditya P. pr(C) =false. respectively. and for any C in Sf. such that for each C in St. S= St ∪ Sf. C is referred to as a true constraint when pr(C) is true and a false constraint otherwise. pr(C) =true.

u=1. u=1. v=0> The following test case does not satisfy C for pr. >). Ltd Predicate constraints: Example Consider the predicate pr: b∧ (r<s) ∨ (u≥v) and a constraint C: (t. =. r=1.Copyright © 2013 Dorling Kindersley (India) Pvt. v=2> Contents Foundations of Software Testing 2E Author: Aditya P. s=2. s=1. The following test case satisfies C for pr. r=1. <b=true. <b=true. Mathur 183 .

4. Mathur 184 .3: Predicate testing criteria Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.4 Tests using predicate syntax 4. Ltd 4.

Mathur 185 Copyright © 2013 Dorling Kindersley (India) Pvt. BRO. We will discuss three such criteria named BOR. we want to generate a test set T such that •  T is minimal and •  T guarantees the detection of any fault in the implementation of pr. Ltd Predicate testing: criteria . Contents Foundations of Software Testing 2E Author: Aditya P.Given a predicate pr. and BRE. faults correspond to the fault model we discussed earlier.

T is referred to as a BOR-adequate test set and sometimes written as TBOR. Mathur 186 . Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Predicate testing: BOR testing criterion A test set T that satisfies the BOR testing criterion for a compound predicate pr.Copyright © 2013 Dorling Kindersley (India) Pvt. guarantees the detection of single or multiple Boolean operator faults in the implementation of pr.

Contents Foundations of Software Testing 2E Author: Aditya P.A test set T that satisfies the BRO testing criterion for a compound predicate pr. Ltd Predicate testing: BRO testing criterion . guarantees the detection of single Boolean operator and relational operator faults in the implementation of pr. Mathur 187 Copyright © 2013 Dorling Kindersley (India) Pvt. T is referred to as a BRO-adequate test set and sometimes written as TBRO.

T is referred to as a BRE-adequate test set and sometimes written as TBRE. Mathur 188 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Predicate testing: BRE testing criterion A test set T that satisfies the BRE testing criterion for a compound predicate pr. guarantees the detection of single Boolean operator. and arithmetic expression faults in the implementation of pr. Contents Foundations of Software Testing 2E Author: Aditya P. relational expression.

Mathur 189 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.Let Tx.BRE}. and arithmetic expression fault. be a test set derived from predicate pr. x∈{BOR. relational operator fault. p(t)≠ pf(t). BRO. Tx is said to guarantee the detection of faults in pf if for some t∈Tx. Ltd Predicate testing: guaranteeing fault detection . Let pf be another predicate obtained from pr by injecting single or multiple faults of one of three kinds: Boolean operator fault.

f) t3: <a=1. b=0. a<b is true and c<d is also true.Guaranteeing fault detection: example Copyright © 2013 Dorling Kindersley (India) Pvt. c=1. d=0 >. t2: <a=1. Ltd Let pr=a<b ∧ c>d Constraint set S={(t.f). b=2. t). t2. (f. b=2.e. i. t)} Let TBOR={t1. Satisfies (t. Mathur 190 . t1: <a=1. t) Contents Foundations of Software Testing 2E Author: Aditya P. Satisfies (f. (t. t3} is a BOR adequate test set that satisfies S. d=2 >. c=1. d=0 >. c=1. Satisfies (t. t).

Ltd Guaranteeing fault detection: In class exercise Generate single Boolean operator faults in pr: a<b ∧ c>d and show that T guarantees the detection of each fault. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 191 .Copyright © 2013 Dorling Kindersley (India) Pvt.

and BRE adequate tests Contents Foundations of Software Testing 2E Author: Aditya P.4. BRO.Copyright © 2013 Dorling Kindersley (India) Pvt.4 Tests using predicate syntax 4. Ltd 4.1: BOR. Mathur 192 .

b)|a∈A and b∈B} The onto product of two sets A and B is defined as: A⊗B={(u. Ltd Algorithms for generating BOR.} Note that A⊗B is a minimal set. BRO. Contents Foundations of Software Testing 2E Author: Aditya P. and BRE adequate tests Define the cross product of two sets A and B as: A×B={(a. such that each element of A appears at least once as u and each element of B appears once as v. Mathur 193 .v)|u∈A. v∈B.Copyright © 2013 Dorling Kindersley (India) Pvt.

=. f). Mathur 194 . <} A×B={(t.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Set products: Example Let A={t. (=. (>. (t. (=.<)} A⊗B ={(t.<)} Any other possibilities for A⊗B? Contents Foundations of Software Testing 2E Author: Aditya P. <). (>. (=. (>.<). >} and B={f. <).f). f). f).

Generation of BOR constraint set Copyright © 2013 Dorling Kindersley (India) Pvt. ∧ a<b c>d Contents Foundations of Software Testing 2E Author: Aditya P. Ltd See page 184 for a formal algorithm. generate syntax tree of pr. An illustration follows. Mathur 195 . We want to generate TBOR for pr: a<b ∧ c>d First.

and SNf is the false constraint. Mathur 196 . Contents Foundations of Software Testing 2E Author: Aditya P. where SNt is the true constraint set. we use the following notation: SN= SNt ∪ SNf is the constraint set.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Generation of the BOR constraint set Given node N in the syntax tree for predicate pr.

) Copyright © 2013 Dorling Kindersley (India) Pvt. (f)}. (f)} SN2= N2 {(t). and so on for convenience.Generation of the BOR constraint set (contd. N3 N1 SN1= ∧ c>d a<b {(t). label each leaf node with the constraint set {(t). We label the nodes as N1. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Second. N2. (f)} Notice that N1 and N2 are direct descendants of N3 which is an AND-node. Mathur 197 .

(f)} = {(f. Ltd Third.Generation of the BOR constraint set (contd. t)} ∧ N2 N1 SN3f = (SN1f ×{t2})∪({t1}× SN2f c>d {(t).t). Mathur 198 Copyright © 2013 Dorling Kindersley (India) Pvt. SN3={(t. f)} SN3t = SN1t ⊗ SN2t ={(t)} ⊗ {(t)}={(t.{(t. t). For an AND node. in this . f)} Contents Foundations of Software Testing 2E Author: Aditya P. t)}∪{(t.) case N3. (f)} a<b = ({(f)} ×{(t)})∪({(t)}× {(f)}) = {(f. compute the constraint set for the next higher node in the syntax tree. t). f)} N3 {(t). (f. (t. the formulae used are the following.

b=0. Mathur 199 Copyright © 2013 Dorling Kindersley (India) Pvt. c=6. b=2. d=2> (t. TBOR ={t1.Generation of TBOR the AST(pr). We can now generate a test set using the BOR constraint set associated with the root node. we have computed the BOR constraint set for the root node of . SN3={(t. (f)} a<b {(t). c=1. b=2. t). (f.t). c=6. (f)} t1=<a=1. t) t3=<a=1. f)} SN3 contains a sequence of three constraints and N3 hence we get a minimal test set consisting of three test cases. Here is one possible test set. t3} ∧ N2 N1 c>d {(t). t2. d=5> (f. Ltd As per our objective. f) Contents Foundations of Software Testing 2E Author: Aditya P. t) t2=<a=1. d=5> (t. (t.

Ltd Generation of BRO constraint set . Mathur 200 Copyright © 2013 Dorling Kindersley (India) Pvt.See pages 187-188 for a formal algorithm. Recall that a test set adequate with respect to a BRO constraint set for predicate pr. An illustration follows. Contents Foundations of Software Testing 2E Author: Aditya P. guarantees the detection of all combinations of single or multiple Boolean operator and relational operator faults.

(>)} relop: ≤ St={(<). (=).BRO constraint set The BRO constraint set S for relational expression e1 relop e2: Copyright © 2013 Dorling Kindersley (India) Pvt. (>)} relop: < St={(<)} Sf={(=). (=)} Sf={(>)} Note: tN denotes an element of StN and fN denotes an element of SfN Foundations of Software Testing 2E Author: Aditya P. (<)} Sf={(<)} relop: = St={(=)} Sf={(<). (<)} Separation of S into its true (St) and false (Sf)components: relop: > St={(>)} relop: ≥ St={(>). Mathur 201 Contents . (=)} Sf={(=). Ltd S={(>).

Ltd pr: (a+b<c)∧!p ∨ (r>s) Step 1: Construct the AST for the given predicate. ∨ N4 ∧ r>s N1 a+b<c N6 ! p N5 N3 N2 Contents Foundations of Software Testing 2E Author: Aditya P.BRO constraint set: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 202 .

(=). (<)} N3 ! p Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 203 .) ∨ N4 N6 ∧ r>s N1 a+b<c {(>). (=). Ltd Step 2: Label each leaf node with its constraint set S. (<)} N2 {(t).BRO constraint set: Example (contd. N5 {(>). (f)} Contents Foundations of Software Testing 2E Author: Aditya P.

f).) StN3=SN2f={(f)} Copyright © 2013 Dorling Kindersley (India) Pvt. (=. Ltd Step 2: Traverse the tree and compute constraint set for each internal node. (=. f)} SfN4= (SfN1 × {(tN3)}) ∪ ({(tN1)} × SfN3) =({(>. Mathur 204 . f). f)} ∪ {(<. (<.BRO constraint set: Example (contd.=)} ×{(f)}) ∪ {(<)} ×{(t)}) ={(>. t)} Contents Foundations of Software Testing 2E Author: Aditya P. f). t)} ={(>. SfN3=SN2t= {(t)} StN4=SN1t ⊗ SN3t={(<)} ⊗{(f)}={(<.

f).∨ {(<. Mathur 205 Copyright © 2013 Dorling Kindersley (India) Pvt. (<. (<)} N6 ! p N5 {(>). f). Ltd BRO constraint set: Example (contd. (>. (<)} N3 {(f). f). (=). (=). (=.) . (f)} Contents Foundations of Software Testing 2E Author: Aditya P. t)} N4 ∧ r>s N1 a+b<c {(>). {t)} N2 {(t).

f).f)} ×{(>)}) ={(<.=)} ∪ {(>.f.>)} ={(<.t.t)} ⊗{(=).(<.(>.) Copyright © 2013 Dorling Kindersley (India) Pvt.(<.=). Mathur 206 .=).=)} StN6= (StN4 × {(fN5)})∪ ({(fN4)} × StN5) =({(<.f). SfN6=SfN4 ⊗ SfN5 ={(>.f.f. (=. Ltd Next compute the constraint set for the rot node (this is an OR-node).f.BRO constraint set: Example (contd. f)} ={(>.(<)}={(<.>)} Contents Foundations of Software Testing 2E Author: Aditya P.f)} ×{(=)}) ∪ {(>.f.<).(=.f.

(=). (f)} Foundations of Software Testing 2E N5 Author: Aditya P.(>. Mathur Contents 207 . f).f. Ltd BRO constraint set: Example (contd.=). {t)} ! p N2 {(t).<). (<)} N3 {(f). (<.t.=).=).(<.f.f. (=). (<. (=. f). (<)} r>s {(>).) pr: (a+b<c)∧!p ∨ (r>s) {(>.f.Constraint set for Copyright © 2013 Dorling Kindersley (India) Pvt. (=. f). t)} ∨ N6 N4 ∧ N1 a+b<c {(>).>)} {(<. (>.

=). {(>.(>. Mathur 208 .=).<).Copyright © 2013 Dorling Kindersley (India) Pvt.f.>)} Contents Foundations of Software Testing 2E Author: Aditya P. (=.f. construct TBRO . Ltd BRO constraint set: In-class exercise Given the constraint set for pr: (a+b<c)∧!p ∨ (r>s).f. (<.=).t.f.(<.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 209 .4 Tests using predicate syntax 4. Ltd 4.5: BOR constraints for non-singular expressions Contents Foundations of Software Testing 2E Author: Aditya P.4.

Test generation procedures described so far are for singular predicates. and their mutually singular components. Contents Foundations of Software Testing 2E Author: Aditya P. their respective disjunctive normal forms (DNF). Ltd BOR constraints for non-singular expressions . Recall that a singular predicate contains only one occurrence of each variable. We will now learn how to generate BOR constraints for non-singular predicates. Mathur 210 Copyright © 2013 Dorling Kindersley (India) Pvt. First. let us look at some non-singular expressions.

de Copyright © 2013 Dorling Kindersley (India) Pvt.Predicate (pr) DNF Mutually singular components in pr ab(b+c) abb+abc a. b(b+c) a(bc+ bd) abc+abd a. !b+!c+ cde a(bc+!b+de) abc+a!b+ade a. bc+!b. Ltd Non-singular expressions and DNF: Examples Contents Foundations of Software Testing 2E Author: Aditya P. (bc+bd) a(!b+!c)+cde a!ba +a!c+cde a. Mathur 211 .

Ltd Generating BOR constraints for non-singular expressions We proceed in two steps. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 212 . we examine the procedure to generate BOR constraint set for a non-singular predicate. Next.Copyright © 2013 Dorling Kindersley (India) Pvt. First we examine the Meaning Impact (MI) procedure for generating a minimal set of constraints from a possibly non-singular predicate.

The MI procedure is on page 193 and is illustrated next. the MI procedure produces a set of constraints SE that guarantees the detection of missing or extra NOT (!) operator faults in the implementation of E.Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 213 . Ltd Meaning Impact (MI) procedure Given Boolean expression E in DNF.

For example bc is the same as b∧c. For example. b. Ltd Consider the non-singular predicate: a(bc+!bd). Mathur 214 .MI procedure: An Example Copyright © 2013 Dorling Kindersley (India) Pvt. Note that a. a could represent r<s. Recall that + is the Boolean OR operator. and as per common convention we have omitted the Boolean AND operator. ! is the Boolean NOT operator. Its DNF equivalent is: E=abc+a!bd. Contents Foundations of Software Testing 2E Author: Aditya P. c. Each literal represents a condition. and d are Boolean variables and also referred to as literals.

t. are to be interpreted similarly.c.t.t). Te1 ={(t.f.t. Contents Foundations of Software Testing 2E Author: Aditya P. and d. Step 1: Construct a constraint set Te1 for e1 that makes e1 true. Similarly construct Te2 for e2 that makes e2 true.t. The second element.f.MI procedure: Example (contd. and others. respectively.) Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Step 0: Express E in DNF notation.t).f. Mathur 215 . b. where e1=abc and e2=a!bd.t)} Note that the four t’s in the first element of Te1 denote the values of the Boolean variables a. (t. we can write E=e1+e2.t. Clearly.f)} Te2 ={(t. (t.

t)} Contents Foundations of Software Testing 2E Author: Aditya P.t).t.t.MI procedure: Example (contd. This gives us TSei and TSej. Note that this step will lead TSei ∩TSej =∅. (t.f. Mathur 216 .t.) Copyright © 2013 Dorling Kindersley (India) Pvt. Hence we get: TSe1 ={(t. (t. remove the constraints that are in any other Tej.t).f. There are no common constraints between Te1 and Te2 in our example.t.f)} TSe2 ={(t.f. Ltd Step 2: From each Tei .t.

Also.t).t.t. StE is minimal. Mathur 217 . Ltd Step 3: Construct StE by selecting one element from each Te.f)} Note that for each constraint x in StE we get E(x)=true.f. (t. StE ={(t. Check it out! Contents Foundations of Software Testing 2E Author: Aditya P.) Copyright © 2013 Dorling Kindersley (India) Pvt.f.MI procedure: Example (contd.

Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd MI procedure: Example (contd. obtain terms by complementing each literal. Mathur 218 . derive constraints Fe that make e true. one at a time.) Step 4: For each term in E. We get the following six sets. e11= !abc e21= a!bc e31= ab!c e12= !a!bd e22= abd e32= a!b!d From each term e above.

(t. Mathur 219 . (t.f.t.f)} Contents Foundations of Software Testing 2E Author: Aditya P.f.t.f)} Fe12= {(f.t.f.t.t). Ltd MI procedure: Example (contd.t).f.f. (t. (f.f.f.t.Copyright © 2013 Dorling Kindersley (India) Pvt.t.t).f.t).t.f).t.t. (t.f)} Fe21= {(t.) Fe11= {(f.t)} Fe32= {(t.f)} Fe31= {(t.t.f.f. (f.t).t)} Fe22= {(t.t.t.f.t.

f)} FSe31= FSe13 Constraints common to Te1 and Te2 are removed. FSe11= FSe11 FSe21= {(t. Mathur 220 Copyright © 2013 Dorling Kindersley (India) Pvt.) .t)} FSe32= FSe13 Contents Foundations of Software Testing 2E Author: Aditya P.f.Step 5: Now construct FSe by removing from Fe any constraint that appeared in any of the two sets Te constructed earlier. Ltd MI procedure: Example (contd.t.t. FSe12= FSe12 FSe22= {(t.f.

t)} Note: Each constraint in StE makes E true and each constraint in SfE makes E false.t.t. Mathur 221 Copyright © 2013 Dorling Kindersley (India) Pvt.t).t.t.t).f.t.f). (f. (t.f. Check it out! We are now done with the MI procedure. Ltd Step 6: Now construct SfE by selecting one constraint from each Fe .t. (f.) SfE ={(f. Contents Foundations of Software Testing 2E Author: Aditya P. (t. (t.f.t)} Step 7: Now construct SE= StE ∪SfE SE={{(t.MI procedure: Example (contd.t.t.f).f.t).t.f).t. (t.f.f.f. (f.f.t.f). (t.t.f).

Contents Foundations of Software Testing 2E Author: Aditya P. The entire procedure is described on page 195. Mathur 222 Copyright © 2013 Dorling Kindersley (India) Pvt. The BOR-MI-CSET procedure using the MI procedure described earlier.The BOR-MI-CSET procedure takes a non-singular expression E as input and generates a constraint set that guarantees the detection of Boolean operator faults in the implementation of E. Ltd BOR-MI-CSET procedure . We illustrate it with an example.

Contents Foundations of Software Testing 2E Author: Aditya P.BOR-MI-CSET: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 223 . Ltd Consider a non-singular Boolean expression: E= a(bc+!bd) Mutually non-singular components of E: e1=a e2=bc+!bd We use the BOR-CSET procedure to generate the constraint set for e1 (singular component) and MI-CSET procedure for e2 (non-singular component).

Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Sfe1={f} Recall that Ste1 is true constraint set for e1 and Sfe1 is false constraint set for e1. Ltd BOR-MI-CSET: Example (contd.) For component e1 we get: Ste1={t}. Mathur 224 .

BOR-MI-CSET: Example (contd.t)} Contents Foundations of Software Testing 2E Author: Aditya P. We can write e2=u+v where u=bc and v=! bd. As per Step 1 of the MI-CSET procedure we obtain: Tu={(t.t. (f.t).f)} Tv={(f.f. Let us now apply the MI-CSET procedure to obtain the BOR constraint set for e2. Ltd Component e2 is a DNF expression.t).t. Mathur 225 .) Copyright © 2013 Dorling Kindersley (India) Pvt. (t.t.

t)} Next we apply Step 4 to u and v. Mathur 226 Copyright © 2013 Dorling Kindersley (India) Pvt.BOR-MI-CSET: Example (contd. We obtain the following complemented expressions from u and v: One possible alternative. (f. Ltd Applying Steps 2 and 3 to Tu and Tv we obtain: .) TSu=Tu TSv=Tv Ste2={(t. Can you think of other alternatives? u1=!bc u2=b!c v1=bd v2=!b!d Contents Foundations of Software Testing 2E Author: Aditya P. t.f).t.

f)} FSv1={(t.f)} Contents Foundations of Software Testing 2E Author: Aditya P.f). (t.f).f)} Fv1={(t.f. Mathur 227 .f.t.t. (t.t.BOR-MI-CSET: Example (contd.t)} Fv2={(f.t.) Fu1={(f.t. (f. (t. (f.f. (f.f.t).f.t)} FSv2={(f.t.t).f.f)} Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Continuing with Step 4 we obtain: Next we apply Step 5 to the F constraint sets to obtain: FSu1={(f.f)} Fu2=(t.f)} FSu2=(t.t).f.f.t).

f.t)}.t.BOR-MI-CSET: Example (contd.f).f. (f.t.t)}. Combing the true and false constraint sets for e2 we get: Se2={(t. t). t. Mathur 228 .f). Ltd Applying Step 6 to the FS sets leads to the following Sfe2={(f.) Copyright © 2013 Dorling Kindersley (India) Pvt.f). (t. {(f. Contents Foundations of Software Testing 2E Author: Aditya P. (t.t.

Ltd Summary: from BOR-CSET procedure.f.f).) Ste1={(t)} Sfe1={(f)} Ste2={(t.t. (f. We now apply Step 2 of the BOR-CSET procedure to obtain the constraint set for the entire expression E.t. Mathur 229 .t)} from MI-CSET procedure. t. t)} Copyright © 2013 Dorling Kindersley (India) Pvt. (t. Contents Foundations of Software Testing 2E Author: Aditya P.BOR-MI-CSET: Example (contd.f). Sfe2={(f.

(t.) StN3=StN1 ⊗ StN22 AND node.(f)} b ∧ c !b d Apply MI-CSET Foundations of Software Testing 2E Author: Aditya P.t.t)} N1 a ∧ {(t).t)} N2 ∨ {(t.(t.f. Mathur Contents 230 Copyright © 2013 Dorling Kindersley (India) Pvt.t.t. (f.f.t.f).BOR-MI-CSET: Example (contd. (f. (t.t.t.t. SfN3=(SfN1 × {t2})∪({t1} × SfN2) N3 ∧ {(t.t. t).t).f).f). Ltd Obtained by applying Step 2 of BOR-CSET to an .f).t. (t.f).f.f. t. (f.

Predicate testing procedures covered are excellent means to generate tests to ensure that each condition is tested adequately. Mathur 231 Copyright © 2013 Dorling Kindersley (India) Pvt.Most requirements contain conditions under which functions are to be executed. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Summary .

…action n. Ltd Usually one would combine equivalence partitioning. action 2. Contents Foundations of Software Testing 2E Author: Aditya P. BVA.Summary (contd. Apply predicate testing Apply eq. Mathur 232 .) Copyright © 2013 Dorling Kindersley (India) Pvt. boundary value analysis. and predicate testing procedures to generate tests for a requirement of the following type: if condition then action 1. partitioning. and predicate testing if there are nested conditions.

Mathur 233 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Chapter 5 Test Generation from Finite State Models Updated: July 16. 2013 Foundations of Software Testing 2E Contents Author: Aditya P.

Mathur 234 . Ltd Learning Objectives UIO method is not covered in these slides. It is left for the students to read on their own (Section 5. Contents Foundations of Software Testing 2E Author: Aditya P.8).§  What are Finite State Models? §  The W method for test generation §  The Wp method for test generation §  Automata theoretic versus control-flow based test generation Copyright © 2013 Dorling Kindersley (India) Pvt.

nuclear plant protection systems. Ltd §  elevator designs. etc). transmission.Where are these methods used? Conformance testing of communications protocols--this is where it all started. e. Copyright © 2013 Dorling Kindersley (India) Pvt. §  Testing of any system/subsystem modeled as a finite state machine. automobile components (locks. steam boiler control. stepper motors.g. etc. Mathur Contents 235 . Generation of tests from FSM specifications assists in testing the conformance of implementations to the corresponding FSM model. Alert: It will be a mistake to assume that the test generation methods described here are applicable only to protocol testing! Foundations of Software Testing 2E Author: Aditya P.) §  Finite state machines are widely used in modeling of all kinds of systems.

Ltd 5.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 236 .2 Finite State Machines Contents Foundations of Software Testing 2E Author: Aditya P.

is an abstract representation of behavior exhibited by some systems. For example. abbreviated as FSM. An FSM is derived from application requirements.A finite state machine. Ltd What is a Finite State Machine? . Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 237 Copyright © 2013 Dorling Kindersley (India) Pvt. a network protocol could be modeled using an FSM.

Real time requirements. performance requirements. Mathur 238 . Ltd What is a Finite State Machine? Not all aspects of an application’s requirements are specified by an FSM.Copyright © 2013 Dorling Kindersley (India) Pvt. and several types of computational requirements cannot be specified by an FSM. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 239 . Note that FSMs are a part of UML 2.0 design notation. The role assigned to an FSM depends on whether it is a part of the requirements specification or of the design specification. Contents Foundations of Software Testing 2E Author: Aditya P.Requirements or design specification? Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd An FSM could serve any of two roles: as a specification of the required behavior and/ or as a design artifact according to which an application is to be implemented.

WEB applications. Teller machines. safety software modeling in nuclear plants. Contents Foundations of Software Testing 2E Author: Aditya P.Modeling GUIs. Ltd Where are FSMs used? . While the FSM’s considered in examples are abstract machines. Mathur 240 Copyright © 2013 Dorling Kindersley (India) Pvt. pacemakers. and many more. they are abstractions of many real-life machines. network protocols.

the reverse is not true. While FSMs can be modeled using statecharts. Ltd FSM and statcharts . Mathur 241 Copyright © 2013 Dorling Kindersley (India) Pvt. The term “state diagram” is often used to denote a graphical representation of an FSM or a statechart. Contents Foundations of Software Testing 2E Author: Aditya P.Note that FSMs are different from statecharts. Techniques for generating tests from FSMs are different from those for generating tests from statecharts.

O). q0. Mathur 242 . X is a finite set of input symbols also known as the input alphabet. Q is a finite set states. δ: Q x X→ Q is a next-state or state transition function. Ltd An FSM (Mealy) is a 6-tuple: (X.FSM (Mealy machine): Formal definition Copyright © 2013 Dorling Kindersley (India) Pvt. where:. Y is a finite set of output symbols also known as the output alphabet. δ. Y. Q. and O: Q x X→ Y is an output function Contents Foundations of Software Testing 2E Author: Aditya P. q0 in Q is the initial state.

X . Y. Q. Mathur 243 . q0. O.Copyright © 2013 Dorling Kindersley (India) Pvt. Y. q0. δ. where:. Ltd FSM (Moore machine): Formal definition An FSM (Moore) is a 7-tuple: (X. and δ are the same as in FSM (Mealy) O: Q → Y is an output function F∈Q is the set of final or accepting or terminating states. F). Contents Foundations of Software Testing 2E Author: Aditya P. Q.

Copyright © 2013 Dorling Kindersley (India) Pvt.) Mealy machines are due to G. Mathur 244 . Mealy (1955 publication) Moore machines are due to E. Moore (1956 publication) Contents Foundations of Software Testing 2E Author: Aditya P. F. Ltd FSM: Formal definition (contd. H.

Test generation from FSMs Our focus FSM Test generation algorithm Test generation for application Application Test inputs Blue: Generated Test driver data Pass/fail Test inputs Oracle Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Requirements FSM based Test inputs Application Observed behavior Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 245 .

Contents Foundations of Software Testing 2E Author: Aditya P. In any case. an embedded system contains one or more computers for processing inputs. an .Embedded systems automobile has several embedded computers to perform various tasks. An embedded system can be as simple as a child's musical keyboard or as complex as the flight controller in an aircraft. For example. Ltd Many real-life devices have computers embedded in them. engine control being one example. Such devices are also known as embedded systems. Mathur 246 Copyright © 2013 Dorling Kindersley (India) Pvt. Another example is a computer inside a toy for processing inputs and generating audible and visual responses.

The response of an embedded system to its inputs depends on its current state. Contents Foundations of Software Testing 2E Author: Aditya P.An embedded computer often receives inputs from its environment and responds with appropriate actions. Ltd Specifying embedded systems . While doing so. It is this behavior of an embedded system in response to inputs that is often modeled by a finite state machine (FSM). it moves from one state to another. Mathur 247 Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 248 . Ltd Simple three state lamp behavior: (a) Lamp switch can be turned clockwise. Contents Foundations of Software Testing 2E Author: Aditya P. (b) Lamp switch can be turned clockwise and counterclockwise.FSM: lamp example Copyright © 2013 Dorling Kindersley (India) Pvt.

(b) INIT: Initialize num. INIT. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Machine to convert a sequence of decimal digits to an integer: (a) Notice the ADD. Mathur 249 . ADD. and OUT actions. OUT: Output num.FSM: Actions with state transitions Copyright © 2013 Dorling Kindersley (India) Pvt. ADD: Add to num.

Q.Copyright © 2013 Dorling Kindersley (India) Pvt. O). where: X is a finite set of input symbols also known as the input alphabet. δ. q0. Q is a finite set states. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 250 . Y. Ltd FSM: Formal definition An FSM is a quintuple: (X. Y is a finite set of output symbols also known as the output alphabet.

Ltd q0 in Q is the initial state. Mathur 251 . Also.FSM: Formal definition (contd. sometimes it is convenient to add F⊆ Q as a set of final or accepting states while specifying an FSM.) Copyright © 2013 Dorling Kindersley (India) Pvt. and O: Q x X→ Y is an output function. In some variants of FSM more than one state could be specified as an initial state. δ: Q x X→ Q is a next-state or state transition function. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 252 Copyright © 2013 Dorling Kindersley (India) Pvt. i is also known as the input portion of the edge and o its output portion. Each edge is labeled i/o where i denotes an input symbol that belongs to the input alphabet X and o denotes an output symbol that belongs to the output alphabet O. Ltd State diagram representation of FSM . Contents Foundations of Software Testing 2E Author: Aditya P. Each node is labeled with the state it represents.A state diagram is a directed graph that contains nodes representing states and edges representing state transitions and output functions. Each directed edge in a state diagram connects two states.

Mathur 253 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 5.2 Tabular representation Contents Foundations of Software Testing 2E Author: Aditya P.2.

The table consists of two sub-tables that consist of one or more columns each. The leftmost sub table is the output or the action sub-table. The rightmost sub-table is the next state sub-table.A table is often used as an alternative to the state diagram to represent the state transition function δ and the output function O. Mathur 254 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Tabular representation of FSM . The rows are labeled by the states of the FSM. Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd The table given below shows how to represent functions δ and O for the DIGDEC machine. Mathur 255 .Tabular representation of FSM: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.

3 Properties of FSM Contents Foundations of Software Testing 2E Author: Aditya P.2. Mathur 256 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 5.

Completely specified: An FSM M is said to be completely specified if from each state in M there exists a transition for each input symbol. Mathur 257 Copyright © 2013 Dorling Kindersley (India) Pvt. Strongly connected: An FSM M is considered strongly connected if for each pair of states (qi . Ltd Properties of FSM . qj) there exists an input sequence that takes M from state qi to state qj. Contents Foundations of Software Testing 2E Author: Aditya P.

O2) be two FSMs. O1) and M2=(X. s) for all s in V. m20. T1. Ltd Properties of FSM: Equivalence . Contents Foundations of Software Testing 2E Author: Aditya P. m10.V-equivalence: Let M1=(X. be the states of machines M1 and M2. Let qi and qj. i≠ j.e. Y. T2. Q2. Y. Q1. Mathur 258 Copyright © 2013 Dorling Kindersley (India) Pvt. qi and qj are considered V-equivalent if O1(qi. Let V denote a set of non-empty strings over the input alphabet X i. s)=O2(qj. V⊆ X+. respectively.

yield identical output sequences. If qi and qj are not equivalent then they are said to be distinguishable. respectively. States qi and qj are said to be equivalent if O1(qi. Thus. This definition of equivalence also applies to states within a machine. Mathur 259 Copyright © 2013 Dorling Kindersley (India) Pvt. r) for any set V. Ltd Properties of FSM: Distinguishable . Contents Foundations of Software Testing 2E Author: Aditya P. states qi and qj are considered V-equivalent if M1 and M2 . when excited in states qi and qj. machines M1 and M2 could be the same machine.Stated differently. r)=O2(qj.

Ltd Machine equivalence: Machines M1 and M2 are said to be equivalent if (a) for each .Properties of FSM: Machine Equivalence state σ in M1 there exists a state σ ' in M2 such that σ and σ ' are equivalent and (b) for each state σ in M2 there exists a state σ ' in M1 such that σ and σ ' are equivalent. Mathur 260 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Machines that are not equivalent are considered distinguishable. Minimal machine: An FSM M is considered minimal if the number of states in M is less than or equal to any other FSM equivalent to M.

Q2.k-equivalence: Let M1=(X. when excited by any input of length k. yield identical output sequences. Y. O1) and M2=(X. Y. Mathur 261 Copyright © 2013 Dorling Kindersley (India) Pvt. T1. T2. m20. m10. Ltd Properties of FSM: k-equivalence . O2) be two FSMs. Q1. Contents Foundations of Software Testing 2E Author: Aditya P. States qiε Q1 and qjε Q2 are considered k-equivalent if.

Ltd States that are not k-equivalent are considered k-distinguishable. Once again. It is also easy to see that if two states are k-distinguishable for any k>0 then they are also distinguishable for any n≥ k. Mathur 262 .) Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.Properties of FSM: k-equivalence (contd. If M1 and M2 are not kdistinguishable then they are said to be k-equivalent. M1 and M2 may be the same machines implying that kdistinguishability applies to any pair of states of an FSM.

Ltd Example: Completely specified machine Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 263 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd 5.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 264 .4 A fault model Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 265 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Faults in implementation . Hence tests generated from an FSM target faults related to the FSM itself. What faults are targeted by the tests generated using an FSM? Contents Foundations of Software Testing 2E Author: Aditya P.An FSM serves to specify the correct requirement or design of an application.

Mathur 266 .Fault model a/0 q0 q0 a/1 b/1 q0 a/1 b/1 a/1 b/1 q1 q1 q1 b/0 b/0 b/0 Correct design Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd a/1 Operation error a/1 Transfer error Contents Foundations of Software Testing 2E Author: Aditya P.

) q1 b/0 Extra state error Missing state error Contents Foundations of Software Testing 2E Author: Aditya P.a/1 b/1 a/0 q0 q0 a/1 q2 a/1 b/0 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Fault model (contd. Mathur 267 .

5 Characterization set 5.Copyright © 2013 Dorling Kindersley (India) Pvt.6 The W-method Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 268 . Ltd 5.

Minimality: An FSM M is considered minimal if the number of states in M is less than or equal to any other FSM equivalent to M. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 269 Copyright © 2013 Dorling Kindersley (India) Pvt. Completely specified: An FSM M is said to be completely specified if from each state in M there exists a transition for each input symbol. Ltd Assumptions for test generation .

Mathur 270 Copyright © 2013 Dorling Kindersley (India) Pvt. Step 2: Construct the characterization set W for M.Z Contents Foundations of Software Testing 2E Author: Aditya P. Step 3: (a) Construct the testing tree for M and (b) generate the transition cover set P from the testing tree. Step 5: Desired test set=P. Ltd Chow’s (W) method .Step 1: Estimate the maximum number of states (m) in the correct implementation of the given FSM M. Step 4: Construct set Z from W and m.

Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 271 . In the absence of any such knowledge.Copyright © 2013 Dorling Kindersley (India) Pvt. let m=|Q|. Ltd Step 1: Estimation of m This is based on a knowledge of the implementation.

s) Contents Foundations of Software Testing 2E Author: Aditya P. s)≠O(qj.Step 2: Construction of the W-set Copyright © 2013 Dorling Kindersley (India) Pvt. W contains a string s such that: O(qi. Each input sequence in W is of finite length. Ltd Let M=(X. Q. q1. δ. Mathur 272 . Y. W is a finite set of input sequences that distinguish the behavior of any pair of states in M. Given states qi and qj in Q. O) be a minimal and complete FSM.

q1)=1101 O(baaa.aa. Mathur 273 . baaa distinguishes state q1 from q2 as O(baaa. Ltd Example of a W-set W={baaa.q1) ≠ O(baaa.q2)=1100 Thus.q2) Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.aaa} O(baaa.

Ltd Steps in the construction of W-set .Step 1: Construct a sequence of k-equivalence partitions of Q denoted as P1. Step 2: Traverse the k-equivalence partitions in reverse order to obtain distinguishing sequence for each pair of states. m>0. Mathur 274 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. …Pm. P2.

is a collection of n finite sets Σk1.A k-equivalence partition of Q. If state u is in Σki and v in Σkj for i≠j. Mathur 275 Copyright © 2013 Dorling Kindersley (India) Pvt. denoted as Pk. then u and v are k-distinguishable. Σk2 … Σkn such that ∪ni=1 Σki =Q States in Σki are k-equivalent. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd What is a k-equivalence partition of Q? .

Current state Output Next state a b a b q1 0 1 q1 q4 q2 0 1 q1 q5 q3 0 1 q5 q1 q4 1 1 q3 q4 q5 1 1 q2 q5 Contents Foundations of Software Testing 2E Author: Aditya P. start with a tabular representation of M. Mathur 276 . construct a 1-equivalence partition. Ltd Given an FSM M.How to construct a k-equivalence partition? Copyright © 2013 Dorling Kindersley (India) Pvt.

q2. q5}. q3} and Σ2 ={q4. Ltd of Σ1={q1. Mathur 277 .Construct 1-equivalence partition Group states identical in their Output entries. This gives us 1-partition P1 consisting Σ 1 2 Current state Output Copyright © 2013 Dorling Kindersley (India) Pvt. Next state a b a b q1 0 1 q1 q4 q2 0 1 q1 q5 q3 0 1 q5 q1 q4 1 1 q3 q4 q5 1 1 q2 q5 Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd Rewrite P1 table. Σ 1 2 Current state Next state a b q1 q11 q42 q2 q11 q52 q3 q52 q11 q4 q31 q42 q5 q21 q52 P1 Table Group number Contents Foundations of Software Testing 2E Author: Aditya P.Construct 2-equivalence partition: Rewrite P1 table is the group number in which lies state qi. Replace a state entry qi by qij where j . Mathur 278 Copyright © 2013 Dorling Kindersley (India) Pvt. Remove the output columns.

Note the change in second subscripts. Mathur 279 Copyright © 2013 Dorling Kindersley (India) Pvt. This . Ltd Group all entries with identical second subscripts under the next state column.Construct 2-equivalence partition: Construct P2 table gives us the P2 table. Σ Current state Next state a b q1 q11 q43 q2 q11 q53 2 q3 q53 q11 3 q4 q32 q43 q5 q21 q53 1 P2 Table Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 280 Copyright © 2013 Dorling Kindersley (India) Pvt. Σ Current state Next state a b q1 q11 q43 q2 q11 q54 2 q3 q54 q11 3 q4 q32 q43 4 q5 q21 q54 1 P3 Table Contents Foundations of Software Testing 2E Author: Aditya P. Note the change in second subscripts.Construct 3-equivalence partition: Construct P3 table gives us the P3 table. Ltd Group all entries with identical second subscripts under the next state column. This .

Mathur 281 Copyright © 2013 Dorling Kindersley (India) Pvt. . we finally arrive at P4 table.Construct 4-equivalence partition: Construct P4 table Σ Current state P4 Table Next state a b 1 q1 q11 q44 2 q2 q11 q55 3 q3 q55 q11 4 q4 q33 q44 5 q5 q22 q55 Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Continuing with regrouping and relabeling.

Contents Foundations of Software Testing 2E Author: Aditya P. Ltd k-equivalence partition: Convergence The process is guaranteed to converge. When the process converges. The next step is to obtain the distinguishing strings for each state. Mathur 282 .Copyright © 2013 Dorling Kindersley (India) Pvt. each state will be in a separate group. and the machine is minimal.

q2) are in the same group in Pi and different groups in Pi+1. Initialize z=ε. Find tables Pi and Pi+1 such that (q1. We get P3 and P4. We update z to z. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Finding distinguishing sequences: Example Let us find a distinguishing sequence for states q1 and q2.Copyright © 2013 Dorling Kindersley (India) Pvt. Find the input symbol that distinguishes q1 and q2 in table P3. This symbol is b. Hence z now becomes b.b. Mathur 283 .

We update z to baa. Update z which now becomes ba. The next states for states q4 and q5 on symbol a are. respectively. respectively.The next states for q1 and q2 on b are. q3 and q2. We move to the P2 table and find the input symbol that distinguishes q4 and q5. These two states are distinguished in P1 by a and b. q4 and q5. Let us select a.) . Let us select a as the distinguishing symbol. Ltd Finding the distinguishing sequences: Example (contd. Mathur 284 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.

respectively. Check that o(q1.) . Ltd Finding the distinguishing sequences: Example (contd.baaa).The next states for q3 and q2 on a are. Mathur 285 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. q1 and q5. This is the farthest we can go backwards through the various tables. baaa is the desired distinguishing sequence for states q1 and q2. Moving to the original state transition table we obtain a as the distinguishing symbol for q1 and q5 We update z to baaa.baaa)≠o(q2.

Ltd Finding the distinguishing sequences: Example (contd. This leads us to the following characterization set for our FSM.) Using the procedure analogous to the one used for q1 and q2. W={a. aa. we can find the distinguishing sequence for each pair of states. aaa.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 286 . baaa} Contents Foundations of Software Testing 2E Author: Aditya P.

Step 2: Construct the characterization set W for M. Ltd Chow’s method: where are we? .Z Contents Foundations of Software Testing 2E Author: Aditya P. Step 5: Desired test set=P.Step 1: Estimate the maximum number of states (m) in the correct implementation Done of the given FSM M. Step 3: (a) Construct the testing tree for M and (b) generate the transition cover set P Next (a) from the testing tree. Mathur 287 Copyright © 2013 Dorling Kindersley (India) Pvt. Step 4: Construct set Z from W and m.

State q0. Contents Foundations of Software Testing 2E Author: Aditya P. Here is how we construct the testing tree. Select a node n at level k. This step is repeated for all nodes at level k. x)=m for x∈ X .Step 3: (a) Construct the testing tree for M path from the initial state to the remaining states in the FSM. This branch is labeled as x. It contains at least one . the initial state. Suppose that the testing tree has been constructed until level k . then n is a leaf node and is not expanded any further. Mathur 288 Copyright © 2013 Dorling Kindersley (India) Pvt. The (k+1)th level is built as follows. is the root of the testing tree. If n is not a leaf node then we expand it by adding a branch from node n to a new node m if δ(n. Ltd A testing tree of an FSM is a tree rooted at the initial state. If n appears at any level from 1 through k .

.Example: Construct the testing tree for M q1 becomes leaf. Ltd Start here. M . . No further expansion possible . Mathur 289 Copyright © 2013 Dorling Kindersley (India) Pvt. initial state is the root. Contents Foundations of Software Testing 2E Author: Aditya P. q4 can be expanded.

Z Contents Foundations of Software Testing 2E Author: Aditya P. (b) from the testing tree. Step 4: Construct set Z from W and m.Step 1: Estimate the maximum number of states (m) in the correct implementation Done of the given FSM M. Step 3: (a) Construct the testing tree for M and (b) generate the transition cover set P Next. Step 2: Construct the characterization set W for M. Mathur 290 Copyright © 2013 Dorling Kindersley (India) Pvt. Step 5: Desired test set=P. Ltd Chow’s method: where are we? .

bab. in the testing tree. baaab. baaaa} Contents Foundations of Software Testing 2E Author: Aditya P. starting at the . The empty string (ε) also belongs to P. Mathur 291 Copyright © 2013 Dorling Kindersley (India) Pvt. a. bb. Ltd A transition cover set P is a set of all strings representing sub-paths. baa. P={ε.Step 3: (b) Find the transition cover set from the testing tree root. baab. baaa. b. ba. Concatenation of the labels along the edges of a sub-path is a string that belongs to P.

Step 2: Construct the characterization set W for M.Z Contents Foundations of Software Testing 2E Author: Aditya P.Step 1: Estimate the maximum number of states (m) in the correct implementation Done of the given FSM M. Mathur 292 Copyright © 2013 Dorling Kindersley (India) Pvt. Step 3: (a) Construct the testing tree for M and (b) generate the transition cover set P Done from the testing tree. Next Step 5: Desired test set=P. Ltd Chow’s method: where are we? . Step 4: Construct set Z from W and m.

Xm-1-n. baaa. Mathur 293 . aa. baaa} ∪ {a. aaa.W ∪ Xm-n.Step 4: Construct set Z from W and m Copyright © 2013 Dorling Kindersley (India) Pvt. aa. aaa. aa. baaa}. ba. baaaa. we get Z = X0.W ={a. baaa} ={a.W ∪ …..W=W For X={a. aa.{a. we have: Z = X0. b}. bbaaa} Contents Foundations of Software Testing 2E Author: Aditya P.W ∪ X1. aaa. W={a. b}. Ltd Given that X is the input alphabet and W the characterization set. aaa. aa. m=6 Z = W ∪ X1. baaa. baa.W For m=n. aaaa. aaa.

Step 2: Construct the characterization set W for M. Ltd Chow’s method: where are we? .Z Done Next Contents Foundations of Software Testing 2E Author: Aditya P. Step 5: Desired test set=P.Step 1: Estimate the maximum number of states (m) in the correct implementation Done of the given FSM M. Mathur 294 Copyright © 2013 Dorling Kindersley (India) Pvt. Step 4: Construct set Z from W and m. Step 3: (a) Construct the testing tree for M and (b) generate the transition cover set P Done from the testing tree.

Z Do the following to test the implementation: 1. there might be variables to be set before it can be exercised with elements of T. Ltd The test inputs based on the given FSM M can now be derived as: T=P. 2.  Generate test cases for the application.  Find the expected response to each element of T.Z Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur Contents 295 .  Execute the application and check if the response matches. Foundations of Software Testing 2E Author: Aditya P.Step 5: Desired test set=P. Note that even though the application is modeled by M. Reset the application to the initial state after each test. 3.

Example 1: Testing an erroneous application Error revealing M(t1)=1101001 M M(t2)=11011 test cases Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 296 . Ltd Correct design t1=baaaaaa t2=baaba M1(t1)=1101001 M2(t2)=11001 M1 Foundations of Software Testing 2E Contents M2 Author: Aditya P.

Ltd Example 2: Extra state. Mathur Contents 297 . Author: Aditya P.M1 M2 t1=baaba M(t1)=11011 M1(t1)=11001 t2=baaa M(t2)=1101 M2(t2)=1100 Foundations of Software Testing 2E Copyright © 2013 Dorling Kindersley (India) Pvt. m=6. N=5.

s where r is in P and s in W. r Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd moves the application from initial state q0 to state qj. s=as’ takes it from qi to state qj or qj’. Then. Mathur 298 .Error detection process: in-class discussion Given m=n. Contents Foundations of Software Testing 2E Author: Aditya P. each test case t is of the form r.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 5. Mathur 299 .7 The Partial W method Contents Foundations of Software Testing 2E Author: Aditya P.

Test generation process is divided into two phases: Phase 1: Generate a test set using the state cover set (S) and the characterization set (W). and connected FSM.Tests are generated from minimal. complete. Phase 2: Generate additional tests using a subset of the transition cover set and state identification sets. Size of tests generated is generally smaller than that generated using the W-method. What is a state cover set? A state identification set? Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 300 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd The partial W (Wp) method .

baaa} S is always a subset of the transition cover set P. there is a string in S that takes M from its initial state to qi. Mathur 301 Copyright © 2013 Dorling Kindersley (India) Pvt. a state cover set S is a finite non-empty set of . S={ε. Ltd Given FSM M with input alphabet X. Also. baa. S is not necessarily unique.State cover set strings over X* such that for each state qi in Q. b. Contents Foundations of Software Testing 2E Author: Aditya P. ba.

] Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 302 Copyright © 2013 Dorling Kindersley (India) Pvt.] (b) O(qi.] (c) No subset of Wi satisfies property (b). for 1≤j≤ n . s)≠ O(qj. s∈ Wi [For each state other than qi. there is a string in Wi that distinguishes qi from qj. s) . 1≤ i≤n [Identification set is a subset of W. an identification set for state qi∈Q is denoted by Wi and has the following properties: (a) Wi⊆ W .Given an FSM M with Q as the set of states. [Wi is minimal. j≠ i . Ltd State identification set .

Mathur 303 Copyright © 2013 Dorling Kindersley (India) Pvt. aa. a} W3={a aa} W4=W5={a. aaa} Foundations of Software Testing 2E 4 Contents Author: Aditya P.x) 1 2 baaa 1 0 3 aa 0 1 4 a 0 1 5 a 0 1 3 aa 0 1 4 a 0 1 5 a 0 1 4 a 0 1 5 a 0 1 5 aaa 1 0 2 3 W1=W2={baaa.Last element of the output string Si Sj X o(Si.x) o(Sj. Ltd State identification set: Example .

W3. W2. Ltd Wp method: Example: Step 1: Compute S. P. b. Mathur 304 . W4. baaa} P={ε. baaa. W5} Contents Foundations of Software Testing 2E Author: Aditya P. bb. baaa} W={W1. Wi.Copyright © 2013 Dorling Kindersley (India) Pvt.W S={ε. a} W3={a aa} W4=W5={a. a. aaa. baab. baaab. ba. baa. aaa} W={a. baaaa} W1=W2={baaa. aa. baa. bab. b. ba. aa. W.

b. baaa}. Ltd Wp method: Example: Step 2: Compute T1 [m=n] T1=S. aaa. W={ε. Mathur 305 .Copyright © 2013 Dorling Kindersley (India) Pvt. aa.{a. baaa} Elements of T1 ensure that the each state of the FSM is covered and distinguished from the remaining states. Contents Foundations of Software Testing 2E Author: Aditya P. ba. baa.

baaa} ={a. baa. ri2. bab. baaaa} Let each element of R be denoted as ri1. where m∈X (the alphabet) Contents Foundations of Software Testing 2E Author: Aditya P.…rik. baaaa}-{ε. baaab. δ(rik. bab. Ltd Wp method: Example: Step 3: Compute R and δ [m=n] ba. ba. baaa. b. baab. m)=qij .R=P-S={ε. bb. bb. b. baaab. baa. Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 306 . a. baab.

a)=q1 δ(q1. δ(q1.W5 ) ∪ ({baaaa}. where Wij is the identification set for state qij. W1 ) ={abaaa. bab)=q5 T2=({a}.W5 ) ∪ ({baab}. bb)=q4 δ(q1. baabaaa} ∪ {baaaba. baaaa)=q1 Copyright © 2013 Dorling Kindersley (India) Pvt.W5 ) ∪ {baaab}.W4 ) ∪ ({bab}. W1 )∪ ({bb}. baaab)=q5 δ(q1.Wp method: Example: Step 4: Compute T2 [m=n] δ(q1. aaa. baaaaa} Contents Foundations of Software Testing 2E Author: Aditya P. babaaa} ∪ {baaba. Wij . Ltd T2=R⊗W=∪k(j=1) (rij}. bbaaa} ∪ {baba. baaabaaa} ∪ {baaaabaaa. baaaaaa. baab)=q5 δ(q1. aa} ∪ {bba. Mathur 307 .

Ltd Wp method: Example: Savings Test set size using the W method= 44 Test set size using the Wp method= 34 (20 from T1+14 from T2) Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 308 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Tests from T2 are applied in phase 2. even when tests from phase cover all transitions. they do not ensure all transition coverage. Mathur 309 . While tests from phase 1 ensure state coverage.Testing using the Wp method Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Testing proceeds in two phases. Contents Foundations of Software Testing 2E Author: Aditya P. Also. Tests from T1 are applied in phase 1. they do not apply the state identification sets and hence not all transfer errors are guaranteed to be revealed by these tests.

1≤i≤ (m-n) T2= T2=R. as follows: T1=S.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Wp method: Both sets T1 and T2 are computed a bit differently. Mathur 310 . where X[m-n] is the set union of Xi . X[m-n] ⊗W Contents Foundations of Software Testing 2E Author: Aditya P. X[m-n].

Mathur 311 . Ltd 5.8 The UIO sequence method [See the text] Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Copyright © 2013 Dorling Kindersley (India) Pvt.9 Automata theoretic versus control flow based techniques Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 312 . Ltd 5.

The W and the Wp methods are considered automata-theoretic methods for test generation. Mathur 313 Copyright © 2013 Dorling Kindersley (India) Pvt. Control theoretic techniques . Ltd Automata-theoretic vs. In contrast. Contents Foundations of Software Testing 2E Author: Aditya P. many books on software testing mention control-theoretic techniques for test generation. Let us understand the difference between the two types of techniques and their fault detection abilities.

Transition cover: A test set T is considered adequate with respect to the branch/ transition cover criterion for an FSM M if the execution of M against each element of T causes each transition in M to be taken at least once Contents Foundations of Software Testing 2E Author: Aditya P. Ltd State cover: A test set T is considered adequate with respect to the state cover criterion for an FSM M if the execution of M against each element of T causes each state in M to be visited at least once. Mathur 314 .Control theoretic techniques Copyright © 2013 Dorling Kindersley (India) Pvt.

a) and tr_2: qk= δ(qi. b) and qi.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 315 . Contents Foundations of Software Testing 2E Author: Aditya P. qj. qk are states in M. where for some input substring ab tr1: qi=δ(qj. Ltd Control theoretic techniques (contd.) Switch cover: A test set T is considered adequate with respect to the 1-switch cover criterion for an FSM M if the execution of M against each element of T causes each pair of transitions (tr1. tr2) in M to be taken at least once.

) . Ltd Control theoretic techniques (contd.Boundary interior cover: A test set T is considered adequate with respect to the boundary-interior cover criterion for an FSM M if the execution of M against each element of T causes each loop (a self-transition) across states to be traversed zero times and at least once. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 316 Copyright © 2013 Dorling Kindersley (India) Pvt. Exiting the loop upon arrival covers the ``boundary" condition and entering it and traversing the loop at least once covers the ``interior" condition.

a correct one (M1) and one with a transfer error . Will the tests generated by the W method reveal this error? Check it out! Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Consider the following machines.Control theoretic technique: Example 1 (M1’). Both machines generate the same output which is 0111. t=abba covers all states but does not not reveal the error. Mathur 317 Copyright © 2013 Dorling Kindersley (India) Pvt.

tr2). abbaab}. There are 12 branch pairs. (tr1. aaba. aabb. Consider the test set: {bb. Does it cover all branches? Does it reveal the error? Are the states in M2 1-distinguishable? Contents Foundations of Software Testing 2E Author: Aditya P. baab. tr5). tr6. a correct one (M2) and one with a transfer error .Control theoretic technique: Example 2 (M2’). Mathur 318 Copyright © 2013 Dorling Kindersley (India) Pvt. tr3). such as (tr1. Ltd Consider the following machines.

t2: abaab}. Is the error revealed by T? Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 319 . Consider T={t1: aab.Control theoretic technique: Example 3 (M3’). Ltd Consider the following machines. T2 causes each loop to be traversed once. T1 causes each state to be entered but loop Copyright © 2013 Dorling Kindersley (India) Pvt. a correct one (M3) and one with a transfer error not traversed.

and minimal. Tests so generated are guaranteed to detect all operation errors. transfer errors. Mathur 320 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Behavior of a large variety of applications can be modeled using finite state . connected. GUIs can also be modeled using FSMs The W and the Wp methods are automata theoretic methods to generate tests from a given FSM model.Summary machines (FSM). and missing/extra state errors in the implementation given that the FSM representing the implementation is complete. What happens if it is not? Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd Automata theoretic techniques generate tests superior in their fault detection ability than their control-theoretic counterparts. Control-theoretic techniques.) Copyright © 2013 Dorling Kindersley (India) Pvt. state cover. Contents Foundations of Software Testing 2E Author: Aditya P. include branch cover. boundary-interior. Mathur 321 .Summary (contd. and n-switch cover. that are often described in books on software testing. The size of tests sets generated by the W method is larger than generated by the Wp method while their fault detection effectiveness are the same.

Ltd Chapter 6 Test Generation: Combinatorial Designs Updated: July 16. Mathur 322 . 2013 Foundations of Software Testing 2E Contents Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 323 . covering arrays and mixed-level covering arrays? §  How to generate mixed-level covering arrays and test configurations from them? Copyright © 2013 Dorling Kindersley (India) Pvt.§  What are test configurations? How do they differ from test sets? §  Why combinatorial design? §  What are Latin squares and mutually orthogonal Latin squares (MOLS)? §  How does one generate test configurations from MOLS? §  What are orthogonal arrays. Ltd Learning Objectives Contents Foundations of Software Testing 2E Author: Aditya P.

1. Mathur 324 . Test configuration and test set Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 6.Copyright © 2013 Dorling Kindersley (India) Pvt.1.

Combinations of factors such as the operating system. and hardware platform. §  Each environment corresponds to a given set of values for each factor. Contents Foundations of Software Testing 2E Author: Aditya P. known as a test configuration. lead to a variety of environments. Mathur 325 Copyright © 2013 Dorling Kindersley (India) Pvt.§  Software applications are often designed to work in a variety of environments. network connection. Ltd Test configuration . §  An environment is characterized by combination of hardware and software.

§  Different versions of operating systems and printer drivers. is one possible configuration.§  Windows XP. §  To ensure high reliability across the intended environments. the application must be tested under as many test configurations. or environments. Ltd Test configuration: Example . Dial-up connection. can be combined to create several test configurations for a printer. Contents Foundations of Software Testing 2E Author: Aditya P. and a PC with 512MB of main memory. The number of such test configurations could be exorbitantly large making it impossible to test the application exhaustively. as possible. Mathur 326 Copyright © 2013 Dorling Kindersley (India) Pvt.

Each test case consists of input values and expected output. Mathur 327 Copyright © 2013 Dorling Kindersley (India) Pvt. §  Techniques we shall learn are useful in deriving test configurations as well as test sets.§  While a test configuration is a combination of factors corresponding to hardware and software within which an application is to operate. a test set is a collection of test cases. Ltd Test configuration and test set . Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 328 Copyright © 2013 Dorling Kindersley (India) Pvt. one corresponding to x and the other to y. each test run of a program often requires at least one value for each variable.§  While testing a program with one or more input variables. a program to find the greatest common divisor of two integers x and y requires two values. Contents Foundations of Software Testing 2E Author: Aditya P. §  For example. Ltd Motivation .

Mathur 329 . Ltd Motivation [2] While equivalence partitioning discussed earlier offers a set of guidelines to design test cases. (b) It lacks guidelines on how to select inputs from various sub-domains in the partition. Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. it suffers from two shortcomings: (a) It raises the possibility of a large number of sub-domains in the partition.

one selects at random a value from each of the subdomains. especially when using uni-dimensional equivalence partitioning. does not account for the possibility of faults in the program under test that arise due to specific interactions amongst values of different input variables.The number of sub-domains in a partition of the input domain increases in direct proportion to the number and type of input variables. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Motivation [3] . Mathur 330 Copyright © 2013 Dorling Kindersley (India) Pvt. and especially so when multidimensional partitioning is used. Such a selection procedure. Once a partition is determined.

Ltd Motivation [4] .While boundary values analysis leads to the selection of test cases that test a program at the boundaries of the input domain. other interactions in the input domain might remain untested. We will learn several techniques for generating test configurations or test sets that are small even when the set of possible configurations or the input domain and the number of sub-domains in its partition. Contents Foundations of Software Testing 2E Author: Aditya P. is large and complex. Mathur 331 Copyright © 2013 Dorling Kindersley (India) Pvt.

2.1. Mathur 332 . Modeling the input and configuration spaces Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 6.Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Modeling: Input and configuration space [1] . The configuration space of P consists of all possible settings of the environment variables under which P could be used. Consider program P that takes two integers x>0 and y>0 as inputs. Mathur 333 Copyright © 2013 Dorling Kindersley (India) Pvt.The input space of a program P consists of k-tuples of values that could be input to P during execution. Contents Foundations of Software Testing 2E Author: Aditya P. The input space of P is the set of all pairs of positive non-zero integers.

and must be able to print to a local or a networked printer.Now suppose that this program is intended to be executed under the Windows and the MacOS operating system. Mathur 334 Copyright © 2013 Dorling Kindersley (India) Pvt. and Z a local or a networked printer. through the Netscape or Safari browsers. The configuration space of P consists of triples (X. Y. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Modeling: Input and configuration space [2] . Y a browser. Z) where X represents an operating system.

Let us assume that each factor may be set at any one from a total of ci. Mathur 335 Copyright © 2013 Dorling Kindersley (India) Pvt. The inputs are also referred to as test parameters or as values. 1≤ i ≤ n values. .Xn. We refer to the inputs as factors. Ltd Factors and levels . X2. Contents Foundations of Software Testing 2E Author: Aditya P. Each value assignable to a factor is known as a level.Consider a program P that takes n inputs corresponding to variables X1. |F| refers to the number of levels for factor F..

(a. c} and {d. we have 2 factors and 3 levels for each factor. For example. e). d). Contents Foundations of Software Testing 2E Author: Aditya P. f). f). Let us say that during an execution of P. X and Y may each assume a value from the set {a. (a. e). (c. namely (a. This leads to a total of 32=9 factor combinations. Ltd Factor combinations . (c. d). b. e. is known as a factor combination. Mathur 336 Copyright © 2013 Dorling Kindersley (India) Pvt. (b. d). f}.A set of values. e). and (c. suppose that program P has two input variables X and Y. respectively. Thus. f). (b. (b. one for each factor.

the number of tests generated for exhaustive testing could be exorbitantly large. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Factor combinations: Too large? . Executing a billion tests might be impractical for many software applications.In general. the total number of factor combinations is nk. For many programs. if a program has 15 factors with 4 levels each. For example. the total number of tests is 415 ~109. Mathur 337 Copyright © 2013 Dorling Kindersley (India) Pvt. for k factors with each factor assuming a value from a set of n values. Suppose now that each factor combination yields one test case.

checks for their validity. Contents Foundations of Software Testing 2E Author: Aditya P. Toppings list. and P. Let us denote these four factors by S. Ltd Example: Pizza Delivery Service (PDS) [1] . Mathur 338 Copyright © 2013 Dorling Kindersley (India) Pvt. respectively. A customer is required to specify the following four items as part of the online order: Pizza size. A. and schedules Pizza for delivery. Delivery address and a home phone number. T.A PDS takes orders online.

In addition. The delivery address consists of customer name. The phone number is a numeric string possibly containing the dash (``--") separator. Ltd Pizza Delivery Service (PDS): Specs . the customer can customize the toppings. Medium. There is a list of 6 toppings from which to select. city. one line of address.Suppose now that there are three varieties for size: Large. Mathur 339 Copyright © 2013 Dorling Kindersley (India) Pvt. and the zip code. and Small. Contents Foundations of Software Testing 2E Author: Aditya P.

The total number of factor combinations is 24+23=24. Suppose we consider 6+1=7 levels for Toppings. Ltd PDS: Input space model . Different types of values for Address and Phone number will further increase the combinations Contents Foundations of Software Testing 2E Author: Aditya P. Number of combinations= 24+5x23+23+5x22=84. Mathur 340 Copyright © 2013 Dorling Kindersley (India) Pvt.

We have three factors in T. Thus. Each of these three factors can be set to any of four levels. Edit.Example: Testing a GUI Copyright © 2013 Dorling Kindersley (India) Pvt. we have a total 43=64 factor combinations. and Format. Ltd The Graphical User Interface of application T consists of three menus labeled File. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 341 .

Ltd The sort utility has several options and makes an interesting example for the identification of factors and levels.Example: The UNIX sort utility Copyright © 2013 Dorling Kindersley (India) Pvt.9x109 combinations.] We have identified a total of 20 factors for the sort command. The command line for sort is given below. sort [-cmu] [-ooutput] [-Tdirectory] [-y [ kmem]] [-zrecsz] [-dfiMnr] [-b] [ tchar] [kkeydef] [+pos1[-pos2]] [file.1 of the book lead to a total of approximately 1. Mathur 342 ... Contents Foundations of Software Testing 2E Author: Aditya P. The levels listed in Table 11.

i. and a browser as a platform. OS.Example: Compatibility testing Copyright © 2013 Dorling Kindersley (India) Pvt. hardware. Here we consider a combination of hardware. operating system. Ltd There is often a need to test a web application on different platforms to ensure that any claim such as ``Application X can be used under Windows and Mac OS X” are valid.e. Given that we want X to work on a variety of hardware. and browser combinations. Mathur 343 . Contents Foundations of Software Testing 2E Author: Aditya P. OS. Let X denote a Web application to be tested for compatibility. and browser. it is easy to obtain three factors.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Compatibility testing: Factor levels Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 344 .

some of these combinations are infeasible.Compatibility testing: Combinations For example. Similarly. Contents Foundations of Software Testing 2E Author: Aditya P. we assume that this is not the case for testing application X. the Safari browser is used on Apple computers and not on the PC in the Dell Series.2 is an OS for the Apple computers and not for the Dell Dimension series PCs. Mac OS10. However. Mathur 345 Copyright © 2013 Dorling Kindersley (India) Pvt. . While various editions of the Windows OS can be used on an Apple computer using an OS bridge such as the Virtual PC. Ltd There are 75 factor combinations.

Note that there is a large number of hardware configurations under the Dell Dimension Series. Contents Foundations of Software Testing 2E Author: Aditya P. memory sizes. and several others. Mathur 346 Copyright © 2013 Dorling Kindersley (India) Pvt. Pentium versus Athelon.The discussion above leads to a total of 40 infeasible factor combinations corresponding to the hardware-OS combination and the hardware-browser combination. Ltd Compatibility testing: Reduced combinations . e.g. These configurations are obtained by selecting from a variety of processor types. in all we are left with 35 platforms on which to test X. Thus. processor speeds.

it will also increase the number of factor combinations. and hence the time to test.While testing against all configurations will lead to more thorough testing of application X. Ltd Compatibility testing: Reduced combinations-2 . Mathur 347 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 348 .2. Ltd 6. Combinatorial test design process Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Modeling of input space or the environment is not exclusive and one might apply either one or both depending on the application under test. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 349 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Combinatorial test design process .

Such an object is also known as a factor covering design. Step 2: The model is input to a combinatorial design procedure to generate a combinatorial object which is simply an array of factors and levels. Ltd Step 1: Model the input space and/or the configuration space. Mathur 350 Copyright © 2013 Dorling Kindersley (India) Pvt. Step 3: The combinatorial object generated is used to design a test set or a test configuration as the requirement might be. The model is expressed in .Combinatorial test design process: steps terms of factors and their respective levels. Steps 2 and 3 can be automated. Contents Foundations of Software Testing 2E Author: Aditya P.

For example. Mathur 351 Contents Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Each combination obtained from the levels listed in Table 6.Combinatorial test design process: test inputs many test inputs.” Two sample test cases are: t1: sort -o afile bfile t2: sort -o cfile dfile Is one of the above tests sufficient? Foundations of Software Testing 2E Author: Aditya P. consider the combination in which all factors are set to ``Unused" except the -o option which is set to ``Valid File" and the file option that is set to ``Exists.1 can be used to generate .

The sequencing of tests generated by most test generation techniques must be determined by the tester and is not a unique characteristic of test generated in combinatorial testing. This sequence too must be determined by the tester. Contents Foundations of Software Testing 2E Author: Aditya P.Combinatorial test design process: summary case. Ltd Combination of factor levels is used to generate one or more test cases. Further. Mathur 352 Copyright © 2013 Dorling Kindersley (India) Pvt. the sequence in which inputs are to be applied to the program under test must be determined by the tester. For each test . the factor combinations do not indicate in any way the sequence in which the generated tests are to be applied to the program under test.

3. Fault model Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 353 . Ltd 6.Copyright © 2013 Dorling Kindersley (India) Pvt.

this invalid state must propagate to a point in the program execution where it is observable and hence is said to reveal the fault. Ltd Fault model . Mathur 354 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Of course.Faults aimed at by the combinatorial design techniques are known as interaction faults. We say that an interaction fault is triggered when a certain combination of t≥1 input values causes the program containing the fault to enter an invalid state.

Ltd t-way interaction faults . For t=2. for any arbitrary value of t. the faults are known as pairwise interaction faults. are known as simple faults. t=1. regardless of the values of other input variables. In general. the faults are known as t--way interaction faults.e. i. Contents Foundations of Software Testing 2E Author: Aditya P.Faults triggered by some value of an input variable. Mathur 355 Copyright © 2013 Dorling Kindersley (India) Pvt.

This is a pairwise interaction fault due to the interaction between factors X and Y. Ltd Pairwise interaction fault: Example . y) when X=x1 and Y=y1.Correct output: f(x. z)-g(x. Mathur 356 Copyright © 2013 Dorling Kindersley (India) Pvt. y. Foundations of Software Testing 2E Contents Author: Aditya P.

the fault is revealed only by the following two of the eight possible input combinations: x=-1. Contents Foundations of Software Testing 2E Author: Aditya P.This fault is triggered by all inputs such that x+y≠x-y and z ≠ 0. However. y=-1. Ltd 3-way interaction fault: Example . Mathur 357 Copyright © 2013 Dorling Kindersley (India) Pvt. z=1 and x=-1. y=1. z=1.

1 ≤ i ≤ k is a specific level for the corresponding factor. Ltd Fault vectors .. A run V is a fault vector for program P if the execution of P against a test case derived from V triggers a fault in P. each at qi. fk. Mathur 358 Copyright © 2013 Dorling Kindersley (India) Pvt.. V is also known as a run. lk). 1≤ i ≤ k levels. where li. Contents Foundations of Software Testing 2E Author: Aditya P. V is considered as a t-fault vector if any t ≤ k elements in V are needed to trigger a fault in P. f2. l2.Given a set of k factors f1. Note that a t-way fault vector for P triggers a t-way fault in P.. a vector V of factor levels is (l1....

(x1. y1. (1. Foundations of Software Testing 2E Contents Author: Aditya P. 1. 0) are two runs. -1. *) is a 2-way fault vector given that the values x1 and y1 trigger the twoway fault. There is a total of eight runs. Mathur 359 Copyright © 2013 Dorling Kindersley (India) Pvt. and z each having two levels. -1.The input domain consists of three factors x. y. 1) and (-1. For example. 1) are three fault vectors that trigger the 3-way fault. Of these eight runs. Ltd Fault vectors: Example . (-1. 1) and (-1.1.

Mathur 360 . Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Goal reviewed The goal of the test generation techniques described in this chapter is to generate a sufficient number of runs such that tests generated from these runs reveal all t-way faults in the program under test.

The number of such runs increases with the value of t.. t +k-1. t+2.. Contents Foundations of Software Testing 2E Author: Aditya P. Hence. one automatically generates some t+1. t is set to 2 and hence the tests generated are expected to reveal pairwise interaction faults. Mathur 361 Copyright © 2013 Dorling Kindersley (India) Pvt. and k-way runs also. there is always a chance that runs generated with t=2 reveal some higher level interaction faults. In many situations. . while generating t-way runs. Of course. Ltd Goal reviewed .

4.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 6. Latin squares Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 362 .

The term ``Latin square" arises from the fact that the early versions used letters from the Latin alphabet A. A Latin square of order n is an n x n matrix such . C. Latin squares of order 3. B. S={A. S={1. 2. etc. 3}. Latin squares of order 2.Latin Squares that no symbol appears more than once in a row and column. Ltd Let S be a finite set of n symbols. in a square arrangement. Mathur 363 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. B}.

For example.Copyright © 2013 Dorling Kindersley (India) Pvt. here is a Latin square M of order 4 constructed by cyclically rotating the first row and placing successive rotations in subsequent rows. Additional rows can be created by permuting the first row. Ltd Larger Latin Squares Larger Latin squares of order n can be constructed by creating a row of n distinct symbols. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 364 .

j)=i+j (mod 4). the Latin square M of order 4 given below is constructed such that M(i. Mathur 365 Copyright © 2013 Dorling Kindersley (India) Pvt. A Latin square based on integers 0. For example. 1≤ (i. j) ≤ 4. Ltd Modulo arithmetic and Latin Squares . 1… n is said to be in standard form if the elements in the top row 0 and the leftmost column are arranged in order.A Latin square of order n>2 can also be constructed easily by doing modulo arithmetic. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 366 . Ltd 6. Mutually orthogonal Latin squares Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.5.

e. Contents Foundations of Software Testing 2E Author: Aditya P. j) is M1(i.Mutually Orthogonal Latin Squares (MOLS) denote.e. the elements in the ith row and jth column of M1 and M2. i. we simply juxtapose the corresponding elements of M1 and M2. it appears exactly once in M. Mathur 367 Copyright © 2013 Dorling Kindersley (India) Pvt. respectively. j)M2(i. j) and M2(i. j) . Let M1(i. then M1 and M2 are said to be mutually orthogonal Latin squares of order n. Ltd Let M1 and M2 be two Latin squares. We now create an n x n matrix M from M1 and M2 such that the L(i. i. If each element of M is unique. j). each of order n.

Mathur 368 Copyright © 2013 Dorling Kindersley (India) Pvt. . Ltd There are no MOLS of order 2. Its elements are unique and hence M1 and M2 are MOLS. Contents Foundations of Software Testing 2E Author: Aditya P.MOLS: Example Juxtaposing the corresponding elements gives us L. MOLS of order 3 follow.

MOLS(n) contains n-1 mutually orthogonal Latin squares. Such a set of MOLS is a complete set. Mathur 369 Copyright © 2013 Dorling Kindersley (India) Pvt. When n is prime. Ltd MOLS: How many of a given order? . N(n)=n-1.MOLS(n) is the set of MOLS of order n. When n is prime or a power of prime. or a power of prime. Numbers 2 and 6 are known as Eulerian numbers after the famous mathematician Leonhard Euler (1707-1783). The number of MOLS of order n is denoted by N(n). Contents Foundations of Software Testing 2E Author: Aditya P. MOLS do not exist for n=2 and n=6 but they do exist for all other values of n>2.

Ltd MOLS: Construction [1] Example: We begin by constructing a Latin square of order 5 given the symbol set S={1. 2. Contents Foundations of Software Testing 2E Author: Aditya P. 4.Copyright © 2013 Dorling Kindersley (India) Pvt. 3. 5}. Mathur 370 .

Mathur 371 Copyright © 2013 Dorling Kindersley (India) Pvt.Next. we obtain M2 by rotating rows 2 through 5 of M1 by two positions to the left. Ltd MOLS: Construction [2] . Contents Foundations of Software Testing 2E Author: Aditya P.

Thus. It is easy to check that indeed the elements of MOLS(5) are mutually orthogonal by superimposing them pairwise. M4}.MOLS: Construction [3] positions. we get MOLS(5)={M1. Mathur 372 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd M3 and M4 are obtained similarly but by rotating the first row of M1 by 3 and 4 . M2. M3. respectively.

For other values of n. There is no general method available to construct the largest possible MOLS(n) for n that is not a prime or a power of prime. Mathur 373 Copyright © 2013 Dorling Kindersley (India) Pvt. the maximum size of MOLS(n) is n-1. Contents Foundations of Software Testing 2E Author: Aditya P.The method illustrated in the previous example is guaranteed to work only when constructing MOLS(n) for n that is prime or a power of prime. The CRC Handbook of Combinatorial Designs gives a large table of MOLS. limitation . Ltd MOLS: Construction.

Mathur 374 . Pairwise designs: Binary factors Contents Foundations of Software Testing 2E Author: Aditya P.6. Ltd 6.Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. Each factor can be at one of two levels.We will now look at a simple technique to generate a subset of factor combinations from the complete set. This assumption will be relaxed later. or binary. factors are considered. Each combination selected generates at least one test input or test configuration for the program under test. Only 2-valued. Mathur 375 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Pairwise designs .

Y2}. Y.Pairwise designs: Example variable. Considering each input variable as a factor. and Z denote the three input variables and {X1. X2}. Mathur 376 Copyright © 2013 Dorling Kindersley (India) Pvt. Z2} their respective sets of values. the total number of factor combinations is 23. {Y1. All possible combinations of these three factors follow. Let X. one corresponding to each input . Each variable can take only one of two distinct values. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Suppose that a program to be tested requires 3 inputs. {Z1.

Z1). It is a balanced design because each value occurs exactly the same number of times. (X2. (Y1. Z1). Y2). (Y1. (X2. Z2). Z2). The following four combinations cover all pairs: The above design is also known as a pairwise design. Y1). There are several sets of four combinations that cover all 12 pairs. (X2. Mathur 377 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Z2). (X1. (X2. Z1). (X1. Z1). . Y2). (X1. (Y2. Y1). Z2). and (Y2.Pairwise designs: Reducing the combinations There are 12 such pairs: (X1. Ltd Now suppose we want to generate tests such that each pair appears in at least one test.

We refer to the inputs as factors. The applet has 5 inputs listed after the next slide with their possible values. Ltd Example: ChemFun applet . Mathur 378 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. For simplicity we assume that each input has exactly two possible values.A Java applet ChemFun allows its user to create an in-memory database of chemical elements and search for an element.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Example: ChemFun applet Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 379 .

Mathur 380 . Ltd Example: ChemFun applet: Factor identification Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd ChemFun applet: Input/Output Input: n=5 factors Output: A set of factor combinations such that all pairs of input values are covered. Mathur 381 .

Ltd Compute the smallest integer k such that n≤ |S2k-1| S2k-1: Set of all binary strings of length 2k-1. Mathur 382 . S3= 3. k>0. Hence the desired integer k=3. Contents Foundations of Software Testing 2E Author: Aditya P. S2k-1= For k=3 we have S5= 10 and for k=2.ChemFun applet: Step 1 Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 383 Copyright © 2013 Dorling Kindersley (India) Pvt. k=3 and we have the following strings in the set S5. Ltd ChemFun applet: Step 2 . We select first five of the 10 strings in S5. Contents Foundations of Software Testing 2E Author: Aditya P. We have.Select any subset of n strings from S2k-1.

This will increase the size of each string from 2k-1 to 2k. Ltd ChemFun applet: Step 3 Append 0's to the end of each selected string. Mathur 384 . Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

is a 0 or a 1. Mathur 385 Copyright © 2013 Dorling Kindersley (India) Pvt. X2.Each combination is of the kind (X1.…. 1≤ i ≤ n. Contents Foundations of Software Testing 2E Author: Aditya P. where the value of each variable is selected depending on whether the bit in column i. Xn). Ltd ChemFun applet: Step 4 .

Ltd The following factor combinations by replacing the 0s and 1s in each column by the corresponding values of each factor. Mathur 386 . Contents Foundations of Software Testing 2E Author: Aditya P.ChemFun applet: Step 4 (contd.) Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd ChemFun applet: tests Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 387 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd ChemFun applet: All tests . Contents Foundations of Software Testing 2E Author: Aditya P.Recall that the total number of combinations is 32. Mathur 388 Copyright © 2013 Dorling Kindersley (India) Pvt. Requiring only pairwise coverage reduces the tests to 6.

Pairwise designs: Multi-valued factors Contents Foundations of Software Testing 2E Author: Aditya P.7. Mathur 389 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 6.

Mathur 390 . Ltd Next we will learn how to use MOLS to construct test configurations when: Contents Foundations of Software Testing 2E Author: Aditya P. •  All factors have the same number of levels.Pairwise designs: Multi-valued factors •  The number of factors is two or more. Copyright © 2013 Dorling Kindersley (India) Pvt. •  The number of levels for each factor is more than two.

One such facility is offered by The Applied Genomics Technology Center (AGTC) at the School of Medicine in Wayne State University. Ltd DNA sequencing is a common activity amongst biologists and other researchers. Contents Foundations of Software Testing 2E Author: Aditya P.Multi-valued factors: Sample problem Copyright © 2013 Dorling Kindersley (India) Pvt. We refer to this software as AGTCS. Mathur 391 . The submission of the sample itself is done using a software application available from AGTC. Several genomics facilities are available that allow a DNA sample to be submitted for sequencing.

Ltd AGTCS is supposed to work on a variety of platforms that differ in their hardware . For simplicity we consider a total of four factors with their respective levels given next. referred to as PI. In addition. must either have a profile already created with AGTCS or create a new one prior to submitting a sample. Contents Foundations of Software Testing 2E Author: Aditya P.Sample problem (contd.) and software configurations. Thus. Mathur 392 Copyright © 2013 Dorling Kindersley (India) Pvt. the hardware platform and the operating system are two factors to be considered while developing a test plan for AGTCS. AGTCS supports only a limited set of browsers. the user of AGTCS.

Ltd DNA sequencing: factors and levels . As PCs and Macs run their dedicated operating systems.There are 64 combinations of the factors listed. the number of combinations reduces to 32. Contents Foundations of Software Testing 2E Author: Aditya P. We want to test under enough configurations so that all possible pairs of factor levels are covered. Mathur 393 Copyright © 2013 Dorling Kindersley (India) Pvt.

We can now proceed to design test configurations in at least two ways. Ltd DNA sequencing: Approach to test design . Mathur 394 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Exercise 6. One way is to treat the testing on PC and Mac as two distinct problems and design the test configurations independently. The approach used in this example is to arrive at a common set of test configurations that obey the constraint related to the operating systems.12 asks you to take this approach and explore its advantages over the second approach used in this example.

and F4’ denote. hardware. Output: A set of factor combinations such that all pairwise combinations are covered. and PI. |F2’|=4. F3’. where F1’. Contents Foundations of Software Testing 2E Author: Aditya P.Input: n=4 factors. respectively. |F4’|=2. Ltd DNA sequencing: Test design algorithm . OS. |F3’|=4. browser. F2’. |F1’|=2. Mathur 395 Copyright © 2013 Dorling Kindersley (India) Pvt.

F3.Test design algorithm: Step 1 Copyright © 2013 Dorling Kindersley (India) Pvt. Let b=|F1|=4 and k=|F2|=4 Contents Foundations of Software Testing 2E Author: Aditya P. F2=F3'. Note that a different assignment is also possible because |F1|=|F4|and |F2|=|F3|. F4=F4'. b=k=4. Ltd Reliable the factors as F1. F2. F3=F1'. Doing so gives us F1=F2'. Mathur 396 . F4 such that |F1|≥|F2| ≥ |F3| ≥ |F4|.

Copyright © 2013 Dorling Kindersley (India) Pvt. Each block contains k rows. Label the columns as F1. F2. Mathur 397 . Contents Foundations of Software Testing 2E Author: Aditya P. … Fn. Ltd Test design algorithm: Step 2 Prepare a table containing 4 columns and b x k=16 rows divided into 4 blocks.

Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. and so on.. 2's in Block 2. k in rows 1 through k (k=4)..) Fill column F1 with 1's in Block 1.. Ltd Test design algorithm: Step 3 (contd. Fill Block 1 of column F2 with the sequence 1. Mathur 398 . 2.

Contents Foundations of Software Testing 2E Author: Aditya P. We choose the following set of MOLS of order 4. Ltd Find MOLS of order 4. Mathur 399 . As 4 is a power of prime. we can use the procedure described earlier.Test design algorithm: Step 4 Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 400 .Test design algorithm: Step 5 From M2 Copyright © 2013 Dorling Kindersley (India) Pvt. An entry marked with an asterisk (*) indicates an invalid level. Ltd From M1 Fill the remaining two columns of the table constructed earlier using columns of M1 for F3 and M2 for F4. A boxed entry in each row indicates a pair that does not satisfy the operating system constraint. Contents Foundations of Software Testing 2E Author: Aditya P.

Solution: One simple way to get rid of the infeasible values is to replace them by an arbitrarily selected feasible value for the corresponding factor. Ltd Test design algorithm: Step 6 [1] . Problem 1: Factors F3 and F4 can only assume values 1 and 2 whereas the table above contains other infeasible values for these two factors. These infeasible values are marked with an asterisk. we need to resolve two problems before we get to the design of test configurations. we can obtain 16 distinct test configurations for AGTCS. Mathur 401 Copyright © 2013 Dorling Kindersley (India) Pvt. However.Using the 16 entries in the table above.. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur Contents 402 Copyright © 2013 Dorling Kindersley (India) Pvt.Problem 2: Some configurations do not satisfy the operating system constraint. Here is an example: F1: Operating system=1(Win 2000) F3: Hardware=2 (Mac) is infeasible. Here we are assume that one is not using Virtual PC on the Mac. Foundations of Software Testing 2E Author: Aditya P. Ltd Test design algorithm: Step 6 [2] . Four such configurations are highlighted in the design by enclosing the corresponding numbers in rectangles.

F4=2). Consider block 3. (F2=3. F2=3). Contents Foundations of Software Testing 2E Author: Aditya P. (F2=3.Delete rows with conflicts?: Obviously we cannot delete these rows as that would leave some pairs uncovered. F4=2). Mathur 403 Copyright © 2013 Dorling Kindersley (India) Pvt. and (F3=1. (F1=3. F3=1). Removing Row~3 will leave the following five pairs uncovered: (F1=3. Ltd Test design algorithm: Step 6 [3] . F4=2).

Test design algorithm: Step 6 [4] Copyright © 2013 Dorling Kindersley (India) Pvt. Step 2: Add new configurations that cover the pairs that are left uncovered when we replace the highlighted rows. Mathur 404 . Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Proposed solution: We follow a two step procedure to remove the highlighted configurations and retain complete pairwise coverage. Step 1: Modify the four highlighted rows so they do not violate the constraint.

Mathur 405 .Test design algorithm: Step 6 [5] F2: Browser F4: PI F3: Hardware Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd F1: OS Contents Foundations of Software Testing 2E Author: Aditya P.

This is in contrast to 32 configurations obtained using a brute force method. Can we remove some rows from the design without affecting pairwise coverage? Contents Foundations of Software Testing 2E Author: Aditya P.Test design algorithm: Design configurations Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd We can easily construct 20 test configurations from the design obtained. Mathur 406 .

Ltd A sufficient number of MOLS might not exist for the problem at hand. While the MOLS approach assists with the generation of a balanced design in that all interaction pairs are covered an equal number of times.Shortcomings of using MOLS Copyright © 2013 Dorling Kindersley (India) Pvt. the number of test configurations is often larger than what can be achieved using other methods. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 407 .

Mathur 408 . Ltd 6.Copyright © 2013 Dorling Kindersley (India) Pvt.8. Orthogonal Arrays Contents Foundations of Software Testing 2E Author: Aditya P.

is an N x k matrix in which the entries are from a finite set S of s symbols such that any N x t sub array contains each t-tuple exactly the same number of times. Foundations of Software Testing 2E Contents Author: Aditya P. Ltd Examine this matrix and extract as many properties as you can: . such as the one above. s. Mathur 409 Copyright © 2013 Dorling Kindersley (India) Pvt. k.Orthogonal arrays An orthogonal array. Such an orthogonal array is denoted by OA(N. t).

2. Mathur 410 Copyright © 2013 Dorling Kindersley (India) Pvt. 2). This array is denoted as OA(4. and F3 to indicate the three factors. F2. Note that the value of parameter k is 3 and hence we have labeled the columns as F1. 3. 2}. It uses symbols from the set {1. Ltd Orthogonal arrays: Example .The following orthogonal array has 4 runs and has a strength of 2. Contents Foundations of Software Testing 2E Author: Aditya P.

1). (2. There is a total of st=22=4 pairs given as (1. (1. 1). 2).Orthogonal arrays: Index as the number of runs and t as the strength of the orthogonal array. It is easy to verify that each of the four pairs appears exactly once in each 4 x 2 sub array. 2). Contents Foundations of Software Testing 2E Author: Aditya P. Ltd The index of an orthogonal array is denoted by λ and is equal to N/st. Mathur 411 Copyright © 2013 Dorling Kindersley (India) Pvt. N is referred to . λ =4/22=1 implying that each pair (t=2) appears exactly once (λ =1) in any 4 x 2 sub array. and (2.

2) and has an index of 1. Each of the four factors can be at any one of 3 levels.What kind of an OA is this? It has 9 runs and a strength of 2. 3. 4. This array is denoted as OA(9. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Orthogonal arrays: Another example . Mathur 412 Copyright © 2013 Dorling Kindersley (India) Pvt.

k. s are determined from the context. Mathur 413 . Ltd Orthogonal array of N runs where k factors take on any value from a set of s symbols. t.Orthogonal arrays: Alternate notations Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Arrays shown earlier are LN denotes an orthogonal array of 9 runs. i. by examining the array itself.e.

Mathur 414 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 6.9. Mixed-level Orthogonal Arrays Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 415 . one encounters more than one factor. Contents Foundations of Software Testing 2E Author: Aditya P.Mixed level Orthogonal arrays Copyright © 2013 Dorling Kindersley (India) Pvt. This is because the design of such arrays assumes that all factors assume values from the same set of s values. each taking on a different set of values. Ltd So far we have seen fixed level orthogonal arrays. Mixed orthogonal arrays are useful in designing test configurations for such applications. In many practical applications.

Mathur 416 .Copyright © 2013 Dorling Kindersley (India) Pvt. k1 factors at s1 levels. Ltd Mixed level Orthogonal arrays: Notation Strength=t. and so on. Total factors: Contents Foundations of Software Testing 2E Author: Aditya P. Runs=N. k2 at s2 levels.

The balance property of orthogonal arrays remains intact for mixed level orthogonal arrays in that any N x t sub array contains each t-tuple corresponding to the t columns. Mathur 417 . which is λ.Mixed level Orthogonal arrays: Index and balance Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd The formula used for computing the index λ of an orthogonal array does not apply to the mixed level orthogonal array as the count of values for each factor is a variable. Contents Foundations of Software Testing 2E Author: Aditya P. exactly the same number of times.

Contents Foundations of Software Testing 2E Author: Aditya P. In the two leftmost columns. Mathur 418 Copyright © 2013 Dorling Kindersley (India) Pvt. In columns 1 and 5. Can you identify some properties? Balance: In any sub array of size 8 x 2. each possible pair occurs exactly the same number of times. each pair occurs exactly twice. each pair also occurs exactly twice. each pair occurs exactly once. In columns 1 and 3.This array can be used to design test configurations for an application that contains 4 factors each at 2 levels and 1 factor at 4 levels. Ltd Mixed level Orthogonal arrays: Example .

Ltd Mixed level Orthogonal arrays: Example This array can be used to generate test configurations when there are six binary factors. Mathur 419 .Copyright © 2013 Dorling Kindersley (India) Pvt. labeled F7 through F9. labeled F1 through F6 and three factors each with four possible levels. Contents Foundations of Software Testing 2E Author: Aditya P.

Hence we can use the following array to generate test configurations: Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Mixed level Orthogonal arrays: Test generation: Pizza delivery We have 3 binary factors and one factor at 3 levels. Mathur 420 .

Mathur 421 Copyright © 2013 Dorling Kindersley (India) Pvt.Check that all possible pairs of factor combinations are covered in the design above. Ltd Test generation: Pizza delivery: Array . What kind of errors will likely be revealed when testing using these 12 configurations? Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 422 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Test generation: Pizza delivery: test configurations Contents Foundations of Software Testing 2E Author: Aditya P.

9. Covering and mixed-level covering arrays Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 6.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 423 .

and when repeatability is not the focus. Mathur 424 . it is not always so in software testing. unless the application is known to behave non-deterministically. or mixed level covering arrays for combinatorial designs.The “Balance” requirement Observation [Dalal and Mallows. Foundations of Software Testing 2E Contents Author: Aditya P. For deterministic applications. there is generally no need for testing it again for the same pair. we can relax the balance requirement and use covering arrays. For example. Ltd essential in statistical experiments. 1998]: The balance requirement is often Copyright © 2013 Dorling Kindersley (India) Pvt. if a software application has been tested once for a given pair of factor levels.

Mathur 425 Copyright © 2013 Dorling Kindersley (India) Pvt. k. Ltd Covering array . k the number factors. we use λ=1. t) is an N x k matrix in which entries are from a finite set S of s symbols such that each N x t sub-array contains each possible t-tuple at least λ times. the number of levels for each factor. N denotes the number of runs. t the strength. Contents Foundations of Software Testing 2E Author: Aditya P.A covering array CA(N. and λ the index While generating test cases or test configurations for a software application. s. s.

a covering array CA(N. We are interested in minimal covering arrays. s. s. covering arrays do not meet the balance requirement that is met by orthogonal arrays. k.While an orthogonal array OA(N. t) covers each possible t-tuple at least λ times in any N x t sub array. k. Thus. t) covers each possible t-tuple λ times in any N x t sub array. This difference leads to combinatorial designs that are often smaller in size than orthogonal arrays. Mathur 426 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Covering array and orthogonal array . Covering arrays are also referred to as unbalanced designs.

2. a covering design with the same parameters requires only 6 runs. requires 8 runs and is denoted by . Mathur 427 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.Covering array: Example OA(8. 2). However. 5. Ltd A balanced design of strength 2 for 5 binary factors.

s2. Mathur 428 Copyright © 2013 Dorling Kindersley (India) Pvt. Q= ∑k i and each N x t sub- i=1 array contains at least one occurrence of each t-tuple corresponding to the t columns.. € Mixed-level covering arrays are generally smaller than mixed-level orthogonal arrays and more appropriate for use in software testing. Ltd Mixed level covering arrays . Contents Foundations of Software Testing 2E Author: Aditya P.A mixed-level covering array is denoted as p and refers to an N x Q matrix of entries such that.… denote the number of levels of each the corresponding factor. s1.

Is the above array balanced? Contents Foundations of Software Testing 2E Author: Aditya P.Comparing this with Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Mixed level covering array: Example we notice a reduction of 6 configurations. Mathur 429 .

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 6. Mathur 430 .10. Arrays of strength >2 Contents Foundations of Software Testing 2E Author: Aditya P.

Contents Foundations of Software Testing 2E Author: Aditya P.Arrays of strength >2 Copyright © 2013 Dorling Kindersley (India) Pvt. Consider the following factors in a pacemaker. Ltd Designs with strengths higher than 2 are sometimes needed to achieve higher confidence in the correctness of software. Mathur 431 .

a total of 54 tests will be required to test for all 3-way interactions of the 5 pacemaker parameters Could a design of strength 2 cover some triples and higher order tuples? Foundations of Software Testing 2E Author: Aditya P. 3. 5. Ltd Due to the high reliability requirement of the pacemaker.Pacemaker example ensure that there are no pairwise or 3-way interaction errors. 3) that has 54 runs for 5 factors each at 3 levels and is of strength 3. Thus. We could use an orthogonal array OA(54. we need a suitable combinatorial object with strength 3. Thus. Mathur Contents 432 Copyright © 2013 Dorling Kindersley (India) Pvt. we would like to test it to .

(b) Number of values (levels) for each parameter. Ltd We will now study a procedure due to Lei and Tai for the generation of mixed level covering arrays. Output: MCA Contents Foundations of Software Testing 2E Author: Aditya P. Inputs: (a) n ≥2: Number of parameters (factors). The procedure is known as In-parameter Order (IPO) procedure. Mathur 433 .Generating mixed level covering arrays Copyright © 2013 Dorling Kindersley (India) Pvt.

11. Generating covering arrays Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 434 . Ltd 6.

Step 2: Horizontal growth. Ltd Consists of three steps: Step 1: Main procedure. Contents Foundations of Software Testing 2E Author: Aditya P.IPO procedure Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 435 . Step 3: Vertical growth.

b2}. and C. c3}.IPO procedure: Example Copyright © 2013 Dorling Kindersley (India) Pvt. We want to generate a mixed level covering array for these three factors. a2. and C from the set {c1. We begin by applying the Main procedure which is the first step in the generation of an MCA using the IPO procedure. Contents Foundations of Software Testing 2E Author: Aditya P.. Mathur 436 . c2. a3}. A assumes values from the set {a1. Ltd Consider a program with three factors A. B from the set {b1. B.

…t6.IPO procedure: main procedure Copyright © 2013 Dorling Kindersley (India) Pvt. Let us denote the elements of as t1. We obtain the following set. Mathur 437 . The entire IPO procedure would terminate at this point if the number of parameters n=2. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Main: Step 1: Construct all runs that consist of pairs of values of the first two parameters. t2. In our case n=3 hence we continue with horizontal growth.

Mathur 438 . At this point T’ is empty as we have not extended any run in T. HG: Step 2: AP is the set of pairs yet to be covered. Ltd HG: Step 1: Compute the set of all pairs AP between parameters A and C. and parameters B and C.IPO procedure: Horizontal growth Copyright © 2013 Dorling Kindersley (India) Pvt. This leads us to the following set of fifteen pairs. Let T’ denote the set of runs obtained by extending the runs in T. Contents Foundations of Software Testing 2E Author: Aditya P.

(b1. c2). t2.Horizontal growth: Extend Copyright © 2013 Dorling Kindersley (India) Pvt. b1. b1. (a1. t2’=(a1. Mathur 439 . c3)} Update pairs remaining to be covered AP={(a1. b2. (a1. c2). c3)} Update T’ which becomes {(a1. 4: Expand t1. (a2. Ltd HG: Steps 3. c2. (a2. (a3. (a2. c1). b1. This gives us: t1’=(a1. c2). c3). c3. c1). b1. c2). (b2. b2. (a2. b1. t3 by appending c1. c3) Update T’ which now becomes {a1. c1). (b2. c1). c3). b2. c2). and t3’=(a2. b1. (a3. c1). c1). (a3. c3)} Contents Foundations of Software Testing 2E Author: Aditya P. c2).

we choose to extend t4 by c1. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 440 . b2) by c1 then we cover two of the uncovered pairs from AP. If we extend t4=(a2. c1). t5. HG: Step 6: Expand t4. t5. Ltd HG. t6 by suitably selected values of C. c1) and (b2. If we extend it by c2 then we cover one pair from AP. t6 as C does not have enough elements. Thus. If we extend it by c3 then we cover one pairs in AP. We find the best way to extend these in the next step. (a2. Step 5: We have not extended t4. namely.Horizontal growth: Optimal extension Copyright © 2013 Dorling Kindersley (India) Pvt.

(a3. c3)} Contents Foundations of Software Testing 2E Author: Aditya P. (a2. (a3. b2. b2. b1. c1) T’={(a1. c1). b1. c2). (b2. c3). c2). c3). (b2. c1)} AP= {(a1. Ltd T’={(a1. (a2. (a2. (a3. (a3. b1. This leads to: t5’=(a3. c1). c2). (a2. c2). b1. c3) and t6’=(a3.Horizontal growth: Update and extend remaining Copyright © 2013 Dorling Kindersley (India) Pvt. (b1. b2. c3). b2. (a1. b2. c1)} AP= {(a1. (a2. Mathur 441 . b2. c3). (a3. c2). c1). c1). c3). b1. (a3. c3)} HG: Step 6: Similarly we extend t5 and t6 by the best possible values of parameter C. (a2. c2). (a1. c3). b1. c2). (b1. c2).

c2). (a3. c2). we have five pairs remaining to be covered. c3). b2. (a2. c3). (a3. (b1. c2). b1. However. (b2. b2. c2). These are: AP= {(a1. (a2. Contents Foundations of Software Testing 2E Author: Aditya P. c1). we have generated six complete runs namely: T’={(a1. (a2. b1. c1)} We now move to the vertical growth step of the main IPO procedure to cover the remaining pairs. c3)} Also. (a1. c1). c3).Horizontal growth: Done Copyright © 2013 Dorling Kindersley (India) Pvt. (a3. b2. b1. Ltd We have completed the horizontal growth step. Mathur 442 .

Note that the value of parameter Y does not matter and hence is indicated as a * which denotes a don’t care value. Ltd For each missing pair p from AP. c3). c2) Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 443 . Let us begin with the pair p= (a1. The run t= (a1. c2) Next . Next .Vertical growth Copyright © 2013 Dorling Kindersley (India) Pvt. c2). *. This is covered by the run (a2. *. consider p=(a3. we will add a new run to T’ such that p is covered. c3) covers pair p. c2). This is covered by the run (a3. *. consider p=(a2.

c2). c1). Thus. Thus. (a2. c1). p is covered without any new run added. b2. b1. *. (a2. c3). c3). We already have (a3. We already have (a1. (a3. b1. b1. (a1. (a3. *. b2. c2). c3). c1). We replace the don’t care entries by an arbitrary value of the corresponding factor and get: T={(a1. Finally. b1. c2). b2. c3).) to get the run (a1. b2. b2. c2). Ltd Next . (a2. consider p=(b2. p is covered without any new run added. c3). b1. c2) and hence we can modify it to get the run (a3. c3) and hence we can modify it . (a3. b1.Vertical growth (contd. c2)} Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 444 Copyright © 2013 Dorling Kindersley (India) Pvt. consider p=(b1. (a1.

Ltd Final covering array Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 445 .MCA(9. 21 32. 2) Run F1(X) F2(Y) F3(Z) 1 1 1 1 2 1 2 2 3 1 2 3 4 2 1 2 5 2 1 3 6 2 2 1 7 3 1 2 8 3 1 3 9 3 2 1 Copyright © 2013 Dorling Kindersley (India) Pvt.

That completes our presentation of an algorithm to generate covering arrays. Contents Foundations of Software Testing 2E Author: Aditya P. Lei and Tai found that the IPO algorithm performs almost as well as AETG in the size of the generated arrays. A detailed analysis of the algorithm has been given by Lei and Tai. Lei and Tai offer several other algorithms for horizontal and vertical growth that are faster than the algorithm mentioned here. Ltd Practicalities . Mathur 446 Copyright © 2013 Dorling Kindersley (India) Pvt.

For example. Publicly available tool: ACTS from Jeff Lie’s group a UT Arlington. AETG is covered by US patent 5. Contents Foundations of Software Testing 2E Author: Aditya P. parameter A might not assume a value a2 when parameter B assumes value b3. Ltd AETG from Telcordia is a commercial tool to generate covering arrays.043. It allows users to specify constraints across parameters.Tools Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 447 .542.

We introduced one algorithm for generating covering arrays. This continues to be a research topic of considerable interest. most useful amongst these are mixed level covering arrays. and mixed-level covering arrays are used as combinatorial objects to generate test configurations/test cases. MOLS. Mathur 448 . covering arrays. Orthogonal arrays. Ltd test cases.Summary Combinatorial design techniques assist with the design of test configurations and Copyright © 2013 Dorling Kindersley (India) Pvt.” combinatorial designs offer a significant reduction in the number of test configurations/test cases. Contents Foundations of Software Testing 2E Author: Aditya P. For software testing. By requiring only pair-wise coverage and relaxing the “balance requirement. Handbooks offer a number covering and mixed level covering arrays.

2013 Foundations of Software Testing 2E Contents Author: Aditya P. Mathur 449 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Chapter 7 .Test Adequacy Measurement and Enhancement: Control and Data flow Updated: July 16.

condition. decision. multiple condition. LCSAJ. Mathur 450 . Ltd §  adequacy and how to use it to enhance tests? §  Control flow based test adequacy.Learning Objectives What is test adequacy? What is test enhancement? When to measure test Copyright © 2013 Dorling Kindersley (India) Pvt. statement. and MC/DC coverage §  Data flow coverage §  Strengths and limitations of code coverage based measurement of test adequacy §  The “subsumes” relation amongst coverage criteria §  Tools for the measurement of code coverage Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd 7. Mathur 451 .1 Test adequacy: basics Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Let R contain n requirements labeled R1. Ltd §  . Also. §  Suppose now that a set T containing k tests has been constructed to test P to determine whether or not it meets all the requirements in R . P has been executed against each test in T and has produced correct behavior. or as: Is T adequate? Contents Foundations of Software Testing 2E Author: Aditya P.What is adequacy? Consider a program P written to meet a set R of functional requirements. We notate such a P and R as ( P. Rn .…. R2. R). Mathur 452 Copyright © 2013 Dorling Kindersley (India) Pvt. §  We now ask: Is T good enough? This question can be stated differently as: Has P been tested thoroughly?.

The determination of whether or not a test set T for program P satisfies criterion C depends on the criterion itself and is explained later." and ``adequate. have the same meaning.Measurement of adequacy In the context of software testing. §  Adequacy is measured for a given test set designed to test P to determine whether or not P meets its requirements. Contents Foundations of Software Testing 2E Author: Aditya P." ``good enough. the terms ``thorough. Ltd §  . §  This measurement is done against a given criterion C . A test set is considered adequate with respect to criterion C when it satisfies C. Mathur 453 Copyright © 2013 Dorling Kindersley (India) Pvt." used in the questions above.

Example R1 Input two integers. R2. Ltd Program sumProduct must meet the following requirements: Contents Foundations of Software Testing 2E Author: Aditya P. R2. say x and y . Copyright © 2013 Dorling Kindersley (India) Pvt. from the standard input device.2 Find and print to the standard output device the product of x and y if x≥ y.1 Find and print to the standard output device the sum of x and y if x<y . Mathur 454 .

Obviously. R ) is considered adequate if for each requirement r in R there is at least one test case in T that tests the correctness of P with respect to r .1. Contents Foundations of Software Testing 2E Author: Aditya P. but not R2. Ltd Suppose now that the test adequacy criterion C is specified as: . Mathur 455 Copyright © 2013 Dorling Kindersley (India) Pvt. y=3> is inadequate with respect to C for program sumProduct.) C : A test T for program ( P.Example (contd. T={t: <x=2. The lone test case t in T tests R1 and R2.2.

we derive a finite set known as the coverage domain and denoted as Ce . A criterion C is a black-box test adequacy criterion if the corresponding coverage domain Ce depends solely on requirements R for the program P under test. Contents Foundations of Software Testing 2E Author: Aditya P. A criterion C is a white-box test adequacy criterion if the corresponding coverage domain Ce depends solely on program P under test. Ltd Black-box and white-box criteria . Mathur 456 Copyright © 2013 Dorling Kindersley (India) Pvt.For each adequacy criterion C .

Coverage We want to measure the adequacy of T. This fraction is also known as the coverage of T with respect to C . The fraction k/n is a measure of the extent to which T is adequate with respect to C . Given that Ce has n≥ 0 elements. P . Contents Foundations of Software Testing 2E Author: Aditya P. Ltd that T covers Ce if for each element e' in Ce there is at least one test case in T that tests e'. and R. T is considered inadequate with respect to C if it covers k elements of Ce where k<n . we say Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 457 . The notion of “tests” is explained later through examples. T is considered adequate with respect to C if it covers all elements in the coverage domain.

R2. R ) is considered adequate if for each requirement r in R there is at least one test case in T that tests the correctness of P with respect to r.1. P. and R is 0.2}.Let us again consider the following criterion: “A test T for program ( P. R2. Hence T is not adequate with respect to C . T covers R1 and R2. Mathur 458 Copyright © 2013 Dorling Kindersley (India) Pvt.” In this case the finite set of elements Ce={R1. The coverage of T with respect to C.2 .1 but not R2. Ltd Example . Contents Foundations of Software Testing 2E Author: Aditya P.66.

R ) is considered adequate if each path in P is traversed at least once. For the given adequacy criterion C we obtain the coverage domain Ce to be the set { p1.” Assume that P has exactly two paths. respectively.Consider the following criterion: “A test T for program ( P. Ltd Another Example . Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 459 Copyright © 2013 Dorling Kindersley (India) Pvt. We refer to these as p1 and p2. p2}. one corresponding to condition x<y and the other to x≥ y.

and R is 0. As T contains only one test for which x<y .) .To measure the adequacy of T of sumProduct against C . only the path p1 is executed. Thus.5 and hence T is not adequate with respect to C. the coverage of T with respect to C. we execute P against each test case in T . P . Contents Foundations of Software Testing 2E Author: Aditya P. We can also say that p1 is tested and p2 is not tested. Mathur 460 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Another Example (contd.

However. these elements must be derived by analyzing the code and not only by an examination of its requirements. Mathur 461 . and hence the coverage domain.Code-based coverage domain Copyright © 2013 Dorling Kindersley (India) Pvt. This assumption is based on a knowledge of the requirements. Ltd In the previous example we assumed that P contains exactly two paths. Contents Foundations of Software Testing 2E Author: Aditya P. to be different from the expected. when the coverage domain must contain elements from the code. Errors in the program and incomplete or incorrect requirements might cause the program.

There is only one path denoted as p1. T={t: <x=2. Contents Foundations of Software Testing 2E Author: Aditya P. Using the path-based coverage criterion C. C but does not reveal the error. Ltd sumProduct1 .r.Example This program is obviously incorrect as per the requirements of sumProduct. we get coverage domain Ce={ p1}. Mathur 462 Copyright © 2013 Dorling Kindersley (India) Pvt. This path traverses all the statements. y=3> }is adequate w.t.

Ce={ p1.t. p2}.r. It has two paths denoted by p1 and p2. T={t: <x=2. the path-based coverage criterion C. Contents Foundations of Software Testing 2E Author: Aditya P.) Copyright © 2013 Dorling Kindersley (India) Pvt. y=3>} is inadequate w. Mathur 463 .Example (contd. Ltd sumProduct2 This program is correct as per the requirements of sumProduct.

Copyright © 2013 Dorling Kindersley (India) Pvt. This does not diminish in any way the need for the measurement of test adequacy as increasing coverage might reveal an error!. Ltd Lesson An adequate test set might not reveal even the most obvious error in a program. Mathur 464 . Contents Foundations of Software Testing 2E Author: Aditya P.

1.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 465 . Ltd 7.3 Test enhancement Contents Foundations of Software Testing 2E Author: Aditya P.

or testing the features in a sequence different from the one used previously. Ltd While a test set adequate with respect to some criterion does not guarantee an error- . Identification of this deficiency helps in the enhancement of the inadequate test set. Mathur 466 Copyright © 2013 Dorling Kindersley (India) Pvt. Enhancement in turn is also likely to test the program in ways it has not been tested before such as testing untested portion. Inadequacy with respect to any criterion often implies test deficiency. Testing the program differently than before raises the possibility of discovering any uncovered errors.Test Enhancement free program. an inadequate test set is a cause for worry. Contents Foundations of Software Testing 2E Author: Aditya P.

y=4>. T' is adequate with respect to the path coverage criterion. Contents Foundations of Software Testing 2E Author: Aditya P. to make T adequate with respect to the path coverage criterion we need to add a test that covers p2.Test Enhancement: Example Copyright © 2013 Dorling Kindersley (India) Pvt. y=1>} Executing sum-product-2 against the two tests in T’ causes paths p1 and p2 are traversed. Mathur 467 . Ltd For sumProduct2. Adding this test to T and denoting the expanded test set by T' we get: T'={t1: <x=3. Thus. y=1>}. t2: <x=3. One test that does so is {<x=3>.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 468 . Ltd Test Enhancement: Procedure Contents Foundations of Software Testing 2E Author: Aditya P.

Copyright © 2013 Dorling Kindersley (India) Pvt. For y<0 the program skips the computation and outputs a suitable error message. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Test Enhancement: Example Consider a program intended to compute xy given integers x and y. Mathur 469 .

Ltd program for at least one zero and one non-zero value of each of the two inputs x and y. t2: <x=1. y=1>. Contents Foundations of Software Testing 2E Author: Aditya P. One such test set is T={t1: <x=0.Test Enhancement: Example (contd. Mathur 470 . Again. y=0}.) Suppose that test T is considered adequate if it tests the exponentiation Copyright © 2013 Dorling Kindersley (India) Pvt. The coverage domain for C can be determined using C alone and without any inspection of the program For C we get Ce={x=0. y≠ 0. one can derive an adequate test set for the program by an examination of Ce. y=0>}. x≠0.

Ltd Test Enhancement: Example: Path coverage .Criterion C of the previous example is a black-box coverage criterion as it does not require an examination of the program under test for the measurement of adequacy Let us now consider the path coverage criterion defined in in an earlier example. Contents Foundations of Software Testing 2E Author: Aditya P. The number of paths depends on the value of y and hence that of count. An examination of the exponentiation program reveals that it has an indeterminate number of paths due to the while loop. Mathur 471 Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. the number of paths can be arbitrarily large.) . This simple analysis of paths in exponentiation reveals that for the path coverage criterion we cannot determine the coverage domain. In case the program contains a loop. Mathur 472 Copyright © 2013 Dorling Kindersley (India) Pvt. The usual approach in such cases is to simplify C and reformulate it as follows: A test T is considered adequate if it tests all paths. Ltd Example: Path coverage (contd. then it is adequate to traverse the loop body zero times and once.Given that y is any non-negative integer.

The elements of Ce’ are enumerated below with respect to flow graph for the exponentiation program.The modified path coverage criterion leads to C‘e={p1. p3}. Mathur 473 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Example: Path coverage (contd. p2.) . Contents Foundations of Software Testing 2E Author: Aditya P.

66. the coverage of T with respect to C' is 2/3=0. Ltd Example: Path coverage (contd. As T does not contain any test with y<0. Thus. p3 remains uncovered. Contents Foundations of Software Testing 2E Author: Aditya P.) . Mathur 474 Copyright © 2013 Dorling Kindersley (India) Pvt.We measure the adequacy of T with respect to C'.

<x=1.) us use t:<x=5. y=-1>. We add t to T. y=1>. Ltd Any test case with y<0 will cause p3 to be traversed. Mathur 475 Copyright © 2013 Dorling Kindersley (India) Pvt. y=0>. The enhanced test set is: T={<x=0. The loop in the enhancement terminates as we have covered all feasible elements of Ce’.Example: Path coverage (contd. y=-1>} Contents Foundations of Software Testing 2E Author: Aditya P. Test t covers path p3 and P behaves correctly. <x=5. Let .

1. Ltd 7. Mathur 476 .4 Infeasibility and test adequacy Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Infeasibility Copyright © 2013 Dorling Kindersley (India) Pvt. There does not exist an algorithm that would analyze a given program and determine if a given element in the coverage domain is infeasible or not. Mathur 477 . it is usually the tester who determines whether or not an element of the coverage domain is infeasible. Contents Foundations of Software Testing 2E Author: Aditya P. Thus. Ltd An element of the coverage domain is infeasible if it cannot be covered by any test in the input domain of the program under test.

Thus. an attempt to enhance a test set by executing a test aimed at covering element e of program P. In some cases simple arguments can be constructed to show that a given element is infeasible. Ltd Feasibility can be demonstrated by executing the program under test against a test case and showing that indeed the element under consideration is covered. Contents Foundations of Software Testing 2E Author: Aditya P. might fail. Mathur 478 .Demonstrating feasibility Copyright © 2013 Dorling Kindersley (India) Pvt. For complex programs the problem of determining infeasibility could be difficult. Infeasibility cannot be demonstrated by program execution against a finite number of test cases.

Infeasible path: Example

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This program inputs two integers x and y, and
computes z. Ce={p1, p2, p3}.

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p1 is infeasible and cannot be traversed by any test case.
This is because when control reaches node 5, condition
y≥0 is false and hence control can never reach node 6.

Thus, any test adequate with respect to the path
coverage criterion for the exponentiation program will
only cover p2 and p3
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Example: Flow graph and paths

Adequacy and infeasibility

considered adequate when all feasible elements in the domain have been covered.

While programmers might not be concerned with infeasible elements, testers
attempting to obtain code coverage are. Prior to test enhancement, a tester usually does
not know which elements of a coverage domain are infeasible. Unfortunately, it is only
during an attempt to construct a test case to cover an element that one might realize
the infeasibility of an element.

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In the presence of one or more infeasible elements in the coverage domain, a test is

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7.1.5 Error detection and test enhancement

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The purpose of test enhancement is to determine test cases that test the untested
parts of a program or exercise the program using uncovered portions of the input
domain. Even the most carefully designed tests based exclusively on requirements
can be enhanced.
The more complex the set of requirements, the more likely it is that a test set designed
using requirements is inadequate with respect to even the simplest of various test
adequacy criteria.

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Test enhancement

Example

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A program to meet the following requirements is to be developed.

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Example (contd.)

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Example (contd.)

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Consider the following program written to meet the requirements stated earlier.

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Example (contd.)

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Example (contd.)

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Example (contd.)

whether or not our program meets its requirements.
T={<request=1, x=2, y=3>, <request=2, x=4>, <request=3>}

For the first two of the three requests the program correctly outputs 8 and 24,
respectively. The program exits when executed against the last request. This program
behavior is correct and hence one might conclude that the program is correct. It will
not be difficult for you to believe that this conclusion is incorrect.

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Suppose now that the following set containing three tests has been developed to test

Let us now evaluate T against the path coverage criterion.
In class exercise: Go back to the example
program and extract the paths not covered by T.

The coverage domain consists of all paths that traverse each of the three loops zero
and once in the same or different executions of the program. This is left as an exercise
and we continue with one sample, and “tricky,” uncovered path.

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Example (contd.)

Example (contd.)

line 10, then the first if at line 12, followed by the statements that compute the
factorial starting at line 20, and then the code to compute the exponential starting at
line 13.

p is traversed when the program is launched and the first input request is to compute
the factorial of a number, followed by a request to compute the exponential. It is easy
to verify that the sequence of requests in T does not exercise p. Therefore T is
inadequate with respect to the path coverage criterion.
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Consider the path p that begins execution at line 1, reaches the outermost while at

Example (contd.)

T’={<request=2, x=4>, <request=1, x=2, y=3>, <request=3>}

When the values in T' are input to our example program in the sequence given, the
program correctly outputs 24 as the factorial of 4 but incorrectly outputs 192 as the value
of 23 .
This happens because T' traverses our “tricky” path which makes the computation of the
exponentiation begin without initializing product. In fact the code at line 14 begins with
the value of product set to 24.
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To cover p we construct the following test:

This example has illustrated a benefit of test enhancement based on code coverage. Execution of the program under test on T' did cover a path that was not covered earlier and revealed an error in the program.) .In our effort to increase the path coverage we constructed T' . Mathur 493 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Example (contd.

Mathur 494 .6 Single and multiple executions Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 7.1.Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd In the previous example we constructed two test sets T and T' . one for each value of request. x=4>. Mathur 495 Copyright © 2013 Dorling Kindersley (India) Pvt. y=3>. are input in a sequence during a single execution of the test program. Should T (or T’) be considered a single test or a sequence of three tests? T’={<request=2. x=2.Multiple executions and T' contain three tests one for each value of variable request. Notice that both T . <request=1. <request=3>} we assumed that all three tests. Hence we consider T as a test set containing one test case and write it as follows: Contents Foundations of Software Testing 2E Author: Aditya P.

) We assumed that all three tests. Hence we consider T as a test set containing one test case and write it. Ltd during a single execution of the test program. are input in a sequence Copyright © 2013 Dorling Kindersley (India) Pvt. one for each value of request.Multiple executions (contd. Mathur 496 . it as follows: T”=T∪T’ Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd 7.1 Statement and block coverage Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.2. Mathur 497 .

Notation: (P. such as the assignment. Mathur 498 Copyright © 2013 Dorling Kindersley (India) Pvt. Recall that a basic block is a sequence of consecutive statements that has exactly one entry point and one exit point. while others are executable. adequacy with respect to the statement coverage and block coverage criteria are defined next. and while statements in C and Java. For any procedural language.Declarations and basic blocks Some of these statements are declarative. such as the #define and int statements in C. if. Ltd Any program written in a procedural language consists of a sequence of statements. R) denotes program P subject to requirement R. . Contents Foundations of Software Testing 2E Author: Aditya P.

T is considered adequate with respect to the statement coverage criterion if the statement coverage of T with respect to (P. Ltd The statement coverage of T with respect to ( P. i. Si is the number of unreachable statements. and Se is the total number of statements in the program. Contents Foundations of Software Testing 2E Author: Aditya P. R) is 1.e. Mathur 499 Copyright © 2013 Dorling Kindersley (India) Pvt. R ) is computed as Sc/(Se-Si) . where .Statement coverage Sc is the number of statements covered. the size of the coverage domain.

The block coverage of T with respect to (P. R) is computed as Bc/(Be -Bi) . Mathur 500 Copyright © 2013 Dorling Kindersley (India) Pvt. the size of the block coverage domain. Contents Foundations of Software Testing 2E Author: Aditya P. R) is 1. T is considered adequate with respect to the block coverage criterion if the statement coverage of T with respect to (P. Bi is the number of unreachable blocks. where Bc is the number of blocks covered. Ltd Block coverage . i.e. and Be is the total number of blocks in the program.

Note: 7b is unreachable. 3. Ltd Let T1={t1:<x=-1. Se=7. and 10 t2: 2. Contents Foundations of Software Testing 2E Author: Aditya P. y=1>} . 4. 9. The statement coverage for T is 6/(7-1)=1 . 4. 9. 7b. 6. 5. 3. 7. 3. 7. 6. R ) with respect to the statement coverage criterion. and 10. 10} Statements covered: t1: 2. (b) Sc=6. y=-1>.t 2:<x=1. Mathur 501 Copyright © 2013 Dorling Kindersley (India) Pvt. 5. Hence we conclude that T1 is adequate for (P.Example: statement coverage Coverage domain: Se={2. 4. Si=1.

Hence T2 is not adequate for (P. Bc=3. Be=5 . Bi=1. 3.Example: block coverage Blocks covered: t1: Blocks 1. R) with respect to the block coverage criterion. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 502 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Coverage domain: Be={1. 2. t3: same coverage as of t1. 4. 2. Block coverage for T2= 3/(5-1)=0. 5 t2. 5 .75.

t. In class exercise: Verify this statement! Contents Foundations of Software Testing 2E Author: Aditya P.Example: block coverage (contd. Mathur 503 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd T1 is adequate w. if test t2 in T1 is added to T2.r. In . block coverage criterion.) class exercise: Verify this statement! Also. we obtain a test set adequate with respect to the block coverage criterion for the program under consideration.

65 is the same as 65% statement coverage. Ltd The formulae given for computing various types of code coverage. one might instead use percentages. Mathur 504 . Contents Foundations of Software Testing 2E Author: Aditya P. while specifying a coverage value. yield a coverage value between 0 and 1. a statement coverage of 0.Coverage values Copyright © 2013 Dorling Kindersley (India) Pvt. For example. However.

2 Conditions and decisions Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 505 .Copyright © 2013 Dorling Kindersley (India) Pvt.2. Ltd 7.

x and x+y are valid conditions. A AND (x<y). and the constants 1 and 0 correspond to. A . x > y . (A AND B).Conditions Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 506 . A OR B . Contents Foundations of Software Testing 2E Author: Aditya P. are all sample conditions. and B are Boolean variables. respectively. Ltd Any expression that evaluates to true or false constitutes a condition. Given that A . true and false. Such an expression is also known as a predicate. Note that in programming language C. and x and y are integers.

≤ >. Mathur 507 . ==. Simple conditions are also referred to as atomic or elementary conditions because they cannot be parsed any further into two or more conditions. It is made up of variables and at most one relational operator from the set {<. Contents Foundations of Software Testing 2E Author: Aditya P. A compound condition is made up of two or more simple conditions joined by one or more Boolean operators. ≥.Simple and compound conditions Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd A simple condition does not use any Boolean operators except for the not operator. ≠ }.

Mathur 508 Copyright © 2013 Dorling Kindersley (India) Pvt. while. and switch statements to serve as contexts for decisions.Conditions as decisions high level languages provide if. Contents Foundations of Software Testing 2E Author: Aditya P. most . Ltd Any condition can serve as a decision in an appropriate context within a program.

Contents Foundations of Software Testing 2E Author: Aditya P. false. Ltd A decision can have three possible outcomes. When the . Mathur 509 Copyright © 2013 Dorling Kindersley (India) Pvt. In some cases the evaluation of a condition might fail in which case the corresponding decision's outcome is undefined. true. and undefined.Outcomes of a decision condition corresponding to a decision to take one or the other path is taken.

the decision at line 6 evaluates to undefined. Mathur 510 Copyright © 2013 Dorling Kindersley (India) Pvt.The condition inside the if statement at line 6 will remain undefined because the loop at lines 2-4 will never terminate. Ltd Undefined condition . Contents Foundations of Software Testing 2E Author: Aditya P. Thus.

Does Cond contain three or four simple conditions? Both answers are correct depending on one's point of view. Ltd Coupled conditions . there are three distinct conditions A . The answer is four when one is interested in the number of occurrences of simple conditions in a compound condition.How many simple conditions are there in the compound condition: Cond=(A AND B) OR (C AND A)? The first occurrence of A is said to be coupled to its second occurrence. and C. B . Mathur 511 Copyright © 2013 Dorling Kindersley (India) Pvt. Indeed. Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd 7. Mathur 512 .3 Decision coverage Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.2.

Contents Foundations of Software Testing 2E Author: Aditya P. a condition becomes a decision only when it is used in the Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd appropriate context such as within an if statement.Conditions within assignments Strictly speaking. Mathur 513 . x<y does not constitute a decision and neither does A*B. At line 4.

This implies that. i.e. the expression in the if or a while statement has evaluated to true in some execution of the program under test and to false in the same or another execution. Ltd A decision is considered covered if the flow of control has been diverted to all . all outcomes of the decision have been taken. Mathur 514 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.Decision coverage possible destinations that correspond to this decision. for example.

Decision coverage: switch statement

more executions of the program under test the flow of control has been diverted to all
possible destinations.

Covering a decision within a program might reveal an error that is not revealed by
covering all statements and all blocks.

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A decision implied by the switch statement is considered covered if during one or

This program inputs an integer x, and if necessary,
transforms it into a positive value before invoking
foo-1 to compute the output z. The program has an
error. As per its requirements, the program is
supposed to compute z using foo-2 when x≥0.

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Decision coverage: Example

Consider the test set T={t1:<x=-5>}. It is adequate
with respect to statement and block coverage
criteria, but does not reveal the error.

Another test set T'={t1:<x=-5> t2:<x=3>} does
reveal the error. It covers the decision whereas T
does not. Check!

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Decision coverage: Example (contd.)

Decision coverage: Computation

revealing an error that is not revealed by a test set adequate with respect to statement
and block coverage.
The decision coverage of T with respect to ( P, R ) is computed as Dc/(De -Di) , where
Dc is the number of decisions covered, Di is the number of infeasible decisions, and
De is the total number of decisions in the program, i.e. the size of the decision coverage
domain.
T is considered adequate with respect to the decisions coverage criterion if the decision
coverage of T with respect to ( P, R ) is 1.
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The previous example illustrates how and why decision coverage might help in

The domain of decision coverage consists of all decisions in the program under test.

Note that each if and each while contribute to one decision whereas a switch
contribute to more than one.

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Decision coverage: domain

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7.2.4 Condition coverage

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Condition coverage

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A decision can be composed of a simple condition such as x<0 , or of a more
complex condition, such as (( x<0 AND y<0 ) OR ( p≥q )).
AND, OR, XOR are the logical operators that connect two or more simple
conditions to form a compound condition.
A simple condition is considered covered if it evaluates to true and false in one or
more executions of the program in which it occurs. A compound condition is
considered covered if each simple condition it is comprised of is also covered.

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7.2.5 Condition/decision coverage

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Decision coverage is concerned with the coverage of decisions regardless of whether
or not a decision corresponds to a simple or a compound condition. Thus, in the
statement

there is only one decision that leads control to line 2 if the compound condition
inside the if evaluates to true. However, a compound condition might evaluate to true
or false in one of several ways.
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Decision and condition coverage

The condition at line 1 evaluates to false when x≥0 regardless of the value of y.
Another condition, such as x<0 OR y<0, evaluates to true regardless of the value of
y, when x<0.
With this evaluation characteristic in view, compilers often generate code that uses
short circuit evaluation of compound conditions.

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Decision and condition coverage (contd)

Decision and condition coverage (contd)

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Here is a possible translation:

We now see two decisions, one corresponding to each simple condition in the if
statement.

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The condition coverage of T with respect to ( P, R ) is computed as Cc/(Ce -Ci) ,
where Cc is the number of simple conditions covered, Ci is the number of infeasible
simple conditions, and |Ce is the total number of simple conditions in the program, i.e.
the size of the condition coverage domain.
T is considered adequate with respect to the condition coverage criterion if the
condition coverage of T with respect to ( P, R ) is 1.

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Condition coverage

An alternate formula where each simple condition contributes 2, 1, or 0 to Cc
depending on whether it is covered, partially covered, or not covered, respectively. is:

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Condition coverage: alternate formula

Condition coverage: Example

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Partial specifications for computing z:

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Condition coverage: Example (contd.)

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Consider the test set:

Check that T is adequate with respect to the
statement, block, and decision coverage criteria
and the program behaves correctly against t1 and
t2.
Cc=1, Ce=2, Ci=0. Hence condition coverage for
T=0.5.
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Condition coverage: Example (contd.)

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Add the following test case to T:
t3: <x=3, y=4>
Check that the enhanced test set T is adequate
with respect to the condition coverage criterion
and possibly reveals an error in the program.
Under what conditions will a possible error at
line 7 be revealed by t3?

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Condition/decision coverage

imply that each simple condition within a compound condition has taken both
values true and false.
Condition coverage ensures that each component simple condition within a
condition has taken both values true and false.
However, as illustrated next, condition coverage does not require each decision to
have taken both outcomes. Condition/decision coverage is also known as branch
condition coverage.
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When a decision is composed of a compound condition, decision coverage does not

Condition/decision coverage: Example

In class exercise: Confirm that T1 is adequate with respect to
to decision coverage but not condition coverage.
In class exercise: Confirm that T2 is adequate with respect to
condition coverage but not decision coverage.
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Consider the following program and two test sets.

Condition/decision coverage: Definition

+Dc)/((Ce -Ci) +(De-Di)) , where Cc is the number of simple conditions covered,
Dc is the number of decisions covered, Ce and De are the number of simple
conditions and decisions respectively, and Ci and Di are the number of infeasible
simple conditions and decisions, respectively.

T is considered adequate with respect to the multiple condition coverage criterion if
the condition coverage of T with respect to ( P, R ) is 1.

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The condition/decision coverage of T with respect to (P, R) is computed as (Cc

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Condition/decision coverage: Example

In class exercise: Check that the following test set is
adequate with respect to the condition/decision
coverage criterion.

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7.2.6 Multiple Condition coverage

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Consider a compound condition with two or more simple conditions. Using condition
coverage on some compound condition C implies that each simple condition within C
has been evaluated to true and false.

However, does it imply that all combinations of the values of the individual simple
conditions in C have been exercised?

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Multiple condition coverage

Multiple condition coverage: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Consider D=(A<B) OR (A>C) composed of two simple conditions A< B and A> C . The four possible combinations of the outcomes of these two simple conditions are enumerated in the table. Mathur 537 . Consider T: Check: Does T cover all four combinations? Check: Does T’ cover all four combinations? Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 538 . Thus. …. the total number of combinations to be covered is n ∑2 ki i=1 € Contents Foundations of Software Testing 2E Author: Aditya P. For example.Multiple condition coverage: Definition Copyright © 2013 Dorling Kindersley (India) Pvt. Assume also that each decision contains k1. decision i will have a total of 2ki combinations. kn simple conditions. Each decision has several combinations of values of its constituent simple conditions. Ltd Suppose that the program under test contains a total of n decisions. k2.

Contents Foundations of Software Testing 2E Author: Aditya P. where Cc is the number of combinations covered.The multiple condition coverage of T with respect to ( P.) . R ) is 1. and Ce is the total number of combinations in the program. R ) is computed as Cc/(Ce Ci) . Mathur 539 Copyright © 2013 Dorling Kindersley (India) Pvt. T is considered adequate with respect to the multiple condition coverage criterion if the condition coverage of T with respect to ( P. Ci is the number of infeasible simple combinations. Ltd Multiple condition coverage: Definition (contd.

Contents Foundations of Software Testing 2E Author: Aditya P. has been left out. . Mathur 540 Copyright © 2013 Dorling Kindersley (India) Pvt. line 3 in the table. computation of S for one of the four combinations.Multiple condition coverage: Example There is an obvious error in the program. Ltd Consider the following program with specifications in the table.

Ltd Is T adequate with respect to decision coverage? Multiple condition coverage? Does it reveal the error? Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 541 .Multiple condition coverage: Example (contd.) Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 542 .) Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd To improve decision coverage we add t3 to T and obtain T’.Multiple condition coverage: Example (contd. Does T’ reveal the error? Contents Foundations of Software Testing 2E Author: Aditya P.

In class exercise: Construct a table showing the simple conditions covered by T’.) . then under what conditions? Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Multiple condition coverage: Example (contd. Mathur 543 Copyright © 2013 Dorling Kindersley (India) Pvt. Do you notice that some combinations of simple conditions remain uncovered? Now add a test to T’ to cover the uncovered combinations. Does your test reveal the error? If yes.

7 LCSAJ coverage Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 7. Mathur 544 .2.

executed one after the other. Y. Z) where X and Y are. and terminated by a jump to the next such pair. proceeds in pairs .Linear Code Sequence and Jump (LCSAJ) where the first element of the pair is a sequence of statements. respectively. An LCSAJ is represented as a triple (X. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 545 Copyright © 2013 Dorling Kindersley (India) Pvt. locations of the first and the last statements and Z is the location to which the statement at Y jumps. A Linear Code Sequence and Jump is a program unit comprised of a textual code sequence that terminates in a jump to the beginning of another code sequence and jump. Ltd Execution of sequential programs that contain at least one condition.

Y. Mathur 546 Copyright © 2013 Dorling Kindersley (India) Pvt. we say that the LCSAJ (X.Consider this program. Y. Ltd Linear Code Sequence and Jump (LCSAJ) . and then jumps to statement Z. follows through to statement Y. When control arrives at statement X. Z) is a jump and Z may be program exit. Contents Foundations of Software Testing 2E Author: Aditya P. The last statement in an LCSAJ (X. Z) is traversed or covered or exercised.

t1 covers (1. Mathur 547 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd LCSAJ coverage: Example 1 .7) and (7. exit). T covers all three LCSAJs. 6. exit) is executed. t2 covers (1.4. 8. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 548 . Ltd LCSAJ coverage: Example 2 In class exercise: Find all LCSAJs Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd LCSAJ coverage: Example 2 (contd. Mathur 549 .Copyright © 2013 Dorling Kindersley (India) Pvt.) Verify: This set covers all LCSAJs. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 550 . R) is .LCSAJ coverage: Definition Copyright © 2013 Dorling Kindersley (India) Pvt. R) is computed as T is considered adequate with respect to the LCSAJ coverage criterion if the LCSAJ coverage of T with respect to (P. Ltd The LCSAJ coverage of a test set T with respect to (P. Contents Foundations of Software Testing 2E Author: Aditya P.

8 Modified condition/decision coverage Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 7.2. Mathur 551 .Copyright © 2013 Dorling Kindersley (India) Pvt.

When a compound condition C contains n simple conditions. Ltd embedded simple conditions. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 552 .Modified Condition/Decision (MC/DC) Coverage Obtaining multiple condition coverage might become expensive when there are many Copyright © 2013 Dorling Kindersley (India) Pvt. the maximum number of tests required to cover C is 2n .

Mathur 553 . Thus. MC/DC coverage is a weaker criterion than the multiple condition coverage criterion.Compound conditions and MC/DC Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd MC/DC coverage requires that every compound condition in a program must be tested by demonstrating that each simple condition within the compound condition has an independent effect on its outcome. Contents Foundations of Software Testing 2E Author: Aditya P.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 554 . Ltd MC/DC coverage: Simple conditions Contents Foundations of Software Testing 2E Author: Aditya P.

2. Ltd 7.9 MC/DC adequate tests for compound conditions Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 555 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Generating tests for compound conditions columns as Test. Contents Foundations of Software Testing 2E Author: Aditya P. from left to right. The remaining entries are empty. Create a table with five columns and four rows. C3 and C. C2 . Mathur 556 Copyright © 2013 Dorling Kindersley (India) Pvt. Label the . The column labeled Test contains rows labeled by test case numbers t1 through t4 . An optional column labeled “Comments” may be added. C1. Ltd Let C=C1 and C2 and C3.

Generating tests for compound conditions (contd.) Copyright © 2013 Dorling Kindersley (India) Pvt. and C of the empty table. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Copy all entries in columns C1 . Mathur 557 . C2 . C3. and C from the table for simple conditions into columns C2.

Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 558 .) Copyright © 2013 Dorling Kindersley (India) Pvt.Generating tests for compound conditions (contd. Ltd Fill the first three rows in the column marked C1 with true and the last row with false.

and false. C3 . true. Mathur 559 Copyright © 2013 Dorling Kindersley (India) Pvt. respectively. We now have a table containing MC/DC adequate tests for C=(C1 AND C2 AND C3) derived from tests for C=(C1 AND C2) . Contents Foundations of Software Testing 2E Author: Aditya P.) . and C with true.Fill the last row under columns labeled C2 . Ltd MC/DC coverage: Generating tests for compound conditions (contd.

15 and 7. Ltd MC/DC coverage: Generating tests for compound conditions (contd.) The procedure illustrated above can be extended to derive tests for any compound condition using tests for a simpler compound condition (solve Exercises 7. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 560 .16).Copyright © 2013 Dorling Kindersley (India) Pvt.

10 Definition of MC/DC coverage Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.2. Ltd 7. Mathur 561 .

•  Each block in P has been covered. •  Each simple condition within a compound condition C in P has been shown to independently effect the outcome of C. Ltd A test set T for program P written to meet requirements R. Mathur 562 Copyright © 2013 Dorling Kindersley (India) Pvt. This is the MC part of the coverage we discussed. is considered adequate . •  Each simple condition in P has taken both true and false values. the following requirements are met. Contents Foundations of Software Testing 2E Author: Aditya P.MC/DC coverage: Definition with respect to the MC/DC coverage criterion if upon the execution of P on each test in T. •  Each decision in P has taken all possible outcomes.

and decision . condition. such as the one in the following statement A= (p<q) OR (x>y) are also included in the set of conditions to be covered. respectively. the MC/DC coverage criterion is a mix of four coverage criteria based on the flow of control. Thus.Analysis coverage. Ltd The first three requirements above correspond to block. it is to be noted that conditions that are not part of a decision. With regard to the second requirement. Mathur 563 Copyright © 2013 Dorling Kindersley (India) Pvt. The fourth requirement corresponds to ``MC" coverage. Contents Foundations of Software Testing 2E Author: Aditya P.

Analysis (contd. In such cases an adequate test set need only demonstrate the independent effect of any one occurrence of the coupled condition Contents Foundations of Software Testing 2E Author: Aditya P. a condition such as (A AND B) OR (C AND A) . It is not possible to keep the first occurrence of A fixed while varying the value of its second occurrence. Here the first occurrence of A is said to be coupled to its second occurrence. Mathur 564 Copyright © 2013 Dorling Kindersley (India) Pvt.) poses a problem. Ltd With regard to the fourth requirement.

denoted by MCc. ei the number of simple conditions shown to have independent affect on the outcome of Ci. ni denote the number of simple conditions Copyright © 2013 Dorling Kindersley (India) Pvt.. C2.Adequacy Let C1. The MC coverage of T for program P subject to requirements R. . Contents Foundations of Software Testing 2E Author: Aditya P. is computed as follows. CN be the conditions in P. Mathur 565 . Test set T is considered adequate with respect to the MC coverage if MCc=1 of T is 1.. and fi the number of infeasible simple conditions in Ci . Ltd in Ci .

1: Invoke fire-1 when (x<y) AND (z * z > y) AND (prev=``East").Example Copyright © 2013 Dorling Kindersley (India) Pvt. R2: The invocation described above must continue until an input Boolean variable becomes true. Mathur 566 .2: Invoke fire-2 when (x<y) AND (z * z ≤ y) OR (current=``South"). R1. R1. Ltd Consider the following requirements: R1. Contents Foundations of Software Testing 2E Author: Aditya P.3: Invoke fire-3 when none of the two conditions above is true.

) Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Example (contd.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 567 .

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Example (contd. Mathur 568 .) Contents Foundations of Software Testing 2E Author: Aditya P.

) Copyright © 2013 Dorling Kindersley (India) Pvt. executed in the given order. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 569 . block.Example (contd. and decision coverage criteria but not with respect to the condition coverage criterion. Ltd Verify that the following set T1 of four tests. is adequate with respect to statement.

Mathur 570 .) Verify that the following set T2. is adequate with respect Copyright © 2013 Dorling Kindersley (India) Pvt. obtained by adding t5 to T1.Example (contd. Note that sequencing of tests is important in this case! Contents Foundations of Software Testing 2E Author: Aditya P. Ltd to the condition coverage but not with respect to the multiple condition coverage criterion.

and t9 to T2 is adequate Copyright © 2013 Dorling Kindersley (India) Pvt.Example (contd. Note again that sequencing of tests is important in this case (especially for t1 and t7)! Contents Foundations of Software Testing 2E Author: Aditya P. obtained by adding t6.) Verify that the following set T3. t7. t8. Mathur 571 . Ltd with respect to MC/DC coverage criterion.

Copyright © 2013 Dorling Kindersley (India) Pvt.2. Mathur 572 . Ltd 7.12 Error detection and MC/DC adequacy Contents Foundations of Software Testing 2E Author: Aditya P.

. Contents Foundations of Software Testing 2E Author: Aditya P.MC/DC adequacy and error detection Missing condition: One or more simple conditions is missing from a compound condition. For example. Mathur 573 Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd We consider the following three types of errors. Incorrect Boolean operator: One or more Boolean operators is incorrect. For example. the correct condition should be (x<y AND z*x ≥ y AND d=``South") has been coded as (x<y OR z*x ≥ y). For example. Mixed: One or more simple conditions is missing and one or more Boolean operators is incorrect. the correct condition is (x<y AND done) which has been coded as (x<y OR done). the correct condition should be (x<y AND done) but the condition coded is (done).

Mathur 574 . Four Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd tests that form an MC/DC adequate set are in the following table. Verify that the following set of four tests is MC/DC adequate but does not reveal the error.Example Suppose that condition C=C1 AND C2 AND C3 has been coded as C'=C1 AND C2. Contents Foundations of Software Testing 2E Author: Aditya P.

However. (Note the emphasis on “likely.”) Contents Foundations of Software Testing 2E Author: Aditya P. The examples also show that an MC/DC adequate test will likely reveal more errors than a decision or condition-coverage adequate test.Several examples in the book show that satisfying the MC/DC adequacy criteria does not necessarily imply that errors made while coding conditions will be revealed. the examples do favor MC/DC over condition coverage. Ltd MC/DC and condition coverage . Mathur 575 Copyright © 2013 Dorling Kindersley (India) Pvt.

condition C2 is not evaluated if C1 evaluates to false. short circuit evaluation. Contents Foundations of Software Testing 2E Author: Aditya P. Thus. The outcome of the above condition does not depend on C2 when C1 is false. Ltd MC/DC and short circuit evaluation . or requires as in C. the combination C1=false and C2=true. or the combination C1=false and C2=false may be infeasible if the programming language allows.Consider C=C1 AND C2. When using short-circuit evaluation. Mathur 576 Copyright © 2013 Dorling Kindersley (India) Pvt.

MC/DC and decision dependence Copyright © 2013 Dorling Kindersley (India) Pvt. the following sequence of statements. for example. Ltd Dependence of one decision on another might also lead to an infeasible combination. Infeasible condition A<5 Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 577 . Consider.

In this case the second decision is not reachable due an error at line 3. however. It may. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 578 Copyright © 2013 Dorling Kindersley (India) Pvt. Consider the following sequence. be feasible.Infeasibility and reachability not feasible and vice versa. A decision might be reachable but . Ltd Note that infeasibility is different from reachability. In the sequence above. both decisions are reachable but the second decision is not feasible.

Mathur 579 . Ltd 7.Copyright © 2013 Dorling Kindersley (India) Pvt.2.15 Tracing test cases to requirements Contents Foundations of Software Testing 2E Author: Aditya P.

it is desirable to ask the . It has the likelihood of revealing errors and ambiguities in the requirements. Advantages of trace back: Assists us in determining whether or not the new test case is redundant. It assists with the process of documenting tests against requirements.Test trace back following question: What portions of the requirements are tested when the program under test is executed against the newly added test case? The task of relating the new test case to the requirements is known as test trace-back. Ltd When enhancing a test set to satisfy a given coverage criterion. Mathur 580 Copyright © 2013 Dorling Kindersley (India) Pvt. See example 7.27. Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd 7.3 Concepts from data flow 7.Copyright © 2013 Dorling Kindersley (India) Pvt.3. Mathur 581 .1 Definitions and uses Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 582 . Contents Foundations of Software Testing 2E Author: Aditya P. Test adequacy criteria based on the flow of data are useful in improving tests that are adequate with respect to control-flow based criteria. Let us look at an example. This is in contrast to criteria based on “flow of control” that we have examined so far. Ltd We will now examine some test adequacy criteria based on the flow of “data” in a program.Basic concepts Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 583 .Example: Test enhancement using data flow Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Here is an MC/DC adequate test set that does not reveal the error.

) line 6. Foundations of Software Testing 2E Author: Aditya P. To do so one requires a test that causes conditions at lines 5 and 8 to be true. An MC/DC adequate test does not force the execution of this path and hence the divide by zero error is not revealed. Ltd Neither of the two tests force the use of z defined on .Example (contd. at line 9. Mathur Contents 584 Copyright © 2013 Dorling Kindersley (India) Pvt.

Example (contd. Would an LCSAJ adequate test also reveal the error? Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 585 . Ltd error.) Verify that the following test set covers all def-use pairs of z and reveals the Copyright © 2013 Dorling Kindersley (India) Pvt.

defines three variables. Statement x=y+z defines variable x and uses variables y and z. x+y) uses variables x and y.Definitions and uses Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 586 . &x. Statement scanf(``%d %d". Statement printf(``Output: %d \n". y. Ltd A program written in a procedural language. contains variables. Variables are defined by assigning values to them and are used in expressions. A[10]. Declaration int x. Contents Foundations of Software Testing 2E Author: Aditya P. such as C and Java. &y) defines variables x and y.

Ltd Definitions and uses (contd.) A parameter x passed as call-by-value to a function. serves as a definition and use of x Contents Foundations of Software Testing 2E Author: Aditya P. is considered as a use of. Mathur 587 .Copyright © 2013 Dorling Kindersley (India) Pvt. A parameter x passed as call-by-reference. x. or a reference to.

The first of the above statements defines a pointer variable z the second defines y and uses z the third defines x through the pointer variable z and the last defines y and uses x accessed through the pointer variable z.Definitions and uses: Pointers Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 588 . Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Consider the following sequence of statements that use pointers.

Alternate: second statement defines A[i] and not the entire array A. Contents Foundations of Software Testing 2E Author: Aditya P. and y. Ltd Arrays are also tricky.Definitions and uses: Arrays The first statement defines variable A. x. The second statement defines A and uses i . Consider the following declaration and two statements in C: . Mathur 589 Copyright © 2013 Dorling Kindersley (India) Pvt. The choice of whether to consider the entire array A as defined or the specific element depends upon how stringent is the requirement for coverage analysis.

Ltd 7.2 C-use and p-use Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 590 .Copyright © 2013 Dorling Kindersley (India) Pvt.3.

where the ``c" in c-use stands for computational. Ltd Uses of a variable that occur within an expression as part of an assignment statement. and in subscript expressions. How many c-uses of x can you find in the following statements? Answer: 5 Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 591 . are classified as c-use.c-use Copyright © 2013 Dorling Kindersley (India) Pvt. as a parameter within a function call. in an output statement.

is considered as a p-use. Mathur 592 . Ltd statement such as an if and a while. How many p-uses of z and x can you find in the following statements? Answer: 3 (2 of z and 1 of x) Contents Foundations of Software Testing 2E Author: Aditya P. The ``p" in p-use stands for predicate.p-use The occurrence of a variable in an expression used as a condition in a branch Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. Ltd p-use: possible confusion Consider the statement: The use of A is clearly a p-use. Is the use of x in the subscript. Mathur 593 .Copyright © 2013 Dorling Kindersley (India) Pvt. a c-use or a p-use? Discuss.

their definitions flow into this block from some other block. Ltd Consider the basic block While there are two definitions of p in this block.C-uses within a basic block Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 594 . Contents Foundations of Software Testing 2E Author: Aditya P. and uses. We are concerned with global definitions. Note that y and z are global uses. The first definition of p is considered local to the block while the second definition is global. only the second definition will propagate to the next block.

Ltd 7. Mathur 595 .3.4 Data flow graph Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. edges. also known as def-use graph. An example follows. and all paths thorough the control flow graph are preserved in the data flow graph. Mathur 596 .Data flow graph Copyright © 2013 Dorling Kindersley (India) Pvt. captures the flow of definitions (also known as defs) across basic blocks in a program. It is similar to a control flow graph of a program in that the nodes. Ltd A data-flow graph of a program.

Mathur 597 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Each block becomes a node in the def-use graph (this is similar to the control flow graph). Attach defs. We use di(x) to refer to the definition of variable x at node i. c-uses and p-uses in each . ui(x) refers to the use of variable x at node i. c-use and p-use to each node in the graph. find its basic blocks. Label each edge with the condition which when true causes the edge to be taken. compute defs. Similarly.Example block. Ltd Given a program.

Ltd Example (contd. Mathur 598 .Copyright © 2013 Dorling Kindersley (India) Pvt.) Unreachable node Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 599 .5 Def-clear paths Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 7.3.

definition of z at node 2 is live at node 5 while that at node 1 is not live at node 5. is a def-clear path for x. Path 2-5 is def-clear for variable z defined at node 2 and used at node 5. Path 1-2-5 is NOT def-clear for variable z defined at node 1 and used at node 5. Contents Foundations of Software Testing 2E Author: Aditya P. without redefining x anywhere else along the path. Ltd defined and ending at a node at which x is used. Mathur 600 . Thus.Def-clear path Any path starting from a node at which variable x is Copyright © 2013 Dorling Kindersley (India) Pvt.

16 Find def-clear paths for defs and uses of x and z. Ltd P7. Which definitions are live at node 4? Foundations of Software Testing 2E Author: Aditya P.Def-clear path (another example) Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 601 Contents .

Mathur 602 .3. Ltd 7.6 def-use pairs Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 603 Copyright © 2013 Dorling Kindersley (India) Pvt. k) must also be taken during some executions. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Def of a variable at line l1 and its use at line l2 constitute a def-use pair. dcu (di(x)) denotes the set of all nodes where di(x)) is live and used. uj(x)) is covered when a def-clear path that includes nodes i to node j is executed. dpu (di(x)) denotes the set of all edges (k. l) and x is used at node k. If uj(x)) is a p-use then all edges of the kind (j. l) such that there is a def-clear path from node i to edge (k. We say that a def-use pair (di(x). l1 and l2 .Def-use pairs can be the same.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Def-use pairs (example) Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 604 .

However. Ltd Def-use pairs are items to be covered during testing. Mathur 605 Copyright © 2013 Dorling Kindersley (India) Pvt. Analysis of the data flow graph can reveal a minimal set of def-use pairs whose coverage implies coverage of all def-use pairs.Def-use pairs: Minimal set of a def-use pair implies coverage of another def-use pair. Exercise: Analyze the def-use graph shown in the previous slide and determine a minimal set of def-uses to be covered. coverage . Contents Foundations of Software Testing 2E Author: Aditya P. in some cases.

v2…vn each defined at di nodes. Mathur 606 . Contents Foundations of Software Testing 2E Author: Aditya P. Ltd CU: total number of c-uses in a program. PU: total number of p-uses.Data flow based adequacy Copyright © 2013 Dorling Kindersley (India) Pvt. Given a total of n variables v1.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 607 .4 Adequacy criteria based on data flow 7. c-use coverage Contents Foundations of Software Testing 2E Author: Aditya P.4.1. Ltd 7.

Mathur 608 .Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd C-use coverage Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 609 Copyright © 2013 Dorling Kindersley (India) Pvt.16 is executed against the following test: t1: <x=5. k... q. End) covers the c-use at node z of x defined at node q given that (k …. . Ltd C-use coverage: path traversed . count=1> Contents Foundations of Software Testing 2E Author: Aditya P. y=-1. . z.. z) is def clear with respect to x Exercise: Find the c-use coverage when program c-use of x P7.Path (Start. ..

Copyright © 2013 Dorling Kindersley (India) Pvt.4. Ltd 7.4 Adequacy criteria based on data flow 7. Mathur 610 .2 p-use coverage Contents Foundations of Software Testing 2E Author: Aditya P.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd p-use coverage Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 611 .

Mathur 612 Copyright © 2013 Dorling Kindersley (India) Pvt. count=3> Contents Foundations of Software Testing 2E Author: Aditya P.16 is executed against the following test: t2: <x=-2. y=-1.Exercise: Find the p-use coverage when program P7. Ltd p-use coverage: paths traversed .

4. all-uses coverage Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 7.4 Adequacy criteria based on data flow 7.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 613 .3.

16? Contents Foundations of Software Testing 2E Author: Aditya P. to all-uses coverage for P7. Mathur 614 .t.Copyright © 2013 Dorling Kindersley (India) Pvt. t2} adequate w. Ltd All-uses coverage Exercise: Is T={t1.r.

then some c. if this path is infeasible.and c-uses Coverage of a c. However.and p-uses that require this path to be traversed might also be infeasible. Infeasible uses are often difficult to determine without some hint from a test tool. Ltd Infeasible p. Mathur 615 .Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.or a p-use requires a path to be traversed through the program.

Ltd Infeasible c-use: Example . Mathur 616 Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Show that this c-use is infeasible.Consider the c-use at node 4 of z defined at node 5.

4.4 k-dr chain coverage Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 617 . Ltd 7.Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 618 . Some of these are more powerful in their error-detection effectiveness than the c-. (b) Data context and ordered data context coverage. Ltd There exist several other adequacy criteria based on data flows. Contents Foundations of Software Testing 2E Author: Aditya P. Examples: (a) def-use chain or k-dr chain coverage. and all-uses criteria. p-.Other data-flow based criteria Copyright © 2013 Dorling Kindersley (India) Pvt. These are alternating sequences of def-use for one or more variables.

Ltd 7.6 The “subsumes” relation Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 619 .

Mathur 620 Copyright © 2013 Dorling Kindersley (India) Pvt. what can we conclude about the adequacy of T with respect to another criterion C2? Effectiveness: Given a test set T that is adequate with respect to criterion C. Ltd Subsumes relation . what can we expect regarding its effectiveness in revealing errors? Contents Foundations of Software Testing 2E Author: Aditya P.Subsumes: Given a test set T that is adequate with respect to criterion C1.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 621 . Ltd Subsumes relationship Contents Foundations of Software Testing 2E Author: Aditya P.

multiple condition. Mathur 622 . k-dr chain. Control flow based: statement. all-uses. Contents Foundations of Software Testing 2E Author: Aditya P.Summary Copyright © 2013 Dorling Kindersley (India) Pvt. Many more exist. Ltd We have introduced the notion of test adequacy and enhancement. p-uses. Many more exist. elementary data context. decision. condition. Data flow based: c-use. data context. and LCSAJ coverage. MC/DC. Two types of adequacy criteria considered: one based on control flow and the other on data flow.

such as PaRTe. Several other commercial tools. Several test organizations believe that code coverage is useful at unit-level. Contents Foundations of Software Testing 2E Author: Aditya P.Summary (contd. are available.) during testing and displays it in a user-friendly manner. xSUDS is one such set of tools. This is a myth and needs to be shattered. and Bullseye. Incremental assessment of code coverage and enhancement of tests can allow the application of coverage-based testing to large programs. Ltd Use of any of the criteria discussed here requires a test tool that measures coverage . Cobertura. Mathur 623 Copyright © 2013 Dorling Kindersley (India) Pvt.

) . Ltd Summary (contd. Mathur 624 Copyright © 2013 Dorling Kindersley (India) Pvt.Even though coverage is not guaranteed to reveal all program errors. Tests derived using black-box approaches can almost always be enhanced using one or more of the assessment criteria discussed. it is the perhaps the most effective way to assess the amount of code that has been tested and what remains untested. Contents Foundations of Software Testing 2E Author: Aditya P.

Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 625 . 2013 Foundations of Software Testing 2E Contents Author: Aditya P. Ltd Chapter 8 Test Adequacy Measurement and Enhancement Using Mutation Updated: July 18.

§  Mutation operators §  Tools for mutation testing Contents Foundations of Software Testing 2E Author: Aditya P.Learning Objectives What is test adequacy? What is test enhancement? When to measure test Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 626 . Ltd §  adequacy and how to use it to enhance tests? §  What is program mutation? §  Competent programmer hypothesis and the coupling effect. §  Strengths and limitations of test adequacy based on program mutation.

P has been executed against each test in T and has produced correct behavior. Mathur 627 Copyright © 2013 Dorling Kindersley (India) Pvt. R).…. R2. Let R contain n requirements labeled R1. Ltd §  .What is adequacy? Consider a program P written to meet a set R of functional requirements. Also. Rn . or as: Is T adequate? Contents Foundations of Software Testing 2E Author: Aditya P. We notate such a P and R as ( P. §  Suppose now that a set T containing k tests has been constructed to test P to determine whether or not it meets all the requirements in R . §  We now ask: Is T good enough? This question can be stated differently as: Has P been tested thoroughly?.

Ltd What is program mutation? . Now suppose we do the following: Changed to P P’ What behavior do you expect from P’ against tests in T? Contents Foundations of Software Testing 2E Author: Aditya P.§  Suppose that program P has been tested against a test set T and P has not failed on any test case in T. Mathur 628 Copyright © 2013 Dorling Kindersley (India) Pvt.

Contents Foundations of Software Testing 2E Author: Aditya P. In this case we say that T is unable to distinguish P and P’. In this case we say that t distinguishes P’ from P. Or. Ltd What is program mutation? [2] . that t has killed P’. Hence P’ is considered live in the test process. §  There might be a test t in T such that P(t)≠P’(t). Mathur 629 Copyright © 2013 Dorling Kindersley (India) Pvt. §  There might be not be any test t in T such that P(t)≠P’(t).§  P’ is known as a mutant of P.

Contents Foundations of Software Testing 2E Author: Aditya P. Ltd What is program mutation? [3] . We will refer to program mutation as mutation. §  A non-equivalent and live mutant offers the tester an opportunity to generate a new test case and hence enhance T. §  If P’ is not equivalent to P but no test in T is able to distinguish it from P then T is considered inadequate. Mathur 630 Copyright © 2013 Dorling Kindersley (India) Pvt.§  If there does not exist any test case t in the input domain of P that distinguishes P from P’ then P’ is said to be equivalent to P.

§ 

Given a test set T for program P that must meet requirements R, a test adequacy
assessment procedure proceeds as follows.

§ 

Step 1: Create a set M of mutants of P. Let M={M0, M1…Mk}. Note that we have
k mutants.

§ 

Step 2: For each mutant Mi find if there exists a t in T such that Mi(t) ≠P(t). If
such a t exists then Mi is considered killed and removed from further
consideration.
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Test adequacy using mutation [1]

§ 

Step 3: At the end of Step 2 suppose that k1 ≤ k mutants have been killed and (kk1) mutants are live.
Case 1: (k-k1)=0: T is adequate with respect to mutation.
Case 2: (k-k1)>0 then we compute the mutation score (MS) as follows:
MS=k1/(k-e)
where e is the number of equivalent mutants. Note: e ≤ (k-k1).
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Test adequacy using mutation [2]

§ 

One has the opportunity to enhance a test set T after having assessed its
adequacy.

§ 

Step 1: If the mutation score (MS) is 1, then some other technique, or a different
set of mutants, needs to be used to help enhance T.

§ 

Step 2: If the mutation score (MS) is less than 1, then there exist live mutants that
are not equivalent to P. Each live mutant needs to be distinguished from P.
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Test enhancement using mutation

§ 

Step 3: Hence a new test t is designed with the objective of distinguishing at least
one of the live mutants; let us say this mutant is m.

§ 

Step 4: If t does not distinguish m then another test t’ needs to be designed to
distinguish m. Suppose that t does distinguish m.

§ 

Step 5: It is possible that t also distinguishes other live mutants.

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Test enhancement using mutation [2]

§ 

Step 6: Add t to T and re-compute the mutation score (MS).

§ 

Repeat the enhancement process from Step 1.

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Test enhancement using mutation [3]

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§ 

As with any test enhancement technique, there is no guarantee that tests derived
to distinguish live mutants will reveal a yet undiscovered error in P. Nevertheless,
empirical studies have found to be the most powerful of all formal test
enhancement techniques.

§ 

The next simple example illustrates how test enhancement using mutation detects
errors.

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Error detection using mutation

§ 

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Error detection using mutation [2]

Consider the following function foo that is required to return the sum of two
integers x and y. Clearly foo is incorrect.
int foo(int x, y){
return (x-y);

This should be return (x+y)

}

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§ 

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Error detection using mutation [3]

Now suppose that foo has been tested using a test set T that contains two tests:
T={ t1: <x=1, y=0>, t2: <x=-1, y=0>}

§ 

First note that foo behaves perfectly fine on each test in, i.e. foo returns the
expected value for each test case in T. Also, T is adequate with respect to all
control and data flow based test adequacy criteria.
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Error detection using mutation [4]

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Let us evaluate the adequacy of T using mutation. Suppose that the following
three mutants are generated from foo.
M1: int foo(int x, y){

§ 

M2: int foo(int x, y){

M3: int foo(int x, y){

return (x+y);

return (x-0);

return (0+y);

}

}

}

Note that M1 is obtained by replacing the - operator by a + operator, M2 by
replacing y by 0, and M3 by replacing x by 0.
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Error detection using mutation [4]
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Next we execute each mutant against tests in T until the mutant is distinguished
or we have exhausted all tests. Here is what we get.
T={ t1: <x=1, y=0>, t2: <x=-1, y=0>}
Test (t)

foo(t)

M1(t)

M2(t)

M3(t)

t1

1

1

1

0

t2

-1

-1

-1

0

Live

Live

Killed
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Error detection using mutation [5]
After executing all three mutants we find that two are live and one is
distinguished. Computation of mutation score requires us to determine of any of
the live mutants is equivalent.

In class exercise: Determine whether or not the two live mutants are equivalent
to foo and compute the mutation score of T.

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Error detection using mutation [6]

M1: int foo(int x, y){

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Let us examine the following two live mutants.
M2: int foo(int x, y){

return (x+y);

return (x-0);

}

}

Let us focus on M1. A test that distinguishes M1 from foo must
satisfy the following condition:
x-y≠x+y implies y ≠0.
Hence we get t3: <x=1, y=1>
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Error detection using mutation [7]

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Executing foo on t3 gives us foo(t3)=0. However, according to the requirements
we must get foo(t3)=2. Thus, t3 distinguishes M1 from foo and also reveals
the error.
M1: int foo(int x, y){

M2: int foo(int x, y){

return (x+y);

return (x-0);

}

}

In class exercise: (a) Will any test that distinguishes also reveal the error? (b)
Will any test that distinguishes M2 reveal the error?
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Guaranteed error detection

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Sometimes there exists a mutant P’ of program P such that any test t that
distinguishes P’ from P also causes P to fail. More formally:
Let P’ be a mutant of P and t a test in the input domain of P. We say
that P’ is an error revealing mutant if the following condition holds
for any t.
P’(t) ≠P(t) and P(t) ≠R(t), where R(t) is the expected response of P
based on its requirements.
Is M1 in the previous example an error revealing mutant? What about
M2?
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Contents Foundations of Software Testing 2E Author: Aditya P. Ltd A test case t that distinguishes a mutant m from its parent program P program must satisfy the following three conditions: Condition 1: Reachability: t must cause m to follow a path that arrives at the mutated statement in m. Condition 2: Infection: If Sin is the state of the mutant upon arrival at the mutant statement and Sout the state soon after the execution of the mutated statement. then Sin≠ Sout. Mathur 645 .Distinguishing a mutant Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 646 . Ltd Distinguishing a mutant [2] Condition 3: Propagation: If difference between Sin and Sout must propagate to the output of m such that the output of m is different from that of P.Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Exercise: Show that in the previous example both t1 and t2 satisfy the above three conditions for M3.

Equivalent mutants The problem of deciding whether or not a mutant is equivalent to its Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. empirical studies have shown that one can expect about 5% of the generated mutants to the equivalent to the parent program. •  Identifying equivalent mutants is generally a manual and often time consuming--as well as frustrating--process. Ltd •  parent program is undecidable. However. Mathur 647 . •  The number of equivalent mutants can vary from one program to another. Hence there is no way to fully automate the detection of equivalent mutants.

and practitioners that any “coverage” based technique. int p=0.A misconception There is a widespread misconception amongst testing educators. return(x+p*y) } Foundations of Software Testing 2E Author: Aditya P. Consider the following programs. researchers. Mathur Contents 648 . y){ int p=0. y){ Correct program int foo(int x. return(x+p*y) else } p=p-1. will not be Program under test int foo(int x. including mutation. Ltd able to detect errors due to missing path. p=p+1. if(x<y) Missing else Copyright © 2013 Dorling Kindersley (India) Pvt. if(x<y) p=p+1.

in other words M is an error revealing mutant. Mathur 649 . (b) Suppose T is decision adequate for foo. Is T guaranteed to reveal the error? Contents Foundations of Software Testing 2E Author: Aditya P. Is T guaranteed to reveal the error? (c) Suppose T is def-use adequate for foo.A misconception [2] Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd (a)  Suggest at least one mutant M of foo that is guaranteed to reveal the error.

•  Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Mutant operators A mutant operator O is a function that maps the program under test to a set of k (zero or more) mutants of P. Mathur 650 . M1 O(P) M2 …. Mk Contents Foundations of Software Testing 2E Author: Aditya P.

•  For example. An “relational operator replacement” mutant operator replaces relational operator wirh another relational operator. Ltd Mutant operators [2] A mutant operator creates mutants by making simple changes in the program under test.•  Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 651 . the “variable replacement” mutant operator replaces a variable name by another variable declared in the program.

Ltd Mutant operator Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 652 . z=0*y+1.Mutant operators: Examples In P In mutant Variable replacement z=x*y+1. z=(x+1)*y+1. z=x+y-1. x=x*y+1. z=x*(y+1)+1. z=x*x+1. Copyright © 2013 Dorling Kindersley (India) Pvt. Relational operator replacement if (x<y) if(x>y) if(x<=y) Off-by-1 z=x*y+1. z=0. z=x*y-1. Arithmetic operator replacement z=x*y+1. Replacement by 0 z=x*y+1.

[See coupling effect later. is x=z+y. Mathur 653 . A mutant obtained by making two changes is a second order mutant. •  In practice only first order mutants are generated for two reasons: (a) to lower the cost of testing and (b) most higher order mutants are killed by tests adequate with respect to first order mutants. For example.Mutants: First order and higher order •  A mutant obtained by making exactly “one change” is considered first •  Copyright © 2013 Dorling Kindersley (India) Pvt. where the variable replacement operator has been applied twice. Ltd order. a second order mutant of z=x+y.] Contents Foundations of Software Testing 2E Author: Aditya P. Similarly higher order mutants can be defined.

As we shall see later. mutant operators model simple mistakes. Ltd •  programmer •  Several error studies have revealed that programmers--novice and experts--make simple mistakes. the “coupling effect” explains why only simple mistakes are modeled. •  While programmers make “complex mistakes” too. Contents Foundations of Software Testing 2E Author: Aditya P. For example. instead of using x<y+1 one might use x<y. Mathur 654 .Mutant operators: basis A mutant operator models a simple mistake that could be made by a Copyright © 2013 Dorling Kindersley (India) Pvt.

evident that two groups might arrive at a different set of mutation operators for the same programming language. How should we judge whether or not that a set of mutation operators is “good enough?” •  Informal definition: •  Let S1 and S2 denote two sets of mutation operators for language L. Ltd is Thus.Mutant operators: Goodness •  The design of mutation operators is based on guidelines and experience. Based on the effectiveness criteria. Contents Foundations of Software Testing 2E Author: Aditya P. we say that S1 is superior to S2 if mutants generated using S1 guarantee a larger number of errors detected over a set of erroneous programs. Mathur 655 . It Copyright © 2013 Dorling Kindersley (India) Pvt.

We say that one is using “constrained” or “selective” mutation when one uses this small set of mutation operators. Mathur 656 . Ltd •  rather than the complete set of operators.Mutant operators: Goodness [2] Generally one uses a small set of highly effective mutation operators Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. •  Experiments have revealed relatively small sets of mutation operators for C and Fortran.

Ada. •  Languages differ in their syntax thereby offering opportunities for making Copyright © 2013 Dorling Kindersley (India) Pvt. C. and Java. This leads to differences in the set of mutant operators for two languages. •  Mutant operators have been developed for languages such as Fortran. Mathur 657 . Lisp.•  For each programming language one develops a set of mutant operators. [See the text for a comparison of mutant operators across several languages. Ltd Mutant operators: Language dependence mistakes that duffer between two languages.] Contents Foundations of Software Testing 2E Author: Aditya P.

•  Copyright © 2013 Dorling Kindersley (India) Pvt. given an account number. a programmer writes a program P that is in the general neighborhood of the set of correct programs. Contents Foundations of Software Testing 2E Author: Aditya P. while such a situation is unlikely to arise. Of course. Ltd Competent programmer hypothesis (CPH) CPH states that given a problem statement. a programmer is unlikely to write a program that deposits money into an account. Mathur 658 . a devious programmer might certainly write such a program. •  An extreme interpretation of CPH is that when asked to write a program to find the account balance.

Competent programmer hypothesis (CPH) [2] A more reasonable interpretation of the CPH is that the program written Copyright © 2013 Dorling Kindersley (India) Pvt. and if not. Contents Foundations of Software Testing 2E Author: Aditya P. Mistakes will lead to a program that can be corrected by applying one or more first order mutations. at least one sorting algorithm. •  It is Thus. will find one prior to writing the program. and makes use of. •  The CPH assumes that the programmer knows of an algorithm to solve the problem at hand. Mathur 659 . a competent programs knows of. Ltd •  to satisfy a set of requirements will be a few mutants away from a correct program. safe to assume that when asked to write a program to sort a list of numbers.

Lipton.Coupling effect The coupling effect has been paraphrased by DeMillo.. and Copyright © 2013 Dorling Kindersley (India) Pvt. again in the words of DeMillo. Mathur 660 .seemingly simple tests can be quite sensitive via the coupling effect. Lipton and Sayward ``. Ltd •  Sayward as follows: “Test data that distinguishes all programs differing from a correct one by only simple errors is so sensitive that it also implicitly distinguishes more complex errors” •  Stated alternately." Contents Foundations of Software Testing 2E Author: Aditya P.

•  It is during an analysis of the behavior of the mutant in relation to that of its parent that one discovers complex faults. This perturbation takes place at the point of mutation and has the potential of infecting the entire state of the program. Mathur 661 . a non-equivalent mutant forces a slight perturbation in Copyright © 2013 Dorling Kindersley (India) Pvt.Coupling effect [2] For some input. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd •  the state space of the program under test.

Mathur 662 . Two such tools are Proteum for C from Professor Josè Maldonado and muJava for Java from Professor Jeff Offutt. We are not aware of any commercially available tool for mutation testing. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd •  assessment must be done with the help of a tool.Tools for mutation testing As with any other type of test adequacy assessment. See the textbook for a more complete listing of mutation tools. •  There are few mutation testing tools available freely. mutation based Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd •  comparison against that of mutants. §  Management of test set T. Contents Foundations of Software Testing 2E Author: Aditya P. §  A selectable palette of mutation operators. Mathur 663 . §  Generation of mutants.Tools for mutation testing: Features A typical tool for mutation testing offers the following features. §  Execution of the program under test against T and saving the output for Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Tools for mutation testing: Features [2] Mutant execution and computation of mutation score using user identified equivalent mutants. Mathur 664 . allows the application of a subset of mutation operators to a portion of the program under test. §  Incremental mutation testing: i. an advanced mutation tool for Fortran also provided automatic test generation using DeMillo and Offutt’s method.e. Contents Foundations of Software Testing 2E Author: Aditya P.§  Copyright © 2013 Dorling Kindersley (India) Pvt. §  Mothra.

Contents Foundations of Software Testing 2E Author: Aditya P. Ltd §  relatively small units. given a good tool. a class in Java or a small collection of functions in C. one can use mutation to assess adequacy of system tests. §  However. e. §  The following procedure is recommended to assess the adequacy of system tests.g.Mutation and system testing Adequacy assessment using mutation is often recommended only for Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 665 .

This selection is best guided by the operators defined by Eric Wong or Jeff Offutt.] §  Step 3: Apply the operators to the selected unit. Ltd §  secure functioning of the application. Mathur 666 . §  Step 2: Select a small set of mutation operators. [See the text for details. Repeat the following steps for each unit in U. Contents Foundations of Software Testing 2E Author: Aditya P.Mutation and system testing [2] Step 1: Identify a set U of application units that are critical to the safe and Copyright © 2013 Dorling Kindersley (India) Pvt.

If necessary. and perhaps enhanced it. enhance T. Mathur 667 Copyright © 2013 Dorling Kindersley (India) Pvt. use of a limited set of highly effective mutation operators). §  We have now assessed T. §  Step 5: Repeat Steps 3 and 4 for the next unit until all units have been considered.. Note the use of incremental testing and constrained mutation (i.§  Step 4: Assess the adequacy of T using the mutants so generated. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Mutation and system testing [3] .e.

is recommended for applications that must meet stringent availability. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 668 .§  Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Mutation and system testing [4] Application of mutation. safety requirements. security. and other advanced test assessment and enhancement techniques.

Contents Foundations of Software Testing 2E Author: Aditya P. §  Mutation. §  Identification of equivalent mutants is an undecidable problem--similar the identification of infeasible paths in control or data flow based test assessment. Mathur 669 . must be applied incrementally and with assistance from good tools. Ltd §  enhancement of tests.Summary Mutation testing is the most powerful technique for the assessment and Copyright © 2013 Dorling Kindersley (India) Pvt. as with any other test assessment technique.

Summary [2] While mutation testing is often recommended for unit testing. and safe systems. Ltd §  carefully and incrementally. Contents Foundations of Software Testing 2E Author: Aditya P. it can be used for the assessment of system and other types of tests applied to an entire application. secure. Mathur 670 . §  Mutation is a highly recommended technique for use in the assurance of quality of highly available. when done Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Chapter 9 .Test Selection. and Prioritization for Regression Testing Updated: July 17. Minimization. Mathur 671 Copyright © 2013 Dorling Kindersley (India) Pvt. 2013 Foundations of Software Testing 2E Contents Author: Aditya P.

Ltd Learning Objectives What is regression testing? How to select a subset of tests for regression testing? How to select or minimize a set of tests for regression testing? How to prioritize a set of tests for regression testing? Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 672 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd 9. Mathur 673 . What is regression testing? Contents Foundations of Software Testing 2E Author: Aditya P.1.Copyright © 2013 Dorling Kindersley (India) Pvt.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Regression testing Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 674 .

] What are the strengths and shortcomings of this approach? Contents Foundations of Software Testing 2E Author: Aditya P. [This is the TEST-ALL approach.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd What tests to use? Idea 1: All valid tests from the previous version and new tests created to test any added functionality. Mathur 675 .

But what if you have limited resources to run tests and have to meet a deadline? What if running all tests as well as meeting the deadline is simply not possible? Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 676 . Ltd The test-all approach The test-all approach is best when you want to be certain that the the new version works on all tests developed for the previous version and any new tests.Copyright © 2013 Dorling Kindersley (India) Pvt.

We will discuss two of these known as test minimization and test prioritization. Mathur 677 . Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Idea 2: Select a subset Tr of the original test set T such that successful execution of the modified code P’ against Tr implies that all the functionality carried over from the original code P to P‘is intact. Finding Tr can be done using several methods.Test selection Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 678 . Regression test selection: The problem Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 9.Copyright © 2013 Dorling Kindersley (India) Pvt.3.

Ltd Regression Test Selection problem . Mathur 679 Copyright © 2013 Dorling Kindersley (India) Pvt. Such tests are not included in the regression subset Tr. The task of identifying such obsolete tests is known as test revalidation.Given test set T. our goal is to determine Tr such that successful execution of P’ against Tr implies that modified or newly added code in P’ has not broken the code carried over from P. Note that some tests might become obsolete when P is modified to P’. Contents Foundations of Software Testing 2E Author: Aditya P.

Contents Foundations of Software Testing 2E Author: Aditya P.Regression Test Process Test selection Test setup Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 680 . let us look at an overall regression test process. Ltd Now that we know what the regression test selection problem is. Test sequencing Test execution Error correction Output analysis In this chapter we will learn how to select tests for regression testing.

Copyright © 2013 Dorling Kindersley (India) Pvt. Test selection using execution trace Contents Foundations of Software Testing 2E Author: Aditya P.5. Ltd 9. Mathur 681 .

Ltd Step 1: Given P and test set T. This step can be executed while constructing the CFGs of P and P’. Mathur 682 . find the execution trace of P for each test in T. Step 4: Traverse the CFGs and determine the a subset of T appropriate for regression testing of P’. Contents Foundations of Software Testing 2E Author: Aditya P.Overview of a test selection method Copyright © 2013 Dorling Kindersley (India) Pvt. Step 2: Extract test vectors from the execution traces for each node in the CFG of P Step 3: Construct syntax trees for each node in the CFGs of P and P’.

2. Let Tno be the set of all valid tests for P’. Ltd Let G=(N. E) denote the CFG of program P. N is a finite set of nodes and E a finite set of edges connecting the nodes. Contents Foundations of Software Testing 2E Author: Aditya P. It is obtained by discarding all tests that have become obsolete for some reason. Thus. Mathur 683 . Tno contains only tests valid for P’.Execution Trace [1] Copyright © 2013 Dorling Kindersley (India) Pvt. and so on and that Start and End are two special nodes as discussed in Chapter 1. Suppose that nodes in N are numbered 1.

consider the following program. Mathur 684 Copyright © 2013 Dorling Kindersley (India) Pvt.Execution Trace [2] G traversed when P is executed against t. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd An execution trace of program P for some test t in Tno is the sequence of nodes in . As an example.

Ltd Here is a CFG for our example program. Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 685 .Execution Trace [3] Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Now consider the following set of three tests and the corresponding trace. Contents Foundations of Software Testing 2E Author: Aditya P.Execution Trace [4] Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 686 .

Test vector Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd A test vector for node n. is the set of tests that traverse node n in the CFG. For program P we obtain the following test vectors. denoted by test(n). Mathur 687 . Contents Foundations of Software Testing 2E Author: Aditya P.

Syntax trees A syntax tree is constructed for each node of CFG(P) and CFG(P’). Mathur 688 . Contents Foundations of Software Testing 2E Author: Aditya P. Recall that Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd each node represents a basic block. Here sample syntax trees for the example program.

the following three steps are executed to obtain a subset T’ of T for regression testing of P’. Mathur 689 .Test selection [1] Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Given the execution traces and the CFGs for P and P’.

all tests in test (N) are added to T’. If two two nodes N in CFG(P) and N’ in CFG( P’) are found to be syntactically different.Test selection [2] Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd The basic idea underlying the SelectTests procedure is to traverse the two CFGs from their respective START nodes using a recursive descent procedure. The descent proceeds in parallel and the corresponding nodes are compared. Mathur 690 .

Mathur 691 .Test selection example Copyright © 2013 Dorling Kindersley (India) Pvt. t3}. Ltd Suppose that function g1 in P is modified as follows. Try the SelectTests algorithm and check if you get T’={t1. Contents Foundations of Software Testing 2E Author: Aditya P.

Ltd Issues with SelectTests Think: What tests will be selected when only.Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 692 . one declaration is modified? Can you think of a way to select only tests that correspond to variables in the modified declaration? Contents Foundations of Software Testing 2E Author: Aditya P. say.

6. Ltd 9.Copyright © 2013 Dorling Kindersley (India) Pvt. Test selection using dynamic slicing Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 693 .

Let trace(t) be the execution trace of P when executed against test t. L). Mathur 694 Copyright © 2013 Dorling Kindersley (India) Pvt. v. Ltd Dynamic slice . The dynamic slice of P with respect to t and v. denoted as DS(t. Question: What is the dynamic slice of P with respect to v and t if L is not in trace(t)? Contents Foundations of Software Testing 2E Author: Aditya P.Let L be a location in program P and v a variable used at L. is the set of statements in P that (a) lie in trace(t) and (b) effected the value of v at L.

Control and data dependence edges are added from n to the existing nodes in G. Step 3: For each successive statement in trace(t) a new node n is added to G. [Recall from Chapter 2 the definitions of control and data dependence edges. Ltd The DDG is needed to obtain a dynamic slice. There are no edges among these nodes. Step 1: Initialize G with a node for each declaration.] Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 695 .Dynamic dependence graph (DDG) Copyright © 2013 Dorling Kindersley (India) Pvt. Here is how a DDG G is constructed. Step 2: Add to G the first node in trace(t).

7. 6. 7. and 3 respectively. Ltd Construction of a DDG: Example [1] . 2. 2. 5. 1 Contents Foundations of Software Testing 2E Author: Aditya P. 2. 0 and 5. 3. Add a node to G corresponding to statement 1. and for these values f1(x) is 0. 6. 8} Ignore declarations for simplicity. trace(t)={1. 3.Let t: <x=2. 2. Mathur 696 Copyright © 2013 Dorling Kindersley (India) Pvt. y=4> Assume successive values of x to be 2. 4.

7. 2. 4. 2. 6. Also add a data dependence edge from node 3 to node 1 as statement 3 is data dependent on statement 1 and a control edge from node 3 to 2. Ltd trace(t)={1. 3. 7. 5. Mathur 697 Copyright © 2013 Dorling Kindersley (India) Pvt.Construction of a DDG: Example [2] Add another node corresponding to statement 2 in trace(t). 2. 1 2 3 Contents Foundations of Software Testing 2E Author: Aditya P. Also add a data dependence edge from 2 to 1 as statement 2 is data dependent on statement 1. 3. 8} . 6. 1 should be… 3 if(f1(x)==0) 2 Add yet another node corresponding to statement 3 in trace(t).

Construction of a DDG: Example [3] Copyright © 2013 Dorling Kindersley (India) Pvt. 8} Continuing this way we obtain the following DDG for program P and trace(t). 4. 5. 3. Mathur 698 . Ltd trace(t)={1. 2. 6. 2. 6. 3. 7. 2. should be… 3 if(f1(x)==0) Contents Foundations of Software Testing 2E Author: Aditya P. 7.

If no such node exists then the dynamic slice is empty. v. DS(t. Ltd Step 1: Execute P against test t and obtain trace(t). Step 2: Construct the dynamic dependence graph G from P and trace(t). including n.Obtaining dynamic slice (DS) Copyright © 2013 Dorling Kindersley (India) Pvt. n) of all nodes reachable from n. Step 3: Identify in G node n labeled L that contains the last assignment to v. n) is the dynamic slice of P with respect to v at location L and test t. Contents Foundations of Software Testing 2E Author: Aditya P. Step 4: Find in G the set DS(t. other wise execute Step 4. v. Mathur 699 .

8}. This occurs at line 7 as marked. Contents Foundations of Software Testing 2E Author: Aditya P. 7. 3. This gives us the following dynamic slice: {1. First identify the last definition of w in the DDG. We already have the DDG of P for t. Mathur 700 . 6. 5. 2. Traverse the DDG backwards from node 7 and collect all nodes reachable from 7.Obtaining dynamic slice: Example Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Suppose we want to compute the dynamic slice of P with respect to variable w at line 8 and test t shown earlier.

Mathur 701 . .. Which tests from T should be used to obtain a regression test T’ for P’? Find DS(t) for P. Let n1. P’ is the modified program. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Let T be the test set used to test P.nk be the nodes in the CFG of P modified to obtain P’. If any of the modified nodes is in DS(t) then add t to T’. n2.Test selection using dynamic slice Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd In class exercise Suppose line 4 in the example program P shown earlier is modified to obtain P’. Mathur 702 .Copyright © 2013 Dorling Kindersley (India) Pvt. (a)  Should t be included in T’? (b)  Will t be included in T’ if we were to use the execution slice instead of the dynamic slice to make our decision? Contents Foundations of Software Testing 2E Author: Aditya P.

Teasers [1] Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 703 . the variable of interest.] Contents Foundations of Software Testing 2E Author: Aditya P. especially for large programs. How can one reduce the size of the DDG and still obtain the correct DS? The DS contains all statements in trace(t) that had an effect on w. However there could be a statement s in trace(t) that did not have an effect but could affect w if changed. Ltd You may have noticed that a DDG could be huge. How can such statements be identified? [Hint: Read about potential dependence.

how would you select the variable for which to obtain the dynamic slice? Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Suppose statement s in P is deleted to obtain P’? How would you find the tests that should be included in the regression test suite? Suppose statement s is added to P to obtain P’? How would you find the tests that should be included in the regression test suite? In our example we used variable w to compute the dynamic slice. While selecting regression tests.Teasers [2] Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 704 .

Ltd 9.8 Test selection using test minimization Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 705 .Copyright © 2013 Dorling Kindersley (India) Pvt.

Ltd Test minimization [1] Test minimization is yet another method for selecting tests for regression testing. Mathur 706 . To illustrate test minimization.Copyright © 2013 Dorling Kindersley (India) Pvt. main and f. Now suppose that P is tested using test cases t1 and t2. Contents Foundations of Software Testing 2E Author: Aditya P. During testing it was observed that t1 causes the execution of main but not of f and t2 does cause the execution of both main and f. suppose that P contains two functions.

Test minimization [2] Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 707 . the regression test suite consists of only t2. Which of the two test cases should be included in the regression test suite? Obviously there is no need to execute P’ against t1 as it does not cause the execution of f. t2} to a obtain the regression test suite {t2}. Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Now suppose that P’ is obtained from P by making some modification to f. In this example we have used function coverage to minimize a test suite {t1. Thus.

def-use chains. decisions.Copyright © 2013 Dorling Kindersley (India) Pvt. for example. Contents Foundations of Software Testing 2E Author: Aditya P. One uses the following procedure to minimize a test set based on a selected testable entity. Testable entities include. Ltd Test minimization [3] Test minimization is based on the coverage of testable entities in P. program statements. Mathur 708 . and mutants.

.ek be the k testable entities of type TE present in P.A procedure for test minimization Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P. Let e1. Ltd Step 1: Identify the type of testable entity to be used for test minimization.. Step 2: Execute P against all elements of test set T and for each test t in T determine which of the k testable entities is covered. Step 3: Find a minimal subset T’of T such that each testable entity is covered by at least one test in T’. e2. In our previous example TE is function. Mathur 709 .

3 Step3: A minimal test set for regression testing is {t1. Mathur 710 . f1: 1. Ltd blocks for a sample program are shown here for both main and function f1. f1: 1. Contents Foundations of Software Testing 2E Author: Aditya P.Test minimization: Example Step 1: Let the basic block be the testable entity of interest. 3. 2. 3 t1: main: 1. Step 2: Suppose the coverage of the basic blocks when executed against three tests is as follows: t1: main: 1. 2. f1: 1. 3. 3. 3 t2: main: 1. t3}. The basic Copyright © 2013 Dorling Kindersley (India) Pvt.

Is the minimal test set unique? Why or why not? Is test minimization NP hard? How is the traditional set cover problem in mathematics related to the test minimization problem? What criteria should be used to decide the kind of testable entity to be used for minimization? Contents Foundations of Software Testing 2E Author: Aditya P. Ltd Test minimization: Teasers . Mathur 711 Copyright © 2013 Dorling Kindersley (India) Pvt.

Mathur 712 .9 Test selection using test prioritization Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd 9.

Test prioritization Copyright © 2013 Dorling Kindersley (India) Pvt. There is a small chance that if P’ were executed against a discarded test case it would reveal an error in the modification made. In such cases one uses test prioritization. Tests are prioritized based on some criteria. it might not be wise to discard test cases as in test minimization. the one with the next highest coverage m the next higher priority and so on. tests that cover the maximum number of a selected testable entity could be given the highest priority. Contents Foundations of Software Testing 2E Author: Aditya P. For example. When very high quality software is desired. Mathur 713 . Ltd Note that test minimization will likely discard test cases.

In our previous example TE is function.ek be the k testable entities of type TE present in P. For each t in T compute the number of distinct testable entities covered. Contents Foundations of Software Testing 2E Author: Aditya P.. Ltd A procedure for test prioritization Step 1: Identify the type of testable entity to be used for test minimization.Copyright © 2013 Dorling Kindersley (India) Pvt. . Step 3: Arrange the tests in T in the order of their respective coverage. Let e1. Step 2: Execute P against all elements of test set T and for each test t in T. Test with the maximum coverage gets the highest priority and so on. Mathur 714 . e2.

Mathur 715 . In any case test are discarded only after careful consideration that does not depend only on the coverage criteria used.Copyright © 2013 Dorling Kindersley (India) Pvt. The choice is guided by several factors such as the resources available for regression testing and the desired product quality. Ltd Using test prioritization Once the tests are prioritized one has the option of using all tests for regression testing or a subset. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 716 . Tools Contents Foundations of Software Testing 2E Author: Aditya P. Ltd 9.10.Copyright © 2013 Dorling Kindersley (India) Pvt.

Many commercial tools for regression testing simply run the tests automatically. xSuds from Telcordia Technologies can be used for C programs to minimize and prioritize tests. Contents Foundations of Software Testing 2E Author: Aditya P. they do not use any of the algorithms described here for test selection. Such tool are especially useful when all tests are to be rerun.Tools for regression testing Copyright © 2013 Dorling Kindersley (India) Pvt. Mathur 717 . Instead they rely on the tester for test selection. Ltd Methods for test selection described here require the use of an automated tool for all but trivial programs.

Contents Foundations of Software Testing 2E Author: Aditya P. execution of all tests might not be feasible. In such situations one can make use of sophisticated technique for selecting a subset of all tests and hence reduce the time for regression testing. Ltd Summary [1] Regression testing is an essential phase of software product development. Mathur 718 .Copyright © 2013 Dorling Kindersley (India) Pvt. In a situation where test resources are limited and deadlines are to be met.

Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 719 .Summary [2] Copyright © 2013 Dorling Kindersley (India) Pvt. Select tests using dynamic slices [based on execution traces and dynamic slices]. Select tests using code coverage [based on the coverage of testable entities]. Ltd Test selection for regression testing can be done using any of the following methods: Select only the modification traversing tests [based on CFGs]. Select tests using execution slices [based on execution traces].

Use of any of the techniques mentioned here requires access to sophisticated tools. Most commercially available tools are best in situations where test selection is done manually and do not use the techniques described in this chapter. Mathur 720 . Ltd Summary [3] Select tests using a combination of code coverage and human judgment [based on amount of the coverage of testable entities].Copyright © 2013 Dorling Kindersley (India) Pvt. Contents Foundations of Software Testing 2E Author: Aditya P.

Mathur 721 . Ltd Chapter 10 Unit Testing [Under Construction] Contents Foundations of Software Testing 2E Author: Aditya P.Copyright © 2013 Dorling Kindersley (India) Pvt.

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd Chapter 11 Integration Testing [Under Construction] Contents Foundations of Software Testing 2E Author: Aditya P. Mathur 722 .