Ch2 theory

Software Testing and Quality Assurance
Theory and Practice
Chapter 2
Theory of Program Testing

1

Outline of the Chapter
• Basic Concepts in Testing Theory
• Theory of Goodenough and Gerhart
• Theory of Weyuker and Ostrand
• Theory of Gourlay
• Adequacy of Testing
• Limitations of Testing
• Summary

2

Basic Concepts in Testing Theory
• Testing theory puts emphasis on
– Detecting defects through program execution
– Designing test cases from different sources: requirement specification, source
code, and input and output domains of programs
– Selecting a subset of tests cases from the entire input domain
– Effectiveness of test selection strategies
– Test oracles used during testing
– Prioritizing the execution of test cases
– Adequacy analysis of test cases

3

Theory of Goodenough and Gerhart
• Fundamental Concepts
– Let P be a program, and D be its input domain. Let T ⊆ D. P(d) is the result of
executing P with input d.

Figure 2.1: Executing a program with a subset of the input domain.
– OK(d): Represents the acceptability of P(d). OK(d) = true iff P(d) is
acceptable.
– SUCCESSFUL(T): T is a successful test iff ∀t ∈ T, OK(t).
– Ideal Test: T is an ideal test if OK(t), ∀t ∈ T => OK(d), ∀d ∈ D.

4

• Fundamental Concepts (Contd.)
– Reliable Criterion: A test selection criterion C is reliable iff either every test
selected by C is successful, or no test selected is successful.
– Valid Criterion: A test selection criterion C is valid iff whenever P is incorrect,
C selects at least one test set T which is not successful for P.
– Let C denote a set of test predicates. If d ∈ D satisfies test predicate c ∈ C,
then c(d) is said to be true.
– COMPLETE(T, C) ≡ (∀c ∈ C)(∃t ∈ T) c(t) ∧ (∀t ∈ T)(∃c ∈ C) c(t)

• Fundamental Theorem
– (∃T ⊆ D) (COMPLETE(T,C) ∧ RELIABLE(C) ∧ VALID(C) ∧ SUCCESSFUL(T))
=> (∀d ∈ D) OK(d)

5

• Program faults occur due to our
– inadequate understanding of all conditions that a program must deal with.
– failure to realize that certain combinations of conditions require special care.
• Kinds of program faults
– Logic fault
• Requirement fault
• Design fault
• Construction fault
– Performance fault
– Missing control-flow paths
– Inappropriate path selection
– Inappropriate or missing action
• Test predicate: It is a description of conditions and combinations of
conditions relevant to correct operation of the program.

6

• Conditions for Reliability of a set of test predicates C
– Every branching condition must be represented by a condition in C.
– Every potential termination condition must be represented in C.
– Every condition relevant to the correct operation of the program must be
represented in C.

• Drawbacks of the Theory
– Difficulty in assessing the reliability and validity of a criterion.
– The concepts of reliability and validity are defined w.r.t. to a program. The
goodness of a test should be independent of individual programs.
– Neither reliability nor validity is preserved throughout the debugging process.

7

Theory of Weyuker and Ostrand
• d ∈ D, the input domain of program P and T ⊆ D.
• OK(P, d) = true iff P(d) is acceptable.
• SUCC(P, T): T is a successful test for P iff forall ∀t ∈ T, OK(P, t).
• Uniformly valid criterion: Criterion C is uniformly valid iff
– (∀P) [ (∃d ∈ D)(¬OK(P,d)) => (∃T ⊆ D) (C(T) ∧ ¬SUCC(P, T)) ].
• Uniformly reliable criterion: Criterion C is uniformly reliable iff
(∀P) (∀T1, ∀T2 ⊆ D) [ (C(T1) ∧ C(T2)) => (SUCC(P, T1) <==> SUCC(P,T2)) ].
• Uniformly Ideal Test Selection
– A uniformly ideal test selection criterion for a given specification is both
uniformly valid and uniformly reliable.
• A subdomain S is a subset of D.
– Criterion C is revealing for a subdomain S if whenever S contains an input
which is processed incorrectly, then every test set which satisfies C is
unsuccessful.
• REVEALING(C, S) iff
(∃d ∈ S) (¬OK(d)) => (∀T ⊆ S)(C(T) => ¬SUCC(T)) .

8

Theory of Gourlay
• The theory establishes a relationship between three sets of entities
– specifications, programs and tests.
• Notation
– P: The set of all programs (p ∈ P ⊆ P)
– S: The set of all specifications (s ∈ S ⊆ S)
– T: The set of all tests (t ∈ T ⊆ T)
– “p ok(t) s” means the result of testing p with t is judged to be acceptable by s.
– “p ok(T) s” means “p ok(t) s,” ∀t ∈ T.
– “p corr s” means p is correct w.r.t. s.
• A testing system is a collection < P, S, T, corr, ok>, where corr
⊆ P x S and ok ⊆ T x P x S, and ∀p∀s∀t(p corr s => p ok(t) s).
• A test method is a function M: P x S →T
– Program dependent: T = M(P)
– Specification dependent: T = M(S)
– Expectation dependent

9

Theory of Gourlay
• Power of test methods: Let M and N be two test methods.
– For M to be at least as good as N, we want the following to occur:
• Whenever N finds an error, so does M.
• (FM and FN are sets of faults discovered by test sets produced by test
methods M and N, respectively.)
• (TM and TN are test sets produced by test methods M and N, respectively.)
– Two cases: (a) TN ⊆ TM and (b) TM and TN overlap

10

Adequacy of Testing
• Reality: New test cases, in addition to the planned test cases, are
designed while performing testing. Let the test set be T.

• If a test set T does not reveal any more faults, we face a dilemma:
– P is fault-free. OR
– T is not good enough to reveal (more) faults.
 Need for evaluating the adequacy (i.e. goodness) of T.

• Some ad hoc stopping criteria
– Allocated time for testing is over.
– It is time to release the product.
– Test cases no more reveal faults.

11

Adequacy of Testing

Figure 2.4: Context of applying test adequacy.

12

Adequacy of Testing
• Two practical methods for evaluating test adequacy
– Fault seeding
– Program mutation
• Fault seeding
– Implant a certain number (say, X) of known faults in P, and test P with T.
– If k% of the X faults are revealed, T has revealed k% of the unknown faults.
– (More in Chapter 13)
• Program mutation
– A mutation of P is obtained by making a small change to P.
– Some mutations are faulty, whereas the others are equivalent to P.
– T is said to be adequate if it causes every faulty mutations to produce
unexpected results.
– (More in Chapter 3)

13

Limitations of Testing
• Dijkstra’s famous observation
– Testing can reveal the presence of faults, but not their absence.
• Faults are detected by running P with a small test set T, where |T| <<
|D|, where |.| denotes the “size-of” function and “<<“ denoted “much
smaller.”
– Testing with a small test set raises the concern of testing efficacy.
– Testing with a small test set is less expensive.
• The result of each test must be verified with a test oracle.
– Verifying a program output is not a trivial task.
– There are non-testable programs. A program is non-testable if
• There is no test oracle for the program.
• It is too difficult to determine the correct output.

14

Summary
• Theory of Goodenough and Gerhart
• Ideal test, Test selection criteria, Program faults, Test predicates
• Theory of Weyuker and Ostrand
• Uniformly ideal test selection
• Revealing subdomain
• Theory of Gourlay
• Testing system
• Power of test methods (“at least as good as” relation)
• Adequacy of Testing
• Need for evaluating adequacy
• Methods for evaluating adequacy: fault seeding and program mutation
• Limitations of Testing
• Testing is performed with a test set T, s.t. |T| << |D|.
• Dijkstra’s observation
• Test oracle problem
15

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