Lecture Notes in Computer Science 3472

12 Technology of Test-Case Generation 325
a proof fully automatically, in most cases – especially in industrial applications –
the user has to guide the proof search procedure. Such user controlled theorem
provers are called semi-automated or interactive. In this context, automated
theorem proving is the process of constructing a proof by an automated or
semi-automated theorem prover.
The importance of automated theorem proving arises not only from its mathematical
and philosophical relevance, but also from the fact that many problems
in artificial intelligence and formal verification can be reduced to theorem proving.
In industrial environments, however, theorem proving is rarely used because
it is very time consuming and requires expert knowledge. Nevertheless, in some
areas of high-quality productions [Sch00, Sch01], there is no adequate alternative
to formally prove properties of the system under consideration. Moreover, there
exist several attempts to combine theorem proving with model checking in order
to compensate the shortcomings of each other [ORR + 96, CCG + 03]. This can be
seen as further evidence for the importance of automated theorem proving.
In this section, we survey how automated theorem proving technology can
be used in test case generation. Although theorem proving is often used in combination
with other techniques, we focus on approaches where theorem proving
is the dominating technique. The typical method in this context is based on formal
specifications which model the SUT. Such a model resp. specification is then
partitioned into equivalence classes which are assumed to satisfy a uniformity
hypothesis, i.e., they are assumed to represent the same behavior concerning
the test; in particular, test data in the same equivalence class are assumed to
cause the same error (or no error in the case of success). Each such equivalence
class is then interpreted as one test case. Hence, because of the uniformity hypothesis,
it suffices to extract a small amount of test data from each test case.
Note that the test cases generated by the approaches presented in this section are
single instances and not sequences of test cases as would be necessary for testing
reactive systems. Extensions for generating test case sequences are described in
the referred literature or are topic of future research.
Our aim in the following is to show how theorem provers can be applied
to construct such test cases from a given specification. There exist three major
approaches in the literature to address this issue:
• Use theorem provers to generate test cases from Z specifications
• Translate algebraic specifications into logic programs and use Prolog’s theorem
proving mechanisms to generate test cases
• Construct formal proofs from which test cases can be extracted
Since our focus will be on the first item, which is mainly based on the specification
language Z, we start with a short introduction of some basic concepts
of this language.
12.2.1 The Z Specification Language
Z is a commonly used formal specification language based on set theory and
first-order predicate logic [Spi92, PST96, ISO02]. As every formal specification