Lecture Notes in Computer Science 3472

12 Technology of Test-Case Generation 327
represents the value after the application of Increment. 2 This allows us to define
the effect of the operation Increment on Counter in the predicate part. In particular,
if the value of ctr is less than three, then its value after the application
of Increment must be ctr +1, and if ctr is equal to three, then its value after
application of Increment must be zero.
Finally, let us specify another example: the modulo operation of natural
numbers which, given the base and the modulus, returns the common residue.
Modulo
b?, m?, r! :N
b? ≥ m?
(r! < m?) ∨ (m? =0)
∃ k : N • b? =m? ∗ k + r!
This schema declares the three variables b?, m?, and r! to be natural numbers.
The suffixes ‘?’ and ‘!’ indicate input and output parameters respectively.
The predicate part consists of three predicates. The first ensures that the base
must be greater or equal to the modulus. The second ensures that the specified
residue is the common residue, i.e., less than the modulus, or that the modulus
is equal to zero. In the latter case, the residue should be equal to the base which
is greater or equal to the modulus according to the first predicate. The final
predicate specifies the relationship between base, modulus, and residue.
As already mentioned, the above constructs represent only a small fraction
of the whole Z specification language. The reader is referred to [Spi92, PST96,
ISO02] for a more exhaustive introduction to Z. We will use Z specifications as
the basis of test case generation in the following section.
12.2.2 Test Case Generation from Z Specifications
Given a model of the SUT as Z specification, general purpose theorem provers
can be used to support the partitioning of the specification into equivalence
classes. We will now present some approaches of how to construct test cases
from Z specifications in such a way. We start with the disjunctive normal form
approach, a method that allows us to syntactically transform Z specifications into
equivalence classes. Afterwards, we present the classification-tree method which
enables us to integrate semantic aspects into the partitioning process. Finally,
we discuss partitioning heuristics which can be seen as a general theoretical
framework of syntactic and semantic partitioning methods.
The Disjunctive Normal Form Approach. In the approach of Helke et
al. [HNS97], Z specifications are translated into the input language of Isabelle
2 Similarly, a preceding symbol Ξ indicates an implicit declaration of the variables
of Counter which, in contrast to ∆, remain unchanged, that is ctr ′ = ctr.