Lecture Notes in Computer Science 3472

12 Technology of Test-Case Generation 331
(b? > m?) ∧ (k mod 2 = 1) to the predicate part of the original schema, we
obtain after some simplifications the complete test case schema:
Modulo (Partition4)
b?, m?, r! :N
b? > m?
(r! < m?) ∨ (m? =0)
∃ k : N • (k mod 2 = 1) ∧ (b? =m? ∗ k + r!)
This test case can be further refined by transforming it into disjunctive normal
form as in the previous approach, which yields two refined test cases:
(b? > m?) ∧ (r! < m?) ∧∃k : N • ((k mod 2 = 1) ∧ (b? =m? ∗ k + r!))
(b? > m?) ∧ (m? =0)∧∃k : N • ((k mod 2 = 1) ∧ (b? =m? ∗ k + r!))
Note that in addition to the semantic aspects, the classification-tree method
provides a hierarchical structuring of the test cases, while test cases in the pure
disjunctive normal form approach are completely unstructured. An application of
theorem proving to support this approach was presented by Sadeghipour [Sad98].
In particular, the theorem prover Isabelle [Pau94, NPW02] is used as integral
part of a tool environment for test case generation based on the classificationtree
method. The theorem prover is applied, e.g., for simplification tasks, for
checking test data consistency, and for test evaluation.
Partitioning Heuristics. An approach with the objective of building a uniform
theoretical foundation and generalizing the above methods was presented
by Burton et al. [BCM00, Bur00, Bur02]. This approach is based on CADiZ
[TM95, Toy96, Toy98], a Z type checker and theorem prover. Since the input
language of CADiZ is Z itself, there is no necessity for translating Z specifications.
The main difference to the approaches of Helke et al. [HNS97], Singh
et al., and Sadeghipour [SCS97, Sad98] is that various partitioning heuristics
are supported in the work of Burton et al. [BCM00, Bur00, Bur02]. In particular,
the test case generation of Helke et al. [HNS97] is based on disjunctive
normal form partitioning as shown above. This partitioning, however, can be
generalized by other partitioning heuristics such as heuristics based on boundary
value analysis or type analysis. For instance, the predicate m ≥ 0canbe
partitioned by boundary value analysis into (m =0)∨ (m =1)∨ (m > 1),
and the domain of the variable n ∈ Z can be partitioned by type analysis into
(n < 0) ∨ (n =0)∨ (n > 0).
In addition, the partitioning can also be based on fault-based heuristics which
take the experience of faults detected in previous builds into account. For instance,
if the specified operation n 2 is assumed to be incorrectly implemented
as 2 ∗ n, wheren ∈ N, then the partitioning would be (n =1)∨ (n > 2) because
n 2 and 2 ∗ n cannot be distinguished by n =0andn =2.
Of course, these partitionings can also be performed by the classification-tree
method of Singh et al. and Sadeghipour [SCS97, Sad98]. However, as we will see