Lecture Notes in Computer Science 3472

3 State Verification 83
Since the machine M is assumed to be deterministic, there is only one possible
path from each state in G(a/b). If this path ends in a sink it is called linear,
otherwise it is cyclic.
Proposition 3.11 (Naik [Nai97]). If the path from state s in G(a/b) is linear
and of length k, and no other state has a linear path of length ≥ k, then s has
a UIO sequence of length k or k +1.
Proof. Suppose that G(a/b) has no cycles. If we apply input a k to s, then the
output produced will be b k . This is not true for any other state, since all other
paths in G(a/b) have length smaller than k. Thusa k is a UIO sequence of length
k for s.
If, on the other hand, G(a/b) has cycles, consider the input sequence a k+1 .
Applied to states with cyclic paths in G(a/b), it will produce output b k+1 .For
s, the output produced will be b k b ′ , for some output symbol b ′ different from b.
For all other states, the output produced will have fewer than k initial copies of
b. Thusa k+1 is a UIO sequence for s.
It is also easy to see from the first part of the above proof that if all paths in
G(a/b) are linear, then every vertex with a unique path length different from 0
has a UIO sequence.
Depth-First Search in the Pruned UIO Tree. Analysis of the projection graphs
may not produce any UIO sequences at all, or it may, even when combined with
the inference rules, not produce conclusive results for all states of the machine.
In these cases, the pruned UIO tree must be searched. The search can be aborted
as soon as UIO sequences have been found for all vertices except those that were
found to have none during the analysis of the inference graph.
Every time a UIO sequence is found, the inference rules are applied to generate
sequences for other states, and a check is made to determine whether the
search must be continued.
In order to make maximal use of the inference rules, it is desirable to find
at least some sequences as soon as possible. To achieve this, Naik suggests that
the tree be searched depth first, rather than breadth first. This is because if all
states have UIO sequences of approximately the same length, then the breadthfirst
search would go through more or less the whole pruned tree before finding
even one sequence.
The problem with the depth first approach is that longer sequences may be
found before shorter ones for the same state, and when using UIO sequences for
conformance testing (see Chapter4), it is desirable to find sequences that are as
short as possible.
In order to at least partially get around this drawback, Naik suggests what
he calls a hybrid search, which runs in the following way:
(1) Let φ be the root node of the pruned UIO tree.
(2) Visit every child of φ. If a child is terminal, mark it as such. If it represents a
UIO sequence for some state, use the inference rules and determine whether
the search can be terminated.