Lecture Notes in Computer Science 3472

78 Henrik Björklund
Proposition 3.7 ([Nai97]). If GI is the inference graph of Mealy machine M,
and state s does not have any outgoing edges in GI , then s does not have a UIO
sequence.
Proof. In the transition graph G of M, all outgoing edges from s must be converging,
since none of them appear in GI . This means that for every transition
from s to some state t, there is another state s ′ �= s such that s ′ has a transition
with the same label that also leads to t. Thus for any input sequence, there is
some state other than s that produces the same output as s, sos cannot have a
UIO sequence.
3.4 Algorithms
In Section 3.2 it was shown that the problem of determining whether a given
state of a Mealy machine has a UIO sequence is PSPACE-complete. Thus there
is presumably no efficient, polynomial time, algorithm for the general state verification
problem. However, it turns out that in practice, it is solvable for many
machines, and UIO sequences are used as parts of many conformance testing
algorithms. In this section, algorithms for finding UIO sequences are discussed.
The first two have exponential worst-case complexity, but work better in many
practical applications. The third is a genetic algorithm, that is not always guaranteed
to find all UIO sequences.
3.4.1 The Sabnani-Dahbura Algorithm
In 1988 Sabnani and Dahbura presented a test generation procedure for conformance
testing [SD88]. It uses state verification, and includes a procedure for
computing UIO sequences. For their purposes, it is enough to compute sequences
of length at most 2n2 ,wherenis the number of states, if they exist, but the
procedure can be extended to compute longer sequences.
The basic idea when trying to find a UIO sequence for a given state s is
to naively generate all input sequences, in order of ascending length. For each
sequence x , the algorithm then computes λ(s, x ) and tests it for uniqueness,
until a UIO sequence for s is found. The search space of all possible input-output
sequences is thus examined in a breadth-first manner. Since the test generation
procedure of Sabnani and Dahbura [SD88] is only interested in sequences of
length at most 2n2 , the search is terminated at this depth, even if a UIO sequence
has not been found. If we are also interested in longer sequences, another upper
bound, e.g., the n · 2n from Section 3.2, must be supplied to ensure termination.
The algorithm searches the tree of all possible input-output sequences breadth
first, without trying to take advantage of any structural properties of the machine
under test.
Since the problem is PSPACE-complete, we cannot hope to solve the problem
quickly in the general case. The algorithms are, however, intended for practical
use, and therefore it seems like a good idea to analyze the input machine
in order to take advantage of any particularities that can help us. The next
algorithm we consider tries to do this.