Lecture Notes in Computer Science 3472

3 State Verification 85
x = a1a2 ...ak ∈ I ∗ ,letui be the number of states for which a1a2 ...ai is a UIO
sequence (u0 =0).Fori ≥ 1, let ∆ui = ui − ui−1. Also, let vi be the number
of sets in the initial state uncertainty 5 for a1a2 ...ai and ∆vi = vi − vi−1. Now,
for each i ∈{1,...,k} we define
ui +∆ui uie
fi(a1 ...ai) =α
i γ
+ β vi + ∆vi
, (3.3)
i
where α, β, and γ are constants. It is clear that when the number of states a
prefix uniquely identifies grows, the value of the prefix grows exponentially. On
the other hand, an increase in length gives a polynomial reduction in value. The
second term in the function definition rewards having many sets in the initial
state uncertainty, even when not so many UIO sequences have been found yet.
We can now define the fitness of a sequence as the average of the fi -values of its
prefixes:
f (a1 ...ak )= 1
k
k�
fi(a1 ...ai). (3.4)
i=1
After creating an initial population randomly, Guo et al. suggest using roulette
wheel selection to find the parents that are recombined. This means that each
individual has a probability of being selected that is proportional to its fitness
value. Further, uniform crossover is used, which means that in each position,
the value for the offspring is selected with uniform probability from the values
of the parents at the corresponding positions. In order for the population not to
stagnate too quickly, Guo et al. suggest using wild-card characters in the strings
representing sequences.
One drawback of the approach is that the length k of the sequences used
must be set in advance, and the algorithm will not find UIO sequences of length
> k.
Guo et al. only present tests of the algorithm on two small machines, with
reportedly good results. It remains to be be seen in larger scale tests whether
the method is practical.
3.5 Summary
State verification is the problem of verifying that a Mealy machine, placed in a
black box, is in a specified state s. The state diagram of the machine is assumed
to be known. The only way of doing this, short of opening the box, is to feed
input to the machine, and study the output. Therefore, state verification for state
s is only possible if there is an input sequence x such that the output produced
by the machine when it is in state s and is given input x is unique, i.e., there is
no other state that would produce the same output on input x . Such an input
5 The initial state uncertainty of sequence x is a partitioning of the states in the
machine such that if s and t belong to the same set, then λ(s, x) =λ(t, x); see
Chapter 1.