Lecture Notes in Computer Science 3472

30 Sven Sandberg
theorem, each automaton has size O(n log n), so the total size is polynomial in
n but the shortest sequence accepted by all automata is exponential. ⊓⊔
Fig. 1.9. An automaton that accepts exactly the words of positive length divisible by
7. A similar automaton is created for all primes p1,...,pn.
Consider another generalization of synchronizing sequences, where we are
again given a subset Q ⊆ S but now the sequence has to end in Q, thatis,
we want to find a sequence x such that δ(S, x ) ⊆ Q. It is not more difficult
to show that this problem also is PSPACE-complete; however, it can be solved
in time n O(|Q|) , so it is polynomial if the size of Q is bounded by a constant
[Rys83]. Rystsov shows in the same article that several related problems are
PSPACE-complete, and concludes the following result in another paper [Rys92].
Exercise 1.8. A nondeterministic Mealy machine is like a Mealy machine except
δ(s, a) isaset of states. The transition function δ is extended similarly,
so δ(Q, a) = � {δ(s, a) :s ∈ Q} and δ(Q, a1 ...an )=δ(δ(Q, a1,...,an−1), an ).
Show that the synchronizing sequence problem for nondeterministic Mealy machines
is PSPACE-complete. Here, a sequence x is synchronizing for a nondeterministic
machine if |δ(S, x )| =1.
Hint: Use Theorem 1.22.
1.5 Related Topics and Bibliography
The experimental approach to automata theory was initiated by the classical
article by Moore [Moo56], who introduces several testing problems, including
homing sequences and the adaptive version of Algorithm 1. He also shows the
upper bound of n(n − 1)/2 for the length of adaptive homing sequences. The
worst-case length of homing sequences for minimized automata was studied by
Ginsburg [Gin58] and finally resolved by Hibbard [Hib61]. The book by Kohavi
[Koh78] and the article by Gill [Gil61] contain good overviews of the problem.