Lecture Notes in Computer Science 3472

76 Henrik Björklund
Proof. Consider n deterministic finite automata A1,...,An over the alphabet
{0}, such that the transitions of Ai form a cycle of length pi, wherepi is the ith
prime number. The only accepting state of Ai is the predecessor of the initial
state. Thus Ai accepts 0 k if and only if pi | (k + 1). This means that the shortest
string accepted by all automata has length m = � n
i=1 pi − 1. If there was a
string of length m ′ < m, accepted by all automata, then for all i ∈{1,...,n},
we have pi | (m ′ + 1). Thus, by unique prime factorization, m ′ +1= � n
i=1 pi, a
contradiction.
Now note that � n
i=1 pi − 1 ≥ 2 n is exponential in the total representation
size of the automata, which is polynomial in n by Gauss’ prime number theorem,
and apply the reduction used in the proof of Theorem 3.2.1. The sequence r0 m f
will be the shortest UIO sequence for state s, since for any shorter string, the
states of at least one automaton will produce an output of all zeroes.
3.3 Convergence and Inference Graphs
Since we saw in Section 3.2 that it is in general PSPACE-hard to determine
whether a state has a UIO sequence, it is interesting to identify subclasses of
machines and states, for which the question can be answered efficiently. Using
the concept of convergence, first made explicit by Miller and Paul [MP93], Naik
presents a number of results of this kind [Nai97], which are surveyed in this
section.
Definition 3.4 (Convergence [MP93]). Let Mealy machine M have states
t1, t2,...,tk that all have edges to the state s with the same edge label a/b. Say
that
• states t1,...,tk are converging states,
• state s is a convergent state,
• edges (t1, s),...,(tk, s) withlabela/bare converging edges.
If a machine has some converging states it is a converging machine. Otherwise,
the machine is nonconverging.
Using these concepts, Naik invented a way of using a UIO sequence for one
state to infer sequences for other states. The basic construction used is the
following.
Definition 3.5 (Inference Graph [Nai97]). Let G be the transition graph of a
Mealy machine M. Theinference graph of M is the graph GI , obtained from
G by removing all converging edges.
Given graph G for machine M, the inference graph GI is computable in time
O(|E| 2 ), where E is the set of edges of G.
Figure 3.5 shows the inference graph for the machine in Figure 3.3. We see
that the two edges (s3, s1) and(s2, s1) have been taken away. Since they both
end in the same state and have the same label they are converging. The same is
true for the edges (s1, s4) and(s4, s4).
The inference graph is used as follows.