Computing the shortest reset words of synchronizing automata

Abstract

In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to n = 350 states and up to k = 10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word ≈ 2.5 √ n − 5 for binary automata and show that the error of this estimate is sufficiently small. Experiments for automata with more than two input letters show certain trends with the same general pattern.

Reduced checking sequences using unreliable reset

“…Berlinkov [12] claims that a random FSM with n states and q inputs has a synchronizing sequence with probability 1 − Θ(1/n 0.5×q ). This claim was experimentally supported by Kisielewicz et al [13]. Therefore, our method becomes applicable for virtually all FSMs as the size of the FSM increases.…”

“…Based on the knowledge that the given W -set for M is also a W -set for the implementation N, further recognitions for a state s i can be performed by [14] …”

D2-Synchronization in Nondeterministic Automata

“…We have presented a modification of the approach originated in [12] that has allowed us to find shortest proper D 2 -synchronizing words for nondeterministic automata with two input letters and up to 100 states. The size of automata that we are able to analyze may seem modest in comparison with the results of [8] whose authors describe sophisticated methods to compute shortest synchronizing words for deterministic automata with up to 350 states. However, two important nuances should be taken into account.…”

The algebra of row monomial matrices

“…There are several reasons [2], [4], [5], [9], [24] to believe that the length of the shortest synchronizing word for remaining automata with n > 4 (except the sequence of Černy and two examples for n = 5, 6) is essentially less and the gap grows with n. For several classes of automata, one can find some estimations on the length in [2], [11], [7], [13], [15], [25].…”