Abstract
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.
Highlights
Let A = (Q, Σ, δ) be a deterministic finite automaton (DFA), where Q denotes a state set, Σ stands for an input alphabet, and δ : Q × Σ → Q is a transition function defining an action of the letters in Σ on Q
In 1964 Cernyformulated a conjecture concerning an upper bound of the length of the shortest reset word of a synchronizing DFA [4]: the length cannot be larger than (n − 1)2
Evgeny Skvortsov and Yulia Zaks that the probability tends to 1 with n going to infinity.) In terms of automata, Higgins’s result means that a random automaton with an alphabet of size larger than 2n whp has a reset word of length 2n
Summary
Evgeny Skvortsov and Yulia Zaks that the probability tends to 1 with n going to infinity.) In terms of automata, Higgins’s result means that a random automaton with an alphabet of size larger than 2n whp has a reset word of length 2n. The numerical experiment output combined with Higgins’s result offers a new field of research: study of the shortest reset word of random automata, or the most probable length of the shortest reset word, as opposed to the classical problem of the upper bound; and allows to put forward a hypothesis that the most probable length should be much smaller than the upper bound Starting this new line of research we address the following questions. The Wormald’s theorem and the other results used in the work are formulated in Appendix
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