Complexity of Checking Whether Two Automata Are Synchronized by the Same Language

Abstract

Abstract. A deterministic finite automaton A is said to be synchronizing if it has a reset word, i.e. a word that brings all states of the automaton A to a particular one. We prove that it is a PSPACEcomplete problem to check whether the language of reset words for a given automaton coincides with the language of reset words for some particular automaton.

Strongly connected synchronizing automata and the language of minimal reset words

“…• For any regular ideal I having a set of minimal elements M(I) not containing a factor of length ℓ ≤ I 4 + 1 16 , then there is a reset word of length at most (n − 1 2 ) 2 . Proof.…”

“…From this result, as an immediate corollary, we show that Cerny's conjecture holds if this missing factor has length at most 1 4 (n 2 − 3n + 2). Further, a quadratic bound (n − 1 2 ) 2 holds for all the synchronizing automata with n states having set of reset words I satisfying Fact ℓ (M(I)) \ Σ ℓ = ∅ for some ℓ ≤ I 4 + 1 16 . Roughly speaking, a potential counterexample to the Cerny conjecture should be searched in the class of strongly connected synchronizing automata whose set of reset words is an ideal whose set of minimal words contains as factors all the words of length at most 1 4 (n 2 − 3n + 2).…”

Reset Complexity of Ideal Languages Over a Binary Alphabet

“…The complexity of the latter problem has been partially studied in Ref. [9]. It is well known that the equality of the languages recognized by two given DFAs can be checked in time polynomial of the size of automata.…”

“…It is well known that the equality of the languages recognized by two given DFAs can be checked in time polynomial of the size of automata. However, the problem of checking the equality of the languages of reset words of two synchronizing DFAs turns out to be PSPACE-complete [9]. Recall that the problem of checking the equality of languages recognized by two given NFAs is PSPACE-complete as well [13].…”

Representation of (Left) Ideal Regular Languages by Synchronizing Automata

“…One of the obstacles is that MSA is not uniquely defined. Furthermore, the problem of checking, whether a given synchronizing automaton with at least five letters is an MSA for a given ideal language, has recently been shown to be PSPACE-complete [9].…”