Abstract
We prove PSPACE-completeness of checking whether a given ideal language serves as the language of reset words for some automaton with at most four states over a binary alphabet. We compare the reset complexity and the state complexity for languages related to slowly synchronizing automata.
Highlights
Regular languages admit compact representations by different tools: deterministic and nondeterministic finite automata, syntactic monoids, regular expressions, etc
In the present paper we focus on some complexity aspects of the theory of synchronizing automata
Representation via the minimal automaton. This resembles the well-known property of nondeterministic finite automata: for each n ≥ 3, there is an n-state NFA N such that every deterministic finite automaton (DFA) recognizing the same language as N has exactly 2n states [10, 11]
Summary
Regular languages admit compact representations by different tools: deterministic and nondeterministic finite automata, syntactic monoids, regular expressions, etc. This resembles the well-known property of nondeterministic finite automata (for brevity, NFAs): for each n ≥ 3, there is an n-state NFA N such that every DFA recognizing the same language as N has exactly 2n states [10, 11]. The following question arises: does slow synchronization of a given DFA An guarantee that the state complexity of Syn(An) is exponentially smaller than the reset complexity of this language? We study several series of slowly synchronizing automata known from [1] and prove that the considered automata are certificates for an exponential gap between the state complexity and the reset complexity of the corresponding ideal language.
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