We present a new characteristic of a regular ideal language called reset complexity. We find some bounds on the reset complexity in terms of the state complexity of a given language. We also compare the reset complexity and the state complexity for languages related to slowly synchronizing automata and study uniqueness question for automata yielding the minimum of reset complexity.

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.

We study ideal languages generated by a single word. We provide an algorithm to construct a strongly connected synchronizing automaton for which such a language serves as the language of synchronizing words. Also we present a compact formula to calculate the syntactic complexity of this language.

We study representations of ideal languages by means of strongly connected synchronizing automata. For every finitely generated ideal language L we construct such an automaton with at most 2 n states, where n is the maximal length of words in L. Our constructions are based on the De Bruijn graph.

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