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    Synchronizing automata and the language of minimal reset words

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    We study a connection between synchronizing automata and its set MM of minimal reset words, i.e., such that no proper factor is a reset word. We first show that any synchronizing automaton having the set of minimal reset words whose set of factors does not contain a word of length at most 14min⁑{∣u∣:u∈I}+116\frac{1}{4}\min\{|u|: u\in I\}+\frac{1}{16} has a reset word of length at most (nβˆ’12)2(n-\frac{1}{2})^{2} In the last part of the paper we focus on the existence of synchronizing automata with a given ideal II that serves as the set of reset words. To this end, we introduce the notion of the tail structure of the (not necessarily regular) ideal I=Ξ£βˆ—MΞ£βˆ—I=\Sigma^{*}M\Sigma^{*}. With this tool, we first show the existence of an infinite strongly connected synchronizing automaton A\mathcal{A} having II as the set of reset words and such that every other strongly connected synchronizing automaton having II as the set of reset words is an homomorphic image of A\mathcal{A}. Finally, we show that for any non-unary regular ideal II there is a strongly connected synchronizing automaton having II as the set of reset words with at most (kmk)2kmkn(km^{k})2^{km^{k}n} states, where k=∣Σ∣k=|\Sigma|, mm is the length of a shortest word in MM, and nn is the dimension of the smallest automaton recognizing MM (state complexity of MM). This automaton is computable and we show an algorithm to compute it in time O((k2mk)2kmkn)\mathcal{O}((k^{2}m^{k})2^{km^{k}n}).Comment: 17 page
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