Synchronizing Automata with Finitely Many Minimal Synchronizing Words

A synchronizing word for a given synchronizing DFA is called minimal if none of its proper factors is synchronizing. We characterize the class of synchronizing automata having only finitely many minimal synchronizing words (the class of such automata is denoted by FG). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n-5. We also prove that checking whether a given DFA A is in FG is co-NP-hard and provide an algorithm for this problem which is exponential in the number of states A. © 2010 Elsevier Inc. All rights reserved.Author acknowledges support from the Federal Education Agency of Russia, Grant 2.1.1/3537, and from the Russian Foundation for Basic Research, Grants 09-01-12142 and 10-01-00793. This research was initiated with the partial support of GNSAGA during the visit of the author to the Ural State University, Russia

Bicyclic subsemigroups in amalgams of finite inverse semigroups

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2 whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S1|,|S 2|}. Moreover we consider amalgams of finite inverse semigroups respecting the J-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the R-classes to be finite. © 2010 World Scientific Publishing Company

Missing factors of ideals and synchronizing automata

Recently, a series of papers have started to look at Cerný‘s conjecture, and in general at synchronizing automata, from the point of view of the theory of ideals of free monoids. The starting point of such an approach is a simple observation: the set of reset words of an automaton is a two-sided ideal of the free monoid on its alphabet that is also a regular language. We study the relationship between a synchronizing automaton and the sets of (minimal) generators of its reset words. We show that if such set does not contain a word of a certain length, then Cerný‘s conjecture holds

State complexity of code operators

We consider five operators on a regular language. Each of them is a tool for constructing a code (respectively prefix, suffix, bifix, infix) and a hypercode out of a given regular language. We give the precise values of the (deterministic) state complexity of these operators: over a constant-size alphabet for the first four of them and over a quadratic-size alphabet for the hypercode operator. © 2011 World Scientific Publishing Company

Decidability of the word problem in Yamamura's HNN extensions of finite inverse semigroups

We prove that the word problem is decidable in Yamamura’s HNN extensions of finite inverse semigroups, by providing an iterative construction of approximate automata of the Schützenberger automata of words relative to the standard presentation of Yamamura’s HNN-extensions

Finitely generated synchronizing automata

A synchronizing word w for a given synchronizing DFA is called minimal if no proper prefix or suffix of w is synchronizing. We characterize the class of synchronizing automata having finite language of minimal synchronizing words (such automata are called finitely generated). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n - 5. We also prove that checking whether a given DFA A is finitely generated is co-NPhard, and provide an algorithm for this problem which is exponential in the number of states A. © Springer-Verlag Berlin Heidelberg 2009

Synchronizing automata with finitely many minimal synchronizing words

A synchronizing word for a given synchronizing DFA is called minimal if none of its proper factors is synchronizing. We characterize the class of synchronizing automata having only finitely many minimal synchronizing words (the class of such automata is denoted by FG). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n-5. We also prove that checking whether a given DFA A is in FG is co-NP-hard and provide an algorithm for this problem which is exponential in the number of states A. © 2010 Elsevier Inc. All rights reserved