5 research outputs found
Groups and Semigroups Defined by Colorings of Synchronizing Automata
In this paper we combine the algebraic properties of Mealy machines
generating self-similar groups and the combinatorial properties of the
corresponding deterministic finite automata (DFA). In particular, we relate
bounded automata to finitely generated synchronizing automata and characterize
finite automata groups in terms of nilpotency of the corresponding DFA.
Moreover, we present a decidable sufficient condition to have free semigroups
in an automaton group. A series of examples and applications is widely
discussed, in particular we show a way to color the De Bruijn automata into
Mealy automata whose associated semigroups are free, and we present some
structural results related to the associated groups
Complexity of checking whether two automata are synchronized by the same language
A deterministic finite automaton is said to be synchronizing if it has a
reset word, i.e. a word that brings all states of the automaton to a particular
one. We prove that it is a PSPACE-complete problem to check whether the
language of reset words for a given automaton coincides with the language of
reset words for some particular automaton.Comment: 12 pages, 4 figure
Ideal regular languages and strongly connected synchronizing automata
We introduce the notion of a reset left regular decomposition of an ideal regular language, and we prove that the category formed by these decompositions with the adequate set of morphisms is equivalent to the category of strongly connected synchronizing automata. We show that every ideal regular language has at least one reset left regular decomposition. As a consequence, every ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal from which follows a reformulation of Černý's conjecture in purely language theoretic terms. Finally, we present and characterize a subclass of ideal regular languages for which a better upper bound for the reset decomposition complexity holds with respect to the general case
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