Emanuele Rodaro

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.

Recognizing synchronizing automata with finitely many minimal synchronizing words is PSPACE-complete

A deterministic finite-state automaton A is said to be synchronizing if there is a synchronizing word, i.e. a word that takes all the states of the automaton A to a particular one. We consider synchronizing automata whose language of synchronizing words is finitely generated as a two-sided ideal in Σ*. Answering a question stated in [1], here we prove that recognizing such automata is a PSPACE-complete problem.

A geometric approach to (semi)-groups defined by automata via dual transducers

We give a geometric approach to groups defined by automata via the notion of enriched dual of an inverse transducer. Using this geometric correspondence we first provide some finiteness results, then we consider groups generated by the dual of Cayley type of machines. Lastly, we address the problem of the study of the action of these groups on the boundary. We show that examples of groups having essentially free actions without critical points lie in the class of groups defined by the transducers whose enriched duals generate torsion-free semigroup. Finally, we provide necessary and sufficient conditions to have finite Schreier graphs on the boundary yielding to the decidability of the algorithmic problem of the existence of Schreier graphs on the boundary whose cardinalities are bounded from above by some fixed integer.

Regular Ideal Languages and Synchronizing Automata

We introduce the notion of reset left regular decomposition of an ideal regular language and we prove that there is a one-to-one correspondence between these decompositions and strongly connected synchronizing automata. We show that each ideal regular language has at least a reset left regular decomposition. As a consequence each ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to formulate Cernýs conjecture in a pure language theoretic framework.

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|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the

Groups and semigroups defined by colorings of synchronizing automata

\mathcal{J}

STATE COMPLEXITY OF CODE OPERATORS

-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the

MULTILINEAR EQUATIONS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

\mathcal{R}

Amalgams vs Yamamura's HNN-Extensions of Inverse Semigroups

-classes to be finite.

Finitely Generated 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.