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

Equivalence of Mealy and Moore automata

It is proved here that every Mealy automaton is a liomomorphic image of a Moore automaton, and among these Moore automata (up to isomorphism) there exists a unique one which is a homomorphic image of the others. A unique simple Moore automaton M is constructed (up to isomorphism) in the set MO(A) of all Moore automata equivalent to a Mealy automaton A such that M is a homomorphic image of every Moore automaton belonging to MO{A). By the help of this construction, it can be decided in steps |X|k that automaton mappings inducing by states of a k-uniform finite Mealy [Moore] automaton are equal or not. The structures of simple k-uniform Mealy [Moore] automata are described by the results of [1]. It gives a possibility for us to get the k-uniform Mealy [Moore] automata from the simple k-uniform Mealy [Moore] automata. Based on these results, we give a construction for finite Mealy [Moore] automata

Well-Pointed Coalgebras

For endofunctors of varieties preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. The initial algebra consists of all well-pointed coalgebras that are well-founded in the sense of Osius and Taylor. And initial algebras are precisely the final well-founded coalgebras. Finally, the initial iterative algebra consists of all finite well-pointed coalgebras. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems

Applying abstract algebraic logic to classical automata theory : an exercise

In [4], Blok and Pigozzi have shown that a deterministic finite au- tomaton can be naturally viewed as a logical matrix. Following this idea, we use a generalisation of the matrix concept to deal with other kind of automata in the same algebraic perspective. We survey some classical concepts of automata theory using tools from algebraic logic. The novelty of this approach is the understand- ing of the classical automata theory within the standard abstract algebraic logic theory

Constructing Deterministic Parity Automata from Positive and Negative Examples

We present a polynomial time algorithm that constructs a deterministic parity automaton (DPA) from a given set of positive and negative ultimately periodic example words. We show that this algorithm is complete for the class of $\omega$-regular languages, that is, it can learn a DPA for each regular $\omega$-language. For use in the algorithm, we give a definition of a DPA, that we call the precise DPA of a language, and show that it can be constructed from the syntactic family of right congruences for that language (introduced by Maler and Staiger in 1997). Depending on the structure of the language, the precise DPA can be of exponential size compared to a minimal DPA, but it can also be a minimal DPA. The upper bound that we obtain on the number of examples required for our algorithm to find a DPA for $L$ is therefore exponential in the size of a minimal DPA, in general. However we identify two parameters of regular $\omega$-languages such that fixing these parameters makes the bound polynomial.Comment: Changes from v1: - integrate appendix into paper - extend introduction to cover related work in more detail - add a second (more involved) example - minor change