Profinite automata

Profinite automata

Many sequences of p-adic integers project modulo p to p-automatic sequences for every α ≥ 0. Examples include algebraic sequences of integers, which satisfy this property for every prime p, and some cocycle sequences, which we show satisfy this property for a fixed p. For such a sequence, we construct a profinite automaton that projects modulo p to the automaton generating the projected sequenc...

Profinite Methods in Automata Theory

This survey paper presents the success story of the topological approach to automata theory. It is based on profinite topologies, which are built from finite topogical spaces. The survey includes several concrete applications to automata theory. In mathematics, p-adic analysis is a powerful tool of number theory. The p-adic topology is the emblematic example of a profinite topology, a topology ...

Profinite techniques for probabilistic automata and the Markov Monoid algorithm

We consider the value 1 problem for probabilistic automata over finite words. This problem is known to be undecidable. However, different algorithms have been proposed to partially solve it. The aim of this paper is to prove that one such algorithm, called the Markov Monoid algorithm, is optimal. To this end, we develop a profinite theory for probabilistic automata. This new framework gives a t...

Inverse Automata and Profinite Topologies on a Free Group

Profinite Monads, Profinite Equations, and Reiterman's Theorem

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e. classes of finite algebras closed under finite products, subalgebras and quotients. In this paper Reiterman’s theorem is generalised to finite Eilenberg-Moore algebras for a monad T on a variety D of (ordered) algebras: ...