Topological dynamics and recognition of languages

TL;DR

This paper introduces compact automata and monoids, establishing their unique minimal forms and extending language recognition theory beyond finite automata to a topological setting.

Contribution

It defines compact automata and monoids, proving the existence of unique minimal models and generalizing the syntactic monoid concept to a topological framework.

Findings

Every language has a unique minimal compact automaton.

Recognition by compact monoids generalizes finite automata theory.

For rational languages, the theory reduces to classical finite automata results.

Abstract

We define compact automata and show that every language has a unique minimal compact automaton. We also define recognition of languages by compact left semitopological monoids and construct the analogue of the syntactic monoid in this context. For rational languages this reduces to the usual theory of finite automata and finite monoids.