An introduction to finite automata and their connection to logic

TL;DR

This tutorial introduces finite automata, covering determinization, minimization, and their logical connections, including monadic second-order logic and syntactic monoids, highlighting their theoretical equivalences.

Contribution

It provides a comprehensive overview of finite automata and their deep connections to logic, including new insights into the equivalence of automata, logic, and algebraic structures.

Findings

Finite automata are equivalent to monadic second-order logic.

Determinization and minimization procedures are standard.

First-order definability corresponds to aperiodic automata.

Abstract

This is a tutorial on finite automata. We present the standard material on determinization and minimization, as well as an account of the equivalence of finite automata and monadic second-order logic. We conclude with an introduction to the syntactic monoid, and as an application give a proof of the equivalence of first-order definability and aperiodicity.