A Survey on the Local Divisor Technique

TL;DR

This survey reviews the local divisor technique in algebraic automata theory, illustrating its applications across various logical and language-theoretic problems, and introduces the concept of localizable language classes.

Contribution

It provides a comprehensive overview of the local divisor technique and introduces the new concept of localizable language classes to unify existing proof methods.

Findings

Applications to linear temporal logic and rational expressions

Introduction of localizable language classes

Unification of proof techniques for multiple results

Abstract

Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of localizable language class as a new abstract concept which unifies some of the proofs for the results above. The current arXiv-version includes some additional material about codes of bounded synchronization delay as well as some updates concerning related literature.