Automata Minimization: a Functorial Approach

TL;DR

This paper introduces a functorial framework for automata minimization, unifying various automata theory concepts and algorithms through category theory, providing new conditions for minimization and methods to lift adjunctions.

Contribution

It develops a categorical approach to automata minimization, establishing conditions for guaranteed minimization and connecting automata transformations via adjunctions.

Findings

Provided sufficient conditions for automata minimization guarantees

Unified various automata phenomena using a categorical framework

Explained classical minimization algorithms within this new approach

Abstract

In this paper we regard languages and their acceptors - such as deterministic or weighted automata, transducers, or monoids - as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: A) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. B) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. C) We show how this framework can be instantiated to unify several phenomena in automata theory, starting with determinization, minimization and syntactic algebras. We provide explanations of Choffrut's minimization algorithm for subsequential transducers and of Brzozowski's minimization algorithm in this setting.