A Finite Axiomatisation of Finite-State Automata Using String Diagrams

TL;DR

This paper introduces a finite, diagrammatic algebraic framework for finite-state automata, enabling a complete equational theory of language equivalence with Kleene star derived from primitive operations.

Contribution

It presents a novel, fully diagrammatic, finite axiomatisation of finite-state automata where Kleene star is derived, not primitive.

Findings

Complete equational theory for language equivalence

Finite set of axioms for automata

Kleene star decomposed into primitive algebraic blocks

Abstract

We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks.