Tree Automata as Algebras: Minimisation and Determinisation

TL;DR

This paper develops a categorical framework for tree automata using $\

Contribution

It introduces a general minimisation algorithm and a categorical Nerode equivalence for tree automata, extending to side-effects like non-determinism.

Findings

A general minimisation algorithm for categorical tree automata.

A categorical extension of Nerode equivalence relating to minimal automata.

A determinisation procedure capturing side-effects such as non-determinism.

Abstract

We study a categorical generalisation of tree automata, as $\Sigma$-algebras for a fixed endofunctor $\Sigma$ endowed with initial and final states. Under mild assumptions about the base category, we present a general minimisation algorithm for these automata. We then build upon and extend an existing generalisation of the Nerode equivalence to a categorical setting and relate it to the existence of minimal automata. Finally, we show that generalised types of side-effects, such as non-determinism, can be captured by this categorical framework, leading to a general determinisation procedure.