Canonical Automata via Distributive Law Homomorphisms

TL;DR

This paper develops a framework for converting deterministic automata with algebraic structures into succinct automata with side-effects, generalizing classical constructions and discovering new canonical automata for regular languages.

Contribution

It introduces a reverse construction for automata with monadic effects, unifies existing examples, and identifies a new canonical automaton using the free vector space monad.

Findings

Recovered known automata like residual finite-state automaton and atomaton.

Discovered a new canonical automaton for regular languages.

Showed existence of size-minimal succinct automata under certain conditions.

Abstract

The classical powerset construction is a standard method converting a non-deterministic automaton into a deterministic one recognising the same language. Recently, the powerset construction has been lifted to a more general framework that converts an automaton with side-effects, given by a monad, into a deterministic automaton accepting the same language. The resulting automaton has additional algebraic properties, both in the state space and transition structure, inherited from the monad. In this paper, we study the reverse construction and present a framework in which a deterministic automaton with additional algebraic structure over a given monad can be converted into an equivalent succinct automaton with side-effects. Apart from recovering examples from the literature, such as the canonical residual finite-state automaton and the \'atomaton, we discover a new canonical automaton for a regular language by relating the free vector space monad over the two element field to the neighbourhood monad. Finally, we show that every regular language satisfying a suitable property parametric in two monads admits a size-minimal succinct acceptor.