A (co)algebraic theory of succinct automata

TL;DR

This paper develops a unified algebraic framework for automata, generalizing classical constructions to various algebraic structures and automata types, revealing new ways to associate succinct automata with languages.

Contribution

It introduces a general construction linking algebraic structures to automata, extending classical determinization to new automata classes using a coalgebraic approach.

Findings

Unified algebraic framework for automata types

Construction transforming deterministic automata into succinct forms

Application to automata with symmetries and weights

Abstract

The classical subset construction for non-deterministic automata can be generalized to other side-effects captured by a monad. The key insight is that both the state space of the determinized automaton and its semantics---languages over an alphabet---have a common algebraic structure: they are Eilenberg-Moore algebras for the powerset monad. In this paper we study the reverse question to determinization. We will present a construction to associate succinct automata to languages based on different algebraic structures. For instance, for classical regular languages the construction will transform a deterministic automaton into a non-deterministic one, where the states represent the join-irreducibles of the language accepted by a (potentially) larger deterministic automaton. Other examples will yield alternating automata, automata with symmetries, CABA-structured automata, and weighted automata.