Coalgebraic Automata Theory: Basic Results

TL;DR

This paper extends classical automata theory to coalgebraic structures, establishing foundational results for automata on infinite objects and providing constructions for automaton transformations.

Contribution

It generalizes key automata theory results to coalgebras, including closure properties and automaton transformations, for a broad class of set functors.

Findings

Recognizable languages are closed under unions, intersections, and projections.

Nondeterministic automata accept finite coalgebras of the same size as the automaton.

Alternating automata can be transformed into equivalent nondeterministic automata with exponential size bound.

Abstract

We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects. Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.