Generators and Bases for Monadic Closures

TL;DR

This paper advances the categorical theory of automata by developing methods to construct minimal bialgebras and canonical generators over monads, enhancing understanding of non-deterministic automata.

Contribution

It introduces a categorical framework for deriving minimal bialgebras and explores the theory of generators and bases for monad algebras, extending prior automata theory.

Findings

Minimal bialgebras can be obtained via monads on subobject categories.

Develops a theory of generators and bases for monad algebras.

Provides a categorical approach to automata minimization.

Abstract

It is well-known that every regular language admits a unique minimal deterministic acceptor. Establishing an analogous result for non-deterministic acceptors is significantly more difficult, but nonetheless of great practical importance. To tackle this issue, a number of sub-classes of non-deterministic automata have been identified, all admitting canonical minimal representatives. In previous work, we have shown that such representatives can be recovered categorically in two steps. First, one constructs the minimal bialgebra accepting a given regular language, by closing the minimal coalgebra with additional algebraic structure over a monad. Second, one identifies canonical generators for the algebraic part of the bialgebra, to derive an equivalent coalgebra with side effects in a monad. In this paper, we further develop the general theory underlying these two steps. On the one hand, we show that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on an appropriate category of subobjects. On the other hand, we explore the abstract theory of generators and bases for algebras over a monad.