Algebraic Language Theory with Effects

TL;DR

This paper extends algebraic language theory to automata with effects modeled by monads, establishing new recognition frameworks and applying them to probabilistic and weighted automata.

Contribution

It introduces a general algebraic framework for effectful automata, including probabilistic and weighted automata, using monads and monoid morphisms.

Findings

Characterizes languages recognized by effectful automata via monoid morphisms.

Provides a new algebraic approach to probabilistic finite automata.

Generalizes algebraic recognition to weighted automata over semirings.

Abstract

Regular languages -- the languages accepted by deterministic finite automata -- are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we generalize the correspondence between automata and monoids to automata with generic computational effects given by a monad, providing the foundations of an effectful algebraic language theory. We show that, under suitable conditions on the monad, a language is computable by an effectful automaton precisely when it is recognizable by (1) an effectful monoid morphism into an effect-free finite monoid, and (2) a monoid morphism into a monad-monoid bialgebra whose carrier is a finitely generated algebra for the monad, the former mode of recognition being conceptually completely new. Our prime application is a novel algebraic approach to languages computed by probabilistic finite automata. Additionally, we derive new algebraic characterizations for nondeterministic probabilistic finite automata and for weighted finite automata over unrestricted semirings, generalizing previous results on weighted algebraic recognition over commutative rings.