Algebraic Language Theory with Effects

TL;DR

This work generalizes the classical algebraic language theory from regular languages to languages computed by automata with generic effects modeled by a monad $\mathbb{T}$. It introduces two algebraic recognition modes—(i) $\mathbb{T}$-effectful monoid morphisms into finite, effect-free monoids and (ii) ordinary monoid morphisms into finitely generated $\mathbb{T}$-algebras with a monoid structure—and shows these characterize exactly the languages recognized by finite $\mathbb{T}$-automata under suitable conditions. The framework yields new algebraic characterizations for probabilistic languages (via $\mathcal{D}$), nondeterministic probabilistic languages (via $\mathcal{C}$), and weighted languages over unrestricted semirings (via $\mathcal{S}$), including constructions of syntactic convex monoids and finite presentability results. It lays a foundation for extensions to nominal sets and omega-words and hints at connections to profinite approaches, offering a unifying, algebraic view of effectful language computation with broad applicability.

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.

Figures (4)

• Figure 1: Definition of $\bar{\times}$
• Figure 2: Probabilistic monoid morphism
• Figure 3: A PFA
• Figure 4: Witnesses for $X\mathbin{} X$ being (monoidally) finitely generated

Theorems & Definitions (66)

• Example 2
• Theorem 3: rs59
• Definition 4
• Remark 5
• Remark 6
• Definition 7
• Lemma 8
• Definition 10
• Remark 11
• Definition 12