Quantifiers on languages and codensity monads

TL;DR

This paper develops a general framework using codensity monads and duality theory to recognize languages with added quantifiers, extending topo-algebraic techniques beyond regular languages.

Contribution

It introduces a new construction for recognizers with quantifiers and proves a Reutenauer-type theorem using measure-theoretic characterizations of profinite monads.

Findings

General construction for quantifier-based recognizers

Reutenauer-type theorem established

Measure-theoretic characterization of profinite monads

Abstract

This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.