Quantifiers on languages and codensity 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.

30 pages. Presentation improved and details of several proofs added. The main results are unchanged

References in corpus (2)

• Codensity, profiniteness and algebras of semiring-valued measures
• Profinite Monads, Profinite Equations, and Reiterman's Theorem

Cited by in corpus (3)

• Codensity, profiniteness and algebras of semiring-valued measures
• A duality theoretic view on limits of finite structures
• A duality theoretic view on limits of finite structures: Extended version