Monadic Monadic Second Order Logic

TL;DR

This paper investigates the conditions under which regular languages recognized by monads in the category of sets are closed under surjective letter-to-letter homomorphisms, expanding understanding of monadic structures in formal language theory.

Contribution

It provides sufficient conditions for monads to ensure closure of regular languages under homomorphisms and offers numerous examples and counterexamples.

Findings

Identifies conditions for monads to preserve language closure

Provides examples of monads satisfying and not satisfying these conditions

Enhances understanding of monadic structures in regular language recognition

Abstract

One of the main reasons for the correspondence of regular languages and monadic second-order logic is that the class of regular languages is closed under images of surjective letter-to-letter homomorphisms. This closure property holds for structures such as finite words, finite trees, infinite words, infinite trees, elements of the free group, etc. Such structures can be modelled using monads. In this paper, we study which structures (understood via monads in the category of sets) are such that the class of regular languages (i.e. languages recognized by finite algebras) are closed under direct images of surjective letter-to-letter homomorphisms. We provide diverse sufficient conditions for a monad to satisfy this property. We also present numerous examples of monads, including positive examples that do not satisfy our sufficient conditions, and counterexamples where the closure property fails.