On Logics and Homomorphism Closure

TL;DR

This paper explores the relationship between predicate logic and homomorphisms, analyzing the computational complexity of homomorphism closure problems across various logical fragments.

Contribution

It provides a detailed investigation of homomorphism closure problems, including membership, homclosedness, and characterizability, across multiple logical fragments.

Findings

Complexity results for homclosure membership problems

Characterization of homclosedness in different logics

Normal forms for homclosed sentences in certain logical fragments

Abstract

Predicate logic is the premier choice for specifying classes of relational structures. Homomorphisms are key to describing correspondences between relational structures. Questions concerning the interdependencies between these two means of characterizing (classes of) structures are of fundamental interest and can be highly non-trivial to answer. We investigate several problems regarding the homomorphism closure (homclosure) of the class of all (finite or arbitrary) models of logical sentences: membership of structures in a sentence's homclosure; sentence homclosedness; homclosure characterizability in a logic; normal forms for homclosed sentences in certain logics. For a wide variety of fragments of first- and second-order predicate logic, we clarify these problems' computational properties.