Lov\'asz Theorems for Modal Languages

TL;DR

This paper extends Lovász's theorem to characterize equivalences in various modal logics through homomorphism counts on labeled transition systems, linking algebraic counting methods with modal logical equivalences.

Contribution

It identifies classes of labeled transition systems where homomorphism-count indistinguishability captures modal logic equivalences, extending Lovász's theorem beyond finite structures.

Findings

Homomorphism counts characterize positive-existential modal logic equivalence.

Homomorphism counts capture graded and hybrid modal logic equivalences.

Basic modal logic equivalence is not characterized by homomorphism counts.

Abstract

A famous result due to Lov\'{a}sz states that two finite relational structures $M$ and $N$ are isomorphic if, and only if, for all finite relational structures $T$, the number of homomorphisms from $T$ to $M$ is equal to the number of homomorphisms from $T$ to $N$. Since first-order logic (FOL) can describe finite structures up to isomorphism, this can be interpreted as a characterization of FOL-equivalence via homomorphism-count indistinguishability with respect to the class of finite structures. We identify classes of labeled transition systems (LTSs) such that homomorphism-count indistinguishability with respect to these classes, where "counting" is done within an appropriate semiring structure, captures equivalence with respect to positive-existential modal logic, graded modal logic, and hybrid logic, as well as the extensions of these logics with either backward or global modalities. Our positive results apply not only to finite structures, but also to certain well-behaved infinite structures. We also show that equivalence with respect to positive modal logic and equivalence with respect to the basic modal language are not captured by homomorphism-count indistinguishability with respect to any class of LTSs, regardless of which semiring is used for counting.