Lovász Theorems for Modal Languages

A famous result due to Lov\'asz 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. A novelty of our positive results is that they apply not only to finite structures, as with previous Lov\'asz-style theorems, but also to 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.