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.

翻译：Lovász的一个著名结论表明：两个有限关系结构$M$和$N$同构当且仅当对所有有限关系结构$T$，从$T$到$M$的同态个数等于从$T$到$N$的同态个数。由于一阶逻辑（FOL）能描述同构下的有限结构，该结论可被阐释为：在有限结构类上的同态计数不可区分性刻画了FOL等价性。本文识别出带标号迁移系统（LTS）的若干类，使得利用适当半环结构进行计数的同态计数不可区分性能捕获：正存在模态逻辑、分级模态逻辑、混合逻辑，以及这些逻辑添加逆向或全局模态的扩展。我们正面结果的新颖之处在于：不仅适用于有限结构（如先前Lovász型定理），也适用于具良好性质的无限结构。我们还证明：正模态逻辑等价性和基本模态语言等价性无法被任何LTS类上的同态计数不可区分性所刻画——无论使用何种半环进行计数。