A query algorithm based on homomorphism counts is a procedure to decide membership for a class of finite relational structures using only homomorphism count queries. A left query algorithm can ask the number of homomorphisms from any structure to the input structure and a right query algorithm can ask the number of homomorphisms from the input structure to any other structure. We systematically compare the expressive power of different types of left or right query algorithms, including non-adaptive query algorithms, adaptive query algorithms that can ask a bounded number of queries, and adaptive query algorithms that can ask an unbounded number of queries. We also consider query algorithms where the homomorphism counting is done over the Boolean semiring $\mathbb{B}$, meaning that only the existence of a homomorphism is recorded, not the precise number of them.

翻译：基于同态计数的查询算法是一种仅通过同态计数查询来判定有限关系结构类成员资格的程序。左查询算法可以询问从任意结构到输入结构的同态数量，而右查询算法可以询问从输入结构到任意其他结构的同态数量。我们系统比较了不同类型左查询或右查询算法的表达能力，包括非自适应查询算法、可进行有限次查询的自适应查询算法，以及可进行无限次查询的自适应查询算法。同时，我们还考察了在布尔半环 $\mathbb{B}$ 上进行同态计数的查询算法，这意味着仅记录同态的存在性，而非精确数量。