When do homomorphism counts help in query algorithms?
Balder ten Cate, Víctor Dalmau, Phokion G. Kolaitis, Wei-Lin Wu
TL;DR
This work investigates when query algorithms based on counting homomorphisms between fixed reference instances and an input instance can determine whether the input satisfies a property. It shows that left query algorithms over the Boolean semiring are exactly the FO-definable properties closed under homomorphic equivalence, and that for such properties, left-query power over $\mathbb{B}$ matches that over $\mathbb{N}$, so counting does not help in this regime. It further characterizes when both left and right query algorithms over $\mathbb{B}$ exist (precisely for Boolean combinations of Berge-acyclic CQs) and ties these notions to CSPs, FO-definability, and homomorphic equivalence. The results illuminate the limits of counting-based approaches in query evaluation and connect to view determinacy, Datalog boundedness, and semantic frameworks over different semirings, while outlining open questions for extensions to other semirings and to right-query analogues.
Abstract
A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring N of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring B, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over B by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over B and left query algorithms over N. In general, there are properties that admit a left query algorithm over N but not over B. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over B if and only if it admits a left query algorithm over N. In other words and rather surprisingly, homomorphism counts over N do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over B and a right query algorithm over B.