Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts

TL;DR

The paper studies the expressive power of query algorithms that decide classes of finite relational structures using only homomorphism information, comparing left versus right queries, adaptive versus non-adaptive strategies, and running over the naturals ($\\mathbb{N}$) or the Boolean semiring ($\\mathbb{B}$). A central technical tool is a cycle-symmetry lemma that yields a simple formula for $\\hom(A, m\\cdot \\mathsf{C}_n)$ in terms of the invariant $\\gamma(A)$, enabling precise indistinguishability results and lower bounds. The authors establish sharp separations: over $\\mathbb{N}$, many signatures (notably with a binary or higher-arity relation) admit classes not decidable by any bounded adaptive left algorithm, while unbounded adaptivity recovers all classes; conversely, adaptive right queries collapse to a two-query bound. They also show that adaptive right queries over $\\mathbb{B}$ enjoy a topology-based dichotomy (clopen sets) and relate unbounded adaptivity to FO-definability, CSPs, and Datalog via upper envelopes, highlighting deep connections between homomorphism counts, logic, and database queries.

Abstract

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.

Figures (1)

• Figure 1: Summary of our results regarding the relative expressive power of different types of left query algorithms over $\mathbb{N}$. Here, $\mathfrak{N}_k$, $\mathfrak{A}k$, and $\mathfrak{A}{unb}$ denote the collections of classes of structures that admit a non-adaptive left $k$-query algorithm, an adaptive left $k$-query algorithm, and an adaptive left unbounded query algorithm, respectively. No inclusions hold other than those depicted. In particular, $\mathfrak{A}_2\not\subseteq \bigcup_k\mathfrak{N}_k$ and $\mathfrak{N}_{k+1} \nsubseteq \mathfrak A_k$.

Theorems & Definitions (13)

• Definition 1
• Lemma 2: Corollary 1.17 in hell2004graphs
• Definition 3
• Proposition 3
• Definition 4
• Example 5
• Theorem 6: Corollary 5.6 in ten2024homomorphism
• Definition 7
• Example 8
• Example 9