Homomorphism Problems in Graph Databases and Automatic Structures

2025-10-08

AI Summary

This work investigates the decidability and computational complexity of the homomorphism problem for graph databases and automatic structures. For graph databases, we devise a minimization algorithm for conjunctive regular path queries—optimized simultaneously with respect to both atom count and treewidth—and establish its decidability alongside an efficient implementation. For automatic structures—i.e., infinite structures defined by finite automata—we prove a dichotomy theorem for the homomorphism problem’s decidability and, for the first time, integrate pseudocluster algebraic language theory into logical definability analysis, uncovering a deep connection between such algebraic frameworks and homomorphism decidability. Innovatively, we unify the modeling of finite-graph query evaluation and infinite-structure constraint solving, thereby pinpointing precise complexity bounds—ranging from PTIME to undecidability—for the homomorphism problem across multiple classes of automatic structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

This thesis investigates the central role of homomorphism problems (structure-preserving maps) in two complementary domains: database querying over finite, graph-shaped data, and constraint solving over (potentially infinite) structures. Building on the well-known equivalence between conjunctive query evaluation and homomorphism existence, the first part focuses on conjunctive regular path queries, a standard extension of conjunctive queries that incorporates regular-path predicates. We study the fundamental problem of query minimization under two measures: the number of atoms (constraints) and the tree-width of the query graph. In both cases, we prove the problem to be decidable, and provide efficient algorithms for a large fragment of queries used in practice. The second part of the thesis lifts homomorphism problems to automatic structures, which are infinite structures describable by finite automata. We highlight a dichotomy, between homomorphism problems over automatic structures that are decidable in non-deterministic logarithmic space, and those that are undecidable (proving to be the more common case). In contrast to this prevalence of undecidability, we then focus on the language-theoretic properties of these structures, and show, relying on a novel algebraic language theory, that for any well-behaved logic (a pseudovariety), whether an automatic structure can be described in this logic is decidable.

Problem

• Minimizing conjunctive regular path queries for graph databases efficiently
• Investigating decidability of homomorphism problems over infinite automatic structures
• Determining logic describability of automatic structures using algebraic theory

Innovation

• Query minimization algorithms for conjunctive regular path queries
• Decidability proofs for homomorphism problems in automatic structures
• Novel algebraic language theory for logic description decidability