A Dichotomy Theorem for Automatic Structures

2026-02-20

AI Summary

This study investigates the computational complexity of the homomorphism problem for automatic structures, aiming to delineate its decidability boundary. Focusing on (possibly infinite) automatic structures presented by finite automata, the work integrates techniques from automatic structure theory, finite duality analysis, first-order definability, and regular homomorphism modeling to establish a sharp dichotomy: the problem is either decidable in nondeterministic logarithmic space (NL) or undecidable. The central contribution lies in characterizing the decidable cases precisely as those where the target structure exhibits finite duality. This characterization uniformly applies to both the general homomorphism problem and its variant requiring homomorphisms to be regular, thereby demonstrating that both formulations share the same decidability criterion.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Structured Learning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The field of constraint satisfaction problems (CSPs) studies homomorphism problems between relational structures where the target structure is fixed. Classifying the complexity of these problems has been a central quest of the field, notably when both sides are finite structures. In this paper, we study the generalization where the input is an automatic structure -- potentially infinite, but describable by finite automata. We prove a striking dichotomy: homomorphism problems over automatic structures are either decidable in non-deterministic logarithmic space (NL), or undecidable. We show that structures for which the problem is decidable are exactly those with finite duality, which is a classical property of target structures asserting that the existence of a homomorphism into them can be characterized by the absence of a finite set of obstructions in the source structure. Notably, this property precisely characterizes target structures whose homomorphism problem is definable in first-order logic, which is well-known to be decidable over automatic structures. We also consider a natural variant of the problem. While automatic structures are finitely describable, homomorphisms from them into finite structures need not be. This leads to the notion of regular homomorphism, where the homomorphism itself must be describable by finite automata. Remarkably, we prove that this variant exhibits the same dichotomy, with the same characterization for decidability.

Problem

• constraint satisfaction problems
• automatic structures
• homomorphism
• decidability
• finite duality

Innovation

• automatic structures
• dichotomy theorem
• finite duality
• regular homomorphism
• constraint satisfaction problems