Automatic constraint satisfaction problem

2026-04-21

AI Summary

This work investigates infinite constraint satisfaction problems compactly represented by finite automata (AutCSP), aiming to efficiently solve instances encoded with exponential succinctness. By generalizing Schaefer’s dichotomy theorem to AutCSP over the Boolean domain, the paper introduces a polynomial-time decidable algorithm based on automaton polymorphisms. Integrating finite automata theory, polymorphism analysis, and algebraic techniques, the study achieves an efficient procedure for verifying polymorphisms in AutCSP and establishes refined complexity classifications for several AutCSP classes. These results substantially extend the tractable frontier of large-scale constraint problems, enabling efficient solvability for a broader range of exponentially compressed instances.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study constraint satisfaction problems (CSPs) where the constraint languages are defined by finite automata, giving rise to automata-based CSPs. The key notion is the concept of Automatic Constraint Satisfaction Problem ($AutCSP$), where constraint languages and instances are specified by finite automata. The $AutCSP$ captures infinite yet finitely describable sets of relations, enabling concise representations of complex constraints. Studying the complexity of the $AutCSP$s illustrates the interplay between classical CSPs, automata, and logic, sharpening the boundary between tractable and intractable constraints. We show that checking whether an operation is a polymorphism of such a language can be done in polynomial time. Building on this, we establish several complexity classification results for the $AutCSP$. In particular, we prove that Schaefer's Dichotomy Theorem extends to the $AutCSP$ over the Boolean domain, and we provide algorithms that decide tractability of some classes of $AutCSP$s over arbitrary finite domains via automatic polymorphisms. An important part of our work is that our polynomial-time algorithms run on $AutCSP$ instances that can be exponentially more succinct than their standard CSP counterparts.

Problem

• Constraint Satisfaction Problem
• Finite Automata
• Computational Complexity
• Polymorphism
• Dichotomy Theorem

Innovation

• Automatic Constraint Satisfaction Problem
• finite automata
• polymorphisms
• Schaefer's Dichotomy Theorem
• succinct representation