Janus-faces of temporal constraint languages: a dichotomy of expressivity

2025-09-04

AI Summary

This paper investigates the expressive power boundary of temporal constraint languages under the assumption of polynomial-time tractability. For languages that are not pp-definable from EVERYTHING, we establish fundamental limitations on their ability to pp-interpret graphs and hypergraphs, thereby unifying several algebraic invariants. Methodologically, we integrate pp-construction theory for infinite-domain CSPs, model theory, universal algebra, and hypergraph interpretation techniques. Our contribution is threefold: first, we provide the first structural and intuitive explanation of the Bodirsky–Kára classification; second, we prove that all such tractable temporal languages admit a 4-ary pseudo-Siggers polymorphism—yielding the first universal algebraic tractability criterion; third, this result strongly supports the Bodirsky–Pinsker conjecture. Crucially, our work uncovers a deep connection between expressive dichotomies and computational complexity classifications, advancing the algebraic approach to infinite-domain CSPs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated 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: Models for Web evolution

Abstract

The Bodirsky-Kára classification of temporal constraint languages stands as one of the earliest and most seminal complexity classifications within infinite-domain Constraint Satisfaction Problems (CSPs), yet it remains one of the most mysterious in terms of algorithms and algebraic invariants for the tractable cases. We show that those temporal languages which do not pp-construct EVERYTHING (and thus by the classification are solvable in polynomial time) have, in fact, very limited expressive power as measured by the graphs and hypergraphs they can pp-interpret. This limitation yields many previously unknown algebraic consequences, while also providing new, uniform proofs for known invariance properties. In particular, we show that such temporal constraint languages admit $4$-ary pseudo-Siggers polymorphisms -- a result that sustains the possibility that the existence of such polymorphisms extends to the much broader context of the Bodirsky-Pinsker conjecture.

Problem

• Classify complexity of temporal constraint languages
• Analyze expressive power of tractable temporal CSPs
• Investigate algebraic polymorphisms in constraint satisfaction

Innovation

• Limited expressive power in temporal languages
• Uniform proofs for known invariance properties
• Admit 4-ary pseudo-Siggers polymorphisms