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

Johanna Brunar, Michael Pinsker, Moritz Schöbi

TL;DR

The paper addresses the complexity of temporal constraint languages not omni-expressive under the Bodirsky-Pinsker lens, showing that such languages interpret only limited graphs/hypergraphs and thus gain strong algebraic invariants. It develops a uniform framework (via chasing orbits, min-clean tuples, and pseudo-loop constructions) to prove that all not-omni-expressive temporal templates admit 4-ary pseudo-Siggers polymorphisms and related pseudo-loop identities, in particular extending known finite-model results to the ω-categorical temporal setting. The key methodological advance is a decomposition into pp-interpretability and symmetry-driven reductions that yield pseudo-loop conditions across the board, with 2-transitivity shown to be essential for certain hypergraph cases. Collectively, the results support a robust algebraic signature for non-omni-expressive temporal CSPs and bolster the conjectural reach of pseudo-Siggers-type identities in broader Bodirsky-Pinsker contexts.

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.

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

TL;DR

The paper addresses the complexity of temporal constraint languages not omni-expressive under the Bodirsky-Pinsker lens, showing that such languages interpret only limited graphs/hypergraphs and thus gain strong algebraic invariants. It develops a uniform framework (via chasing orbits, min-clean tuples, and pseudo-loop constructions) to prove that all not-omni-expressive temporal templates admit 4-ary pseudo-Siggers polymorphisms and related pseudo-loop identities, in particular extending known finite-model results to the ω-categorical temporal setting. The key methodological advance is a decomposition into pp-interpretability and symmetry-driven reductions that yield pseudo-loop conditions across the board, with 2-transitivity shown to be essential for certain hypergraph cases. Collectively, the results support a robust algebraic signature for non-omni-expressive temporal CSPs and bolster the conjectural reach of pseudo-Siggers-type identities in broader Bodirsky-Pinsker contexts.

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 -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.

Paper Structure

This paper contains 20 sections, 33 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The algebraic approach
  3. Loop lemmata
  4. Our contribution
  5. Is 2-transitivity the new cyclicity?
  6. Preliminaries
  7. Model-theoretic
  8. Graph-theoretic
  9. Algebraic
  10. Temporal constraint languages
  11. Chasing Orbits
  12. Templates with pp
  13. Templates with ll
  14. Proof of the main theorems
  15. Pseudo-loop conditions from pseudo-loops
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathbb G$ be a finite smooth digraph of algebraic length $1$. Either $\mathbb G$ is omni-expressive or $\mathbb G$ contains a loop.

Figures (1)

  • Figure 1: Polymorphisms of temporal constraint languages

Theorems & Definitions (65)

  • Theorem 1.1: BartoKozikNiven
  • Corollary 1.2: KearnesMarkovicMcKenzie
  • Theorem 1.2
  • Corollary 1.2
  • Theorem 1.2
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Bodirsky-Kára classification bodirsky_complexity_2010
  • Definition 2.4
  • ...and 55 more