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Polynomial definability in constraint languages with few subpowers

Jakub Bulín, Michael Kompatscher

TL;DR

The paper investigates when constraint languages admit short pp-definitions and whether this property aligns with the classical few subpowers condition. It proves a main universal-algebraic result: for algebras with a $k$-edge term whose variety is residually finite, all pp-definable relations have length $O(n^{\max(2,k-1)})$, yielding short pp-definitions in many important cases, including all 3-element domains. The work further links short pp-definitions to the subpower membership problem (SMP), showing NP ∩ co-NP containment and highlighting certificates as a vital tool for potential P-time algorithms. It also discusses extensions beyond residual finiteness, interpretability considerations, and open questions about computability and degree bounds, offering a roadmap for strengthening tractability analyses in CSPs through universal algebraic techniques.

Abstract

A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP($Γ$) can be viewed as the problem of deciding the primitive positive theory of $Γ$, and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages $Γ$ is characterized by having few subpowers, that is, the number of $n$-ary relations pp-definable from $Γ$ is bounded by $2^{p(n)}$ for some polynomial $p(n)$. In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to $Γ$ having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers.

Polynomial definability in constraint languages with few subpowers

TL;DR

The paper investigates when constraint languages admit short pp-definitions and whether this property aligns with the classical few subpowers condition. It proves a main universal-algebraic result: for algebras with a -edge term whose variety is residually finite, all pp-definable relations have length , yielding short pp-definitions in many important cases, including all 3-element domains. The work further links short pp-definitions to the subpower membership problem (SMP), showing NP ∩ co-NP containment and highlighting certificates as a vital tool for potential P-time algorithms. It also discusses extensions beyond residual finiteness, interpretability considerations, and open questions about computability and degree bounds, offering a roadmap for strengthening tractability analyses in CSPs through universal algebraic techniques.

Abstract

A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP() can be viewed as the problem of deciding the primitive positive theory of , and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages is characterized by having few subpowers, that is, the number of -ary relations pp-definable from is bounded by for some polynomial . In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers.
Paper Structure (16 sections, 21 theorems, 25 equations, 1 figure)

This paper contains 16 sections, 21 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Few subpowers and short definitions
  4. Polymorphisms
  5. Two canonical examples
  6. Universal-algebraic tools and constructions
  7. Algebras with short pp-definitions
  8. Multi-sorted view
  9. Compact representations
  10. Main result
  11. The subpower membership problem
  12. Discussion
  13. Beyond residual finiteness
  14. Short definitions under pp-interpretability
  15. Efficiently computable short pp-definitions
  16. ...and 1 more sections

Key Result

Theorem 2.1

If $\Gamma$ and $\Delta$ are constraint languages such that $\Delta\subseteq\langle\Gamma\rangle$, then there is a logspace reduction from $\mathop{\mathrm{\mathrm{CSP}}}\nolimits(\Delta)$ to $\mathop{\mathrm{\mathrm{CSP}}}\nolimits(\Gamma)$.

Figures (1)

  • Figure 1: The construction from the proof of Theorem \ref{['theorem:main-result']}

Theorems & Definitions (40)

  • Conjecture 1.1
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proof 1
  • Lemma 2.5
  • Proof 2
  • Lemma 2.6: geiger_closed_1968bodnarcuk_galois_1969jeavons_algebraic_1998
  • Theorem 2.7: berman_varieties_2010idziak_tractability_2010
  • ...and 30 more