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Network Satisfaction Problems Solved by k-Consistency

Manuel Bodirsky, Simon Knäuer

TL;DR

This work addresses whether the network satisfaction problem for finite relation algebras can be solved by the $k$-consistency procedure. It proves undecidability in general, while giving a practical, polynomial-time criterion for a large, important subclass: finite symmetric relation algebras with a normal representation and a flexible atom, via a Siggers-type condition and a binarisation technique that leverages finite binary conservative CSP results. The authors connect NSP to CSP through normal representations, deploy universal-algebraic tools (polymorphisms, atom structures, conservative clones), and derive a concrete tractability boundary: NSP is solvable by $(4,6)$-consistency in the tractable class and NP-complete otherwise. These results yield both a robust negative result about the meta-decidability of NSP and a sharp positive dichotomy for a broad, practically relevant subclass, with implications for qualitative spatial-temporal reasoning and related CSP frameworks.

Abstract

We show that the problem of deciding for a given finite relation algebra A whether the network satisfaction problem for A can be solved by the k-consistency procedure, for some natural number k, is undecidable. For the important class of finite relation algebras A with a normal representation, however, the decidability of this problem remains open. We show that if A is symmetric and has a flexible atom, then the question whether NSP(A) can be solved by k-consistency, for some natural number k, is decidable (even in polynomial time in the number of atoms of A). This result follows from a more general sufficient condition for the correctness of the k-consistency procedure for finite symmetric relation algebras. In our proof we make use of a result of Alexandr Kazda about finite binary conservative structures.

Network Satisfaction Problems Solved by k-Consistency

TL;DR

This work addresses whether the network satisfaction problem for finite relation algebras can be solved by the -consistency procedure. It proves undecidability in general, while giving a practical, polynomial-time criterion for a large, important subclass: finite symmetric relation algebras with a normal representation and a flexible atom, via a Siggers-type condition and a binarisation technique that leverages finite binary conservative CSP results. The authors connect NSP to CSP through normal representations, deploy universal-algebraic tools (polymorphisms, atom structures, conservative clones), and derive a concrete tractability boundary: NSP is solvable by -consistency in the tractable class and NP-complete otherwise. These results yield both a robust negative result about the meta-decidability of NSP and a sharp positive dichotomy for a broad, practically relevant subclass, with implications for qualitative spatial-temporal reasoning and related CSP frameworks.

Abstract

We show that the problem of deciding for a given finite relation algebra A whether the network satisfaction problem for A can be solved by the k-consistency procedure, for some natural number k, is undecidable. For the important class of finite relation algebras A with a normal representation, however, the decidability of this problem remains open. We show that if A is symmetric and has a flexible atom, then the question whether NSP(A) can be solved by k-consistency, for some natural number k, is decidable (even in polynomial time in the number of atoms of A). This result follows from a more general sufficient condition for the correctness of the k-consistency procedure for finite symmetric relation algebras. In our proof we make use of a result of Alexandr Kazda about finite binary conservative structures.
Paper Structure (19 sections, 26 theorems, 23 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 26 theorems, 23 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relation Algebras
  4. The Network Satisfaction Problem
  5. Normal Representations and Constraint Satisfaction Problems
  6. The Universal-Algebraic Approach
  7. Polymorphisms
  8. Atom Structures
  9. Conservative Clones
  10. The Consistency Procedure
  11. The Undecidability of RBCP, CON, and PC
  12. Tractability via -Consistency
  13. The Atom Structure
  14. The Binarisation
  15. No Affine Edges in the Atom Structure
  16. ...and 4 more sections

Key Result

Lemma 2.8

Let $\mathop{\mathrm{\mathbf{A}}}\nolimits_1$ and $\mathop{\mathrm{\mathbf{A}}}\nolimits_2$ be relation algebras. Then the following holds:

Figures (5)

  • Figure 1: Multiplication table of the point algebra $\mathbf{P}$.
  • Figure 2: The statement of Lemma \ref{['lem:taylor']}. The red shape means $(a,b,c)\notin R$, the black arrow means $(a,a,b)\notin R$.
  • Figure 3: The statement of Lemma \ref{['lem:no-cycle']}. The blue shape means $(a',b,c)\in R$, the crossed-out red arrow means $(a',a)$ is not a semilattice edge.
  • Figure 4: Multiplication table of the relation algebra $\mathbf{K}$.
  • Figure 5: Multiplication table of the relation algebra $\mathbf{C}$.

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.8
  • proof
  • ...and 59 more