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A Characterization of Efficiently Compilable Constraint Languages

Authors Christoph Berkholz , Stefan Mengel , Hermann Wilhelm

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Author Details

Christoph Berkholz
  • Technische Universität Ilmenau, Germany
Stefan Mengel
  • Univ. Artois, CNRS, Centre de Recherche en Informatique de Lens (CRIL), France
Hermann Wilhelm
  • Technische Universität Ilmenau, Germany

Funding

  • Berkholz, Christoph: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 414325841.
  • Mengel, Stefan: This work has been partially supported by the ANR project EQUUS ANR-19-CE48-0019.
  • Wilhelm, Hermann: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 414325841.

Cite AsGet BibTex

Christoph Berkholz, Stefan Mengel, and Hermann Wilhelm. A Characterization of Efficiently Compilable Constraint Languages. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.11

Abstract

A central task in knowledge compilation is to compile a CNF-SAT instance into a succinct representation format that allows efficient operations such as testing satisfiability, counting, or enumerating all solutions. Useful representation formats studied in this area range from ordered binary decision diagrams (OBDDs) to circuits in decomposable negation normal form (DNNFs). While it is known that there exist CNF formulas that require exponential size representations, the situation is less well studied for other types of constraints than Boolean disjunctive clauses. The constraint satisfaction problem (CSP) is a powerful framework that generalizes CNF-SAT by allowing arbitrary sets of constraints over any finite domain. The main goal of our work is to understand for which type of constraints (also called the constraint language) it is possible to efficiently compute representations of polynomial size. We answer this question completely and prove two tight characterizations of efficiently compilable constraint languages, depending on whether target format is structured. We first identify the combinatorial property of "strong blockwise decomposability" and show that if a constraint language has this property, we can compute DNNF representations of linear size. For all other constraint languages we construct families of CSP-instances that provably require DNNFs of exponential size. For a subclass of "strong uniformly blockwise decomposable" constraint languages we obtain a similar dichotomy for structured DNNFs. In fact, strong (uniform) blockwise decomposability even allows efficient compilation into multi-valued analogs of OBDDs and FBDDs, respectively. Thus, we get complete characterizations for all knowledge compilation classes between O(B)DDs and DNNFs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
  • Theory of computation → Complexity theory and logic
Keywords
  • constraint satisfaction
  • knowledge compilation
  • dichotomy
  • DNNF

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