skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


This content will become publicly available on April 11, 2026

Title: The Impact of Literal Sorting on Cardinality Constraint Encodings
The effectiveness of satisfiability solvers strongly depends on the quality of the encoding of a given problem into conjunctive normal form. Cardinality constraints are prevalent in numerous problems, prompting the development and study of various types of encoding. We present a novel approach to optimizing cardinality constraint encodings by exploring the impact of literal orderings within the constraints. By strategically placing related literals nearby each other, the encoding generates auxiliary variables in a hierarchical structure, enabling the solver to reason more abstractly about groups of related literals. Unlike conventional metrics such as formula size or propagation strength, our method leverages structural properties of the formula to redefine the roles of auxiliary variables to enhance the solver's learning capabilities. The experimental evaluation on benchmarks from the maximum satisfiability competition demonstrates that literal orderings can be more influential than the choice of the encoding type. Our literal ordering technique improves solver performance across various encoding techniques, underscoring the robustness of our approach.  more » « less
Award ID(s):
2415773
PAR ID:
10629076
Author(s) / Creator(s):
; ; ; ;
Publisher / Repository:
AAAI
Date Published:
Journal Name:
Proceedings of the AAAI Conference on Artificial Intelligence
Volume:
39
Issue:
11
ISSN:
2159-5399
Page Range / eLocation ID:
11327 to 11335
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Berg, Jeremias; Nordström, Jakob (Ed.)
    We present a lightweight reencoding technique that augments propositional formulas containing implicit or explicit exactly-one constraints with auxiliary variables derived from the order encoding. Our approach is based on the observation that many formulas contain clauses where each literal appears only in that clause, and that these unique literal clauses can be replaced by the corresponding sequential counter encoding of exactly-one constraints, which introduces the same variables as the order encoding. We implemented the reencoding in the state-of-the-art SAT solver CaDiCaL with support for proof logging and solution reconstruction. Experiments on SAT Competition benchmarks demonstrate that our technique enables solving dozens of additional formulas. We found that shuffling a formula before reencoding harms performance. To mitigate this issue, we introduce a method that sorts literals within clauses based on the formula structure before applying our reencoding. The same technique also predicts whether reencoding is likely to yield improvements. 
    more » « less
  2. ; ; (, International Conference on Computer Aided Verification (CAV-24))
    Satisfiability (SAT) solvers have been using the same input format for decades: a formula in conjunctive normal form. Cardinality constraints appear frequently in problem descriptions: over 64% of the SAT Competition formulas contain at least one cardinality constraint, while over 17% contain many large cardinality constraints. Allowing general cardinality constraints as input would simplify encodings and enable the solver to handle constraints natively or to encode them using different (and possibly dynamically changing) clausal forms. We modify the modern SAT solver CaDiCaL to handle cardinality constraints natively. Unlike the stronger cardinality reasoning in pseudo-Boolean (PB) or other systems, our incremental approach with cardinality-based propagation requires only moderate changes to a SAT solver, preserves the ability to run important inprocessing techniques, and is easily combined with existing proof-producing and validation tools. Our experimental evaluation on SAT Competition formulas shows our solver configurations with cardinality support consistently outperform other SAT and PB solvers. 
    more » « less
  3. ; (, International Conference on Tools and Algorithms for the Construction and Analysis of Systems)
    Fisman, D.; Rosu, G. (Ed.)
    When augmented with a Pseudo-Boolean (PB) solver, a Boolean satisfiability (SAT) solver can apply apply powerful reasoning methods to determine when a set of parity or cardinality constraints, extracted from the clauses of the input formula, has no solution. By converting the intermediate constraints generated by the PB solver into ordered binary decision diagrams (BDDs), a proof-generating, BDD-based SAT solver can then produce a clausal proof that the input formula is unsatisfiable. Working together, the two solvers can generate proofs of unsatisfiability for problems that are intractable for other proof-generating SAT solvers. The PB solver can, at times, detect that the proof can exploit modular arithmetic to give smaller BDD representations and therefore shorter proofs. 
    more » « less
  4. ; (, International Joint Conference on Artificial Intelligence (IJCAI))
    Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. We complement this by studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification and discuss the interplay between variable/literal and existential/universal quantification. We further identify classes of Boolean formulas and circuits that allow efficient quantification. Literal quantification is more fine-grained than variable quantification, which leads to a refinement of quantified Boolean logic with literal quantification as its primitive. 
    more » « less
  5. ; (, ACM Transactions on Computational Logic)
    In 2006, Biere, Jussila, and Sinz made the key observation that the underlying logic behind algorithms for constructing Reduced, Ordered Binary Decision Diagrams (BDDs) can be encoded as steps in a proof in the extended resolution logical framework. Through this, a BDD-based Boolean satisfiability (SAT) solver can generate a checkable proof of unsatisfiability. Such a proof indicates that the formula is truly unsatisfiable without requiring the user to trust the BDD package or the SAT solver built on top of it. We extend their work to enable arbitrary existential quantification of the formula variables, a critical capability for BDD-based SAT solvers. We demonstrate the utility of this approach by applying a BDD-based solver, implemented by extending an existing BDD package, to several challenging Boolean satisfiability problems. Our results demonstrate scaling for parity formulas as well as the Urquhart, mutilated chessboard, and pigeonhole problems far beyond that of other proof-generating SAT solvers. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email