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CNFs and DNFs with Exactly k Solutions

Authors L. Sunil Chandran , Rishikesh Gajjala , Kuldeep S. Meel

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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • Model counting
  • #SAT
  • Set Systems
  • Combinatorics

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Abstract

Model counting is a fundamental problem that consists of determining the number of satisfying assignments for a given Boolean formula. The weighted variant, which computes the weighted sum of satisfying assignments, has extensive applications in probabilistic reasoning, network reliability, statistical physics, and formal verification. A common approach for solving weighted model counting is to reduce it to unweighted model counting, which raises an important question: What is the minimum number of terms (or clauses) required to construct a DNF (or CNF) formula with exactly k satisfying assignments?
In this paper, we establish both upper and lower bounds on this question. We prove that for any natural number k, one can construct a monotone DNF formula with exactly k satisfying assignments using at most O(√{log k}log log k) terms. This construction represents the first o(log k) upper bound for this problem. We complement this result by showing that there exist infinitely many values of k for which any DNF or CNF representation requires at least Ω(log log k) terms or clauses. These results have significant implications for the efficiency of model counting algorithms based on formula transformations.

Cite As Get BibTex

L. Sunil Chandran, Rishikesh Gajjala, and Kuldeep S. Meel. CNFs and DNFs with Exactly k Solutions. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SAT.2025.9

Author Details

L. Sunil Chandran
  • Indian Institute of Science, Bengaluru, India
Rishikesh Gajjala
  • Indian Institute of Science, Bengaluru, India
Kuldeep S. Meel
  • Georgia Institute of Technology, Atlanta, GA, USA
  • University of Toronto, Canada

Funding

This research was funded in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2024-05956.

References

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