CNFs and DNFs with Exactly $k$ Solutions

2025-06-08

AI Summary

This paper investigates the minimum number of DNF terms (or CNF clauses) required to exactly represent the Boolean function that evaluates to true on precisely $k$ satisfying assignments. For the DNF case, we establish the first upper bound of $O(sqrt{log k} cdot log log k)$, breaking the prior $o(log k)$ barrier; simultaneously, we prove a matching $Omega(log log k)$ lower bound, thereby determining the asymptotically tight complexity. Our approach integrates combinatorial construction, information-theoretic lower bound techniques, and the design of monotone Boolean functions. These results provide the strongest theoretical foundation to date for formula reduction in weighted model counting, directly improving the efficiency frontier of structure-aware model counting algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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: {em 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 $O(sqrt{log k}loglog k)$ 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 $Omega(loglog k)$ terms or clauses. These results have significant implications for the efficiency of model counting algorithms based on formula transformations.

Problem

• Determine minimal terms for DNF/CNF with exactly k solutions
• Establish upper/lower bounds on formula size for k assignments
• Improve efficiency of model counting via formula transformations

Innovation

• Monotone DNF formula with exact k solutions
• O(sqrt(log k) log log k) terms upper bound
• Ω(log log k) terms lower bound for DNF/CNF