Learning CNF formulas from uniform random solutions in the local lemma regime

Weiming Feng; Xiongxin Yang; Yixiao Yu; Yiyao Zhang

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arXiv · 2511.02487

Learning CNF formulas from uniform random solutions in the local lemma regime

Weiming Feng·Xiongxin Yang·Yixiao Yu·Yiyao Zhang

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Abstract

We study the problem of learning a $n$-variables $k$-CNF formula $\Phi$ from its i.i.d. uniform random solutions, which is equivalent to learning a Boolean Markov random field (MRF) with $k$-wise hard constraints. Revisiting Valiant's algorithm (Commun. ACM'84), we show that it can exactly learn (1) $k$-CNFs with bounded clause intersection size under Lov\'asz local lemma type conditions, from $O(\log n)$ samples; and (2) random $k$-CNFs near the satisfiability threshold, from $\widetilde{O}(n^{\exp(-\sqrt{k})})$ samples. These results significantly improve the previous $O(n^k)$ sample complexity. We further establish new information-theoretic lower bounds on sample complexity for both exact and approximate learning from i.i.d. uniform random solutions.

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BibTeX RIS

Weiming Feng, Xiongxin Yang, Yixiao Yu, Yiyao Zhang. 2025-11-04. Learning CNF formulas from uniform random solutions in the local lemma regime. https://arxiv.org/abs/2511.02487

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