arxiv_ml 85% Match Research Paper Theoretical Computer Scientists,Machine Learning Theorists,Researchers in SAT/SMT 19 hours ago

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

graph-neural-networks › graph-learning

📄 Abstract

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.

Key Contributions

This paper significantly improves the sample complexity for learning k-CNF formulas from uniform random solutions. It shows that Valiant's algorithm can learn k-CNFs under local lemma conditions with O(log n) samples and random k-CNFs near the satisfiability threshold with near-optimal sample complexity, establishing new information-theoretic lower bounds.

Business Value

More efficient learning of complex logical formulas can improve automated reasoning, constraint satisfaction solvers, and formal verification tools used in software engineering and hardware design.

Paper Metadata

Innovation Type

Theoretical Improvement

Deployment Feasibility

Theoretical. Focuses on sample complexity bounds, not direct implementation.

Limitations Addressed

High sample complexity ($O(n^k)$) of previous methods for learning CNF formulas from uniform random solutions.

Performance Gains

Reduced sample complexity from $O(n^k)$ to $O(\log n)$ or $\widetilde{O}(n^{\exp(-\sqrt{k})})$ for specific cases.

Technical Tags

k-CNF learninguniform random solutionsBoolean MRFValiant's algorithmLovász local lemmasample complexityinformation theorysatisfiability thresholdcomputational learning theory

Research Topics

Computational Learning TheoryTheoretical Computer ScienceSatisfiability (SAT)Machine Learning TheoryBoolean Functions

Methods & Architectures

Valiant's algorithm analysisInformation-theoretic lower boundsAnalysis under Lovász local lemma conditions

Applications & Tasks

Formal Verification Constraint Satisfaction AI Planning Learning CNF formulasLearning Boolean Markov Random FieldsSample complexity reduction Formula learningModel inferenceSatisfiability checking

Related Fields

Theoretical Computer ScienceMachine Learning TheoryLogicCombinatoricsSatisfiability

Keywords

k-CNFlearningrandom solutionssample complexitylocal lemmaValiant's algorithmBoolean MRFsatisfiabilityinformation theorycomputational learning theory

Academic Context

#Computational Learning Theory#Theoretical Computer Science#Satisfiability (SAT)#Machine Learning Theory#Boolean Functions

Commercial Potential

Target Industries

Software DevelopmentHardware DesignAI Research

Use Case Examples

Efficiently learning complex logical specifications from examples.Improving solvers for constraint satisfaction problems.

Competitive Edge

Provides significant theoretical improvements in sample complexity for a fundamental learning problem.

Resource Requirements

Compute Needs

Theoretical analysis, no direct computation.

Data Requirements

Theoretical samples (uniform random solutions).

Deployment Constraints

N/A (theoretical)

Scalability

N/A (theoretical)

Production Readiness

Maturity Level

Theoretical

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