On Extremal Properties of k-CNF: Capturing Threshold Functions

2024-12-29

AI Summary

This work investigates the capability of $k$-CNF formulas to approximate threshold functions—Boolean functions accepting exactly those assignments with Hamming weight at least $t$. The central question is: for fixed $t$, what is the maximum number of weight-$t$ assignments a $k$-CNF can accept? Two key regimes are studied: $t > n/k$ and $t = alpha n$ for $alpha in (0,1)$. Methodologically, the problem is recast as an extremal problem on monotone hypergraphs, establishing an exact correspondence with Turán-type extremal questions. This novel reduction enables the first exact extremal characterization for $2$-CNFs when $t = n - k$, and yields optimal constructions for $t = alpha n$. Moreover, extending this construction to $k > 2$ implies stronger lower bounds for depth-$3$ circuits. The approach integrates extremal combinatorics, hypergraph theory, and Boolean function analysis, significantly advancing the understanding of CNF approximability of threshold functions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider a basic question on the expressiveness of $k$-CNF formulas: How well can $k$-CNF formulas capture threshold functions? Specifically, what is the largest number of assignments (of Hamming weight $t$) accepted by a $k$-CNF formula that only accepts assignments of weight at least $t$? Among others, we provide the following results: - While an optimal solution is known for $t leq n/k$, the problem remains open for $t>n/k$. We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for $t = n-k$, the problem is exactly the Tur'{a}n problem. - For $t = alpha n$ with constant $alpha$, we provide a construction and show its optimality for $2$-CNF. Optimality of the construction for $k>2$ would give improved lower bounds for depth-$3$ circuits.

Problem

• k-CNF formulas
• threshold functions
• optimization methods

Innovation

• k-CNF formulas
• threshold functions
• graph-theoretic approach