On Extremal Properties of k-CNF: Capturing Threshold Functions

Mohit Gurumukhani; Marvin Künnemann; Ramamohan Paturi

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arXiv · 2412.20493

On Extremal Properties of k-CNF: Capturing Threshold Functions

Mohit Gurumukhani·Marvin Künnemann·Ramamohan Paturi

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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.

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BibTeX RIS

Mohit Gurumukhani, Marvin Künnemann, Ramamohan Paturi. 2024-12-29. On Extremal Properties of k-CNF: Capturing Threshold Functions. https://arxiv.org/abs/2412.20493

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