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On Extremal Properties of k-CNF: Capturing Threshold Functions

Mohit Gurumukhani, Marvin Künnemann, Ramamohan Paturi

TL;DR

This work analyzes how well $k$-CNF formulas can represent threshold functions by studying the extremal quantity $S(n,t,k)$, the maximum number of weight-$t$ assignments a $t$-admissible $k$-CNF can accept. It reveals deep connections to classical combinatorial problems: for $t=n-k$ the problem is equivalent to Turán-type questions, while in the large-width regime it aligns with Steiner systems, and for the linear-threshold regime it admits an adaptive block construction with a proven optimality for $k=2$. The results yield exact bounds in several regimes and imply circuit lower bounds via $F(n,t,k)$, highlighting a rich interface between CNF representations of threshold functions and extremal combinatorics. The findings advance understanding of the combinatorial structure of $k$-CNF solution spaces and their implications for depth-$3$ circuit complexity.

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 Turán problem. - For $t = αn$ with constant $α$, 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.

On Extremal Properties of k-CNF: Capturing Threshold Functions

TL;DR

This work analyzes how well -CNF formulas can represent threshold functions by studying the extremal quantity , the maximum number of weight- assignments a -admissible -CNF can accept. It reveals deep connections to classical combinatorial problems: for the problem is equivalent to Turán-type questions, while in the large-width regime it aligns with Steiner systems, and for the linear-threshold regime it admits an adaptive block construction with a proven optimality for . The results yield exact bounds in several regimes and imply circuit lower bounds via , highlighting a rich interface between CNF representations of threshold functions and extremal combinatorics. The findings advance understanding of the combinatorial structure of -CNF solution spaces and their implications for depth- circuit complexity.

Abstract

We consider a basic question on the expressiveness of -CNF formulas: How well can -CNF formulas capture threshold functions? Specifically, what is the largest number of assignments (of Hamming weight ) accepted by a -CNF formula that only accepts assignments of weight at least ? Among others, we provide the following results: - While an optimal solution is known for , the problem remains open for . We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for , the problem is exactly the Turán problem. - For with constant , we provide a construction and show its optimality for -CNF. Optimality of the construction for would give improved lower bounds for depth- circuits.
Paper Structure (9 sections, 17 theorems, 13 equations, 1 figure)

This paper contains 9 sections, 17 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Bounds on S(n, t, k)
  5. Bounds for Small Thresholds
  6. Bounds for Intermediate Thresholds
  7. Tight Bounds for S(n, t, 2)
  8. Connection to the Turán Problem
  9. Connection to Steiner Systems

Key Result

Theorem 1

For any $n\ge k \ge 2$, we have that $S(n,n-k,k)= \binom{n}{k} - T(n,k+1,k)$.

Figures (1)

  • Figure 1: Illustrates the implication graph of $F$ in the proof of \ref{['lem:k2-monotone']}. In the transformation from $F$ to $F'$, the black dashed edge is deleted and the yellow dashed edges are added.

Theorems & Definitions (20)

  • Theorem 1
  • Conjecture 2
  • Theorem 3: Optimality for $k=2$, Informal Version
  • Lemma 6: MacWilliamsSloane_1977
  • Lemma 7: Putting the block together
  • Theorem 8: Small Thresholds
  • Lemma 9
  • Theorem 10: Intermediate Thresholds
  • Proposition 11: Optimality Implies Circuit Lower Bounds
  • Theorem 12: Tight Bound for 2-CNFs
  • ...and 10 more