Query complexity of Boolean functions on the middle slice of the cube

TL;DR

The paper studies deterministic query complexity for Boolean functions on the middle slice of the Boolean cube, formalized via $D_k(f)$ and $E_k(n)=n-D_k(n)$. It introduces a decision-tree counting framework and proves a non-constructive result: there exists a function on the middle slice with $E_{n/2}(n)\le 7$ for all $n$ (and $\le 5$ for $n\ge 100$), answering a question of Byramji, with a monotonicity-based extension to $E_k(n)$. The method hinges on bounding the number of depth-$n-t$ decision trees by a combinatorial function $g(n,k,t)$, leading to existence results and explicit bounds; the paper also discusses a concrete candidate function and connects the problem to discrepancy-type questions via Disc-max-$d$ and related games. These results advance understanding of information requirements for slice-restricted Boolean functions and illuminate links between query complexity and discrepancy theory, while leaving open questions about explicit constructions and constant bounds for broader regimes.

Abstract

We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e., when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that, somewhat surprisingly, have not been studied before.

Theorems & Definitions (24)

• Theorem 1.1: Byramji byramji
• Theorem 1.2: Byramji byramji
• Theorem 1.3
• Lemma 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3