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

TL;DR

This paper investigates the query complexity of Boolean functions on the middle slice of the hypercube, revealing functions that require nearly all input bits to be queried, and addresses an open question in the field.

Contribution

It demonstrates the existence of Boolean functions with high query complexity on the middle slice, answering a previously open question and proposing a candidate function.

Findings

Existence of functions requiring all but 7 queries on the middle slice

Addresses an open question by Byramji about query complexity

Introduces a candidate function with high query complexity

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.