Block Sensitivity of Minterm-Transitive Functions

TL;DR

This paper investigates the block sensitivity of minterm-transitive Boolean functions, establishing lower bounds and providing improved examples that nearly match these bounds, advancing understanding of their complexity.

Contribution

It extends previous bounds on block sensitivity for minterm-transitive functions and presents an improved example with nearly minimal sensitivity.

Findings

Nonconstant minterm-transitive functions have block sensitivity at least Omega(N^{3/7})

Existing example functions nearly achieve this lower bound

New example function with block sensitivity O(N^{3/7}ln^{1/7}N) is provided

Abstract

Boolean functions with symmetry properties are interesting from a complexity theory perspective; extensive research has shown that these functions, if nonconstant, must have high `complexity' according to various measures. In recent work of this type, Sun gave bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f) = Omega(N^{1/3}), and that there exists such a function for which bs(f) = O(N^{3/7}ln N). His example function belongs to a subclass of transitively invariant functions called the minterm-transitive functions (defined in earlier work by Chakraborty). We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f) = Omega(N^{3/7}). Thus Sun's example function has nearly minimal block sensitivity for this subclass. Second, we give an improved example: a minterm-transitive function for which bs(f) = O(N^{3/7}ln^{1/7}N).