Separations between Combinatorial Measures for Transitive Functions

TL;DR

This paper investigates the differences between various combinatorial complexity measures for transitive Boolean functions, constructing specific examples to demonstrate these separations, which were previously known only for non-transitive functions.

Contribution

The authors modify existing non-transitive functions to create transitive functions that exhibit similar separations between combinatorial measures.

Findings

Constructed transitive functions demonstrating measure separations

Extended known separations from general to transitive functions

Provided methods to modify functions to achieve transitivity

Abstract

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class of functions in the study of Boolean functions. A function $f:\{0,1\}^n \to \{0,1\}$ is called transitive (or weakly-symmetric) if there exists a transitive group $G$ of $S_n$ such that $f$ is invariant under the action of $G$ - that is the function value remains unchanged even after the bits of the input of $f$ are moved around according to some permutation $\sigma \in G$. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. In this work, we study transitive functions in light of several combinatorial measures. We look at the maximum separation between various pairs of measures for transitive functions. Such study for general Boolean functions has been going on for past many years. The best-known results for general Boolean functions have been nicely compiled by Aaronson et. al (STOC, 2021). The separation between a pair of combinatorial measures is shown by constructing interesting functions that demonstrate the separation. But many of the celebrated separation results are via the construction of functions (like "pointer functions" from Ambainis et al. (JACM, 2017) and "cheat-sheet functions" Aaronson et al. (STOC, 2016)) that are not transitive. Hence, we don't have such separation between the pairs of measures for transitive functions. In this paper we show how to modify some of these functions to construct transitive functions that demonstrate similar separations between pairs of combinatorial measures.