Separations between Combinatorial Measures for Transitive Functions

Article Type

Research Article

Publication Title

Discrete Mathematics and Theoretical Computer Science

Abstract

The role of symmetry in Boolean functions f : {0, 1}n → {0, 1} has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of Sn, is an important class of functions in the study of Boolean functions. A function f : {0, 1}n → {0, 1} is called transitive (or weakly-symmetric) if there exists a transitive sub-group G of Sn such that f is invariant under the action of G. In other words, the value of a transitive function remains unchanged even after the input bits of f are moved around according to some permutation σ ∈ G. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. This work investigates relations and separations between various complexity measures for the class of transitive functions. A class of functions called “pointer functions” is well known for demonstrating several state-of-the-art separations for general Boolean functions. The main contribution of this work is to extend this technique to transitive functions, constructing new functions that preserve the structural properties of pointer functions while being transitive. In particular, this allows us to show separations between query complexity and other measures for the class of transitive functions. Our results advance the understanding of the transitivity of Boolean functions and highlight the utility of certain transitive groups, which may be of independent interest in mathematics and theoretical computer science. A comprehensive summary of the relationships between combinatorial measures for transitive functions is presented, modeled after the known table for general Boolean functions.

DOI

10.46298/dmtcs.11133

Publication Date

1-1-2025

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