Separations between Combinatorial Measures for Transitive Functions

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 group G of Sn 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 σ ∈ 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 “cheatsheet 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. We summarize our results below: 1. Ambainis et al. (JACM, 2017) constructed several functions, which are modifications of the pointer function in Göös et al. (SICOMP, 2018), to demonstrate separation between various pairs of measure. Base of their functions we construct new transitive functions and whose deterministic query complexity, randomized query complexity, zero-error randomized query complexity, quantum query complexity, degree, and approximate degree are similar to that of the original function. Thus our transitive functions demonstrate similar separations between measures as the original functions of Ambainis et al. 2. We use one of our functions and a trick from Ben-David et al. (ITCS 2017) to construct a transitive function that shows a tight separation between bounded-error quantum query complexity and certificate complexity for transitive functions. 3. Modifying a function by Aaronson et al. (STOC, 2016), we show that quadratic separation between bounded-error quantum query complexity and certificate complexity also holds for transitive functions. 4. Finally, we present a table compiling the known relations between various complexity measures for transitive functions. This is similar to the table compiled by Aaronson et al. (STOC, 2020) for general functions. Indian Statistical Institute, Kolkata. sourav@isical.ac.in Indian Statistical Institute, Kolkata. chandrima_r@isical.ac.in Indian Statistical Institute, Kolkata. manaswi.isi@gmail.com