🤖 AI Summary
This work investigates the maximal separations among classical combinatorial complexity measures—such as sensitivity, block sensitivity, certificate complexity, decision tree depth, and randomized decision tree complexity—within the class of transitive Boolean functions. Addressing the limitation that existing constructions (e.g., pointer functions and cheat-sheet functions) lack transitive symmetry, we systematically adapt these functions into new families preserving transitivity, leveraging group-theoretic modeling and symmetry-preserving constructions. We rigorously analyze the resulting complexity behavior under transitivity constraints. Our results achieve asymptotically optimal separations for multiple key measure pairs—e.g., $s(f)$ vs. $bs(f)$, $C(f)$ vs. $D(f)$, and $R(f)$ vs. $D(f)$—within the transitive function class. This resolves a long-standing open problem in the study of complexity hierarchies under symmetry constraints and substantially extends the frontiers of Boolean function complexity theory.
📝 Abstract
The role of symmetry in Boolean functions $f:{0,1}^n o {0,1}$ 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 $f:{0,1}^n o {0,1}$ 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.
