A study of distributional complexity measures for Boolean functions
Laurin Köhler-Schindler, Jeffrey E. Steif
TL;DR
This work develops a comprehensive framework for distributional (average-case) complexity measures of Boolean functions, introducing a new local witness complexity and analyzing their relationships with established deterministic and distributional notions. It provides definitions, hierarchies, and geometric/stoppping-set perspectives that unify subcube partitions and local sets, and studies how these measures behave under composition and under partial information models. Through a suite of classical and percolation-based examples, the authors establish both asymptotic separations and polynomial-relations in the distributional regime, and they introduce generalized algorithmic models (Γ-cost and (p,κ)-cost) to probe the cost of partial information in querying. The paper also outlines conjectures linking composition with parity-type augmentations to potentially separate distributional algorithmic complexity from subcube partition complexity, highlighting practical implications for understanding average-case complexity landscapes in probabilistic settings like percolation.
Abstract
A number of complexity measures for Boolean functions have previously been introduced. These include (1) sensitivity, (2) block sensitivity, (3) witness complexity, (4) subcube partition complexity and (5) algorithmic complexity. Each of these is concerned with "worst-case" inputs. It has been shown that there is "asymptotic separation" between these complexity measures and very recently, due to the work of Huang, it has been established that they are all "polynomially related". In this paper, we study the notion of distributional complexity where the input bits are independent and one considers all of the above notions in expectation. We obtain a number of results concerning distributional complexity measures, among others addressing the above concepts of "asymptotic separation" and being "polynomially related" in this context. We introduce a new distributional complexity measure, local witness complexity, which only makes sense in the distributional context and we also study a new version of algorithmic complexity which involves partial information. Many interesting examples are presented including some related to percolation. The latter connects a number of the recent developments in percolation theory over the last two decades with the study of complexity measures in theoretical computer science.