A study of distributional complexity measures for Boolean functions

TL;DR

This paper explores distributional complexity measures for Boolean functions, introducing new concepts and connecting recent percolation theory developments with complexity analysis in theoretical computer science.

Contribution

It studies distributional complexity measures, introduces local witness complexity, and links percolation theory with Boolean function complexity.

Findings

Distributional complexity measures differ from worst-case measures.

New local witness complexity measure is introduced.

Connections between percolation theory and Boolean function complexity.

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.