The Pattern Basis Approach to Circuit Complexity

TL;DR

This paper introduces a novel pattern basis approach to lowerbound circuit complexity of boolean functions, proposing a formal framework that could potentially overcome natural proofs barriers and aid in proving explicit function bounds.

Contribution

It formalizes the concept of pattern bases in vector spaces of truth table properties and explores their potential to derive circuit complexity lower bounds.

Findings

Proposes a new pattern basis formalization for boolean functions

Shows that certain pattern bases could lead to useful complexity formulas

Speculates on methods to find such bases and their implications

Abstract

We describe and motivate a proposed new approach to lowerbounding the circuit complexity of boolean functions, based on a new formalization of "patterns" as elements of a special basis of the vector space of all truth table properties. We prove that a "pattern basis" with certain properties would lead to a useful complexity formula of a specific form, and speculate on how to find such a basis. This formula might take as long to compute on arbitrary functions as a brute-force search among circuits, thus addressing the natural proofs barrier, but has a form amenable to proving lower bounds for well-understood explicit functions.