Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity

TL;DR

This paper explores a novel approach to Boolean complexity lower bounds using cohomology and Grothendieck topologies, aiming to connect topological invariants with circuit depth complexity.

Contribution

It introduces a cohomological framework for modeling Boolean function complexity and proposes bounds for logical operations within this framework, offering new tools for complexity theory.

Findings

Cohomological complexity modeled by Ext groups correlates with circuit depth.

Proposed bounds for AND and negation operations using virtual zero extensions and duality.

Framework shows stability under pullbacks and base change.

Abstract

This paper is motivated by questions such as P vs. NP and other questions in Boolean complexity theory. We describe an approach to attacking such questions with cohomology, and we show that using Grothendieck topologies and other ideas from the Grothendieck school gives new hope for such an attack. We focus on circuit depth complexity, and consider only finite topological spaces or Grothendieck topologies based on finite categories; as such, we do not use algebraic geometry or manifolds. Given two sheaves on a Grothendieck topology, their "cohomological complexity" is the sum of the dimensions of their Ext groups. We seek to model the depth complexity of Boolean functions by the cohomological complexity of sheaves on a Grothendieck topology. We propose that the logical AND of two Boolean functions will have its corresponding cohomological complexity bounded in terms of those of the two functions using ``virtual zero extensions.'' We propose that the logical negation of a function will have its corresponding cohomological complexity equal to that of the original function using duality theory. We explain these approaches and show that they are stable under pullbacks and base change. It is the subject of ongoing work to achieve AND and negation bounds simultaneously in a way that yields an interesting depth lower bound.