Relationship between circuit complexity and symmetry

TL;DR

This paper explores how the symmetry properties of Boolean circuits, viewed as graphs, relate to computational complexity, providing insights into the P vs NP problem through automorphism analysis.

Contribution

It demonstrates that the disparity in automorphism counts between the entire circuit and its subgraphs implies exponential lower bounds for NP-complete problems, linking symmetry to complexity.

Findings

Small number of automorphisms in circuits suggests exponential lower bounds.

Large subgraph automorphisms contrast with overall circuit automorphisms.

Result supports P≠NP without contradicting pseudorandom function existence.

Abstract

It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us the idea that the small number of automorphisms might be a barrier for a polynomial time solution for NP-complete problems. Here I show that by interpreting a Boolean circuit as a graph, the small number of graph automorphisms and the large number of subgraph automorphisms in the circuit establishes the exponential circuit lower bound for NP-complete problems. As this strategy violates the largeness condition in Natural proof, this result shows that P!=NP without any contradictions to the existence of pseudorandom functions.