Relationship between circuit complexity and symmetry

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.