Circuit complexity, proof complexity, and polynomial identity testing

Abstract:

 

We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity. 

In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system 

implies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP). 

As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in 

Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent 

versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds 

on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships 

between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various 

proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we 

highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. 

More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. 

We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, 

with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial 

lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open 

question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege 

cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. 

Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic 

circuit complexity to prove lower bounds in proof complexity.

 

This is joint work with Joshua A. Grochow.