Circuit Complexity, Proof Complexity, and Polynomial Identity Testing: The Ideal Proof System.

We introduce a new and natural algebraic proof system, whose complexity measure is essentially the algebraic circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatzë, proof complexity, and (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 ≠ VP). We also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems. In doing so, we highlight the importance of polynomial identity testing (PIT) in proof complexity. In particular, we use PIT to illuminate AC0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. Finally, we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity. Using the algebraic structure of our proof system, we propose a novel route to such lower bounds. Although such lower bounds remain elusive, this proposal should be contrasted with the difficulty of extending AC0[p] circuit lower bounds to AC0[p]-Frege lower bounds.