Explicit Proofs and The Flip

Abstract

This article describes a formal strategy of geometric complexity theory (GCT) to resolve the {\em self referential paradox} in the $P$ vs. $NP$ and related problems. The strategy, called the {\em flip}, is to go for {\em explicit proofs} of these problems. By an explicit proof we mean a proof that constructs proof certificates of hardness that are easy to verify, construct and decode. The main result in this paper says that (1) any proof of the arithmetic implication of the $P$ vs. $NP$ conjecture is close to an explicit proof in the sense that it can be transformed into an explicit proof by proving in addition that arithmetic circuit identity testing can be derandomized in a blackbox fashion, and (2) stronger forms of these arithmetic hardness and derandomization conjectures together imply a polynomial time algorithm for a formidable explicit construction problem in algebraic geometry. This may explain why these conjectures, which look so elementary at the surface, have turned out to be so hard.