Copmutationally Sound Proofs

Abstract

This paper puts forward a new notion of a proof based on computational complexity and explores its implications for computation at large. Computationally sound proofs provide, in a novel and meaningful framework, answer to old and new questions in complexity theory. In particular, given a random oracle or a new complexity assumption, they enable us to 1. prove that verifying is easier than deciding for all theorems; 2. provides a quite effective way to prove membership in computationally hard languages (such as C-NP-complete ones); and 3. show that every computation possesses a short certificate vouching its correctness. FInally, if a special type of computationally sound proof exists, we show that Blum's notion of program checking can be meaningfully broadened so as to prove that NP-complete languages are checkable.