Transparent Long Proofs: A First PCP Theorem for $\mbox{NP}_{\mathbb R}$

Abstract

We introduce and study the notion of probabilistically checkable proofs for real number algorithms. Our starting point is the computational model of Blum, Shub, and Smale and the real analogue $\mbox{NP}_{\mathbb R}$ of $\mbox{NP}$ in that model. Our main result is, to the best of our knowledge, the first PCP theorem for $\mbox{NP}_{\mathbb R}$. It states $\\mbox{NP}_{\mathbb R} \subseteq \mbox{PCP}_{\mathbb R}(poly,O(1)).$ The techniques used extend ideas from [7] for self-testing and -correcting certain functions over so-called rational domains to more general domains over the real numbers. Thus, independently from real number complexity theory, the paper can be seen as a contribution to constructing self testers and correctors for linear functions over real domains.