Transparent Long Proofs: A First PCP Theorem for NPR

Abstract

We introduce and study the notion of probabilistically checkable proofs (PCP) for real number algorithms. Our starting point is the computational model of Blum, Shub, and Smale (BSS) and the real analogue NPR of NP in that model. We define in a straightforward manner verifiers as well as complexity classes PCPR(r(n),q(n)) for the BSS model. Our main result is, to the best of our knowledge, the first PCP theorem for NPR. It states that each problem in NPR has transparent long proofs, i.e.,NPR \subseteq PCPR(poly,1), where poly denotes the class of univariate polynomial functions. The techniques used extend ideas from [12] for self-testing and self-correcting certain functions over so-called rational domains to more general domains over the real numbers. The latter arise from the particular NPR-complete problem for which we construct a verifier of the required form.