An Algebraic Proof of the Real Number PCP Theorem

Abstract

The PCP theorem has recently been shown to hold as well in the real number model of Blum, Shub, and Smale [3]. The proof given there structurally closely follows the proof of the original PCP theorem by Dinur [7]. In this paper we show that the theorem also can be derived using algebraic techniques similar to those employed by Arora et al. [1, 2] in the first proof of the PCP theorem. This needs considerable additional efforts. Due to severe problems when using low-degree algebraic polynomials over the reals as codewords for one of the verifiers to be constructed, we work with certain trigonometric polynomials. This entails the necessity to design new segmentation procedures in order to obtain well structured real verifiers appropriate for applying the classical technique of verifier composition.

We believe that designing as well an algebraic proof for the real PCP theorem on one side leads to interesting questions in real number complexity theory and on the other sheds light on which ingredients are necessary in order to prove an important result like the PCP theorem in different computational structures.

M. Baartse, K. Meer—Supported under projects ME 1424/7-1 and ME 1424/7-2 by the Deutsche Forschungsgemeinschaft DFG. We gratefully acknowledge the support.