Interactive proof systems: Provers, rounds, and error bounds

Abstract

We introduce generalized multi-prover interactive proof systems and the associated polynomial time complexity classes IP(m, r, 1/h), which depend on the number m of provers, number r of rounds and the value 1/h by which the error is bounded away from one half. In this denotation the class IP(m, r) of languages accepted by ordinary IP-systems with m provers and r rounds appears as IP(m, r, 1/6), whereas we define IP'(m, r) to be the union of all IP(m, r, 1/h) with an arbitrary polynomial h. We prove several simulation theorems that enable us to prove most of the known relations between different IP-classes and a collapse of the IP' hierarchy to essentially only four classes, namely

$$\begin{gathered}IP'(1,1) = IP(1,1) \subseteq IP'(1,poly) = IP(1,poly) = PSPACE \hfill \\\subseteq IP'(2,1) = IP(poly,1) \hfill \\\subseteq IP'(2,2) = IP(poly,poly) = NEXPTIME \hfill \\\end{gathered}$$

Finally we show how to reduce the space needed by an interactive proof system introducing one additional prover.