Real Interactive Proofs for VPSPACE

We study interactive proofs in the framework of real number complexity as introduced by Blum, Shub, and Smale. The ultimate goal is to give a Shamir like characterization of the real counterpart IPR of classical IP. Whereas classically Shamir’s result implies IP = PSPACE = PAT = PAR, in our framework a major difficulty arises from the fact that in contrast to Turing complexity theory the real number classes PARR and PATR differ and space resources considered alone are not meaningful. It is not obvious to see whether IPR is characterized by one of them and if so by which. In recent work the present authors established an upper bound IPR ⊆ MA∃R, where MA∃R is a complexity class satisfying PARR ( MA∃R ⊆ PATR and conjectured to be different from PATR. The goal of the present paper is to complement this result and to prove interesting lower bounds for IPR. More precisely, we design interactive real protocols for a large class of functions introduced by Koiran and Perifel and denoted by UniformVPSPACE0. As consequence, we show PARR ⊆ IPR, which in particular implies co-NPR ⊆ IPR, and P R ⊆ IPR, where Res denotes certain multivariate Resultant polynomials. Our proof techniques are guided by the question in how far Shamir’s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVPSPACE0 are extremely helpful. 1998 ACM Subject Classification F.1.1 Models of Computation; F 1.2 Modes of Computation; F 1.3 Complexity Measures and Classes