Interactive protocols over the reals

Abstract

We introduce the class $ IP_{{\cal R}+} ({\rm resp.}\,IP_{{\cal R}\times}) $ as the class of languages that admit an interactive protocol on the reals when the verifier is a BSS-machine with addition (resp. addition and multiplication). Let $ BIP_{{\cal R}+} ({\rm resp.}\,BIP_{{\cal R}\times}) $ be its restriction when only boolean messages can be exchanged between the prover and the verifier. We prove that the classes $ BIP_{{\cal R}+} $ and $ PAR_{{\cal R}+} $ , the class of languages accepted in parallel polynomial time, coincide. In the case of multiplicative machines, we show that $ BIP_{{\cal R}\times} \subseteq PAR_{{\cal R}\times} $ .¶We also separate $ BIP_{{\cal R} $ from $ IP_{{\cal R} $ in both models by exhibiting a language L which is not in $ PAR_{{\cal R}\times} $ but in $ IP_{{\cal R}+} $ . As a consequence we show that additive quantifier elimination cannot be solved in $ PAR_{{\cal R}\times} $ and that all boolean languages admit an interactive proof with addition and a real constant.