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.