Interactive protocols on the reals

Abstract

We introduce the classes IP _{R +} (resp. IP _{R x}) 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 _{R +} (resp. BIP _{R x}) its restriction when only boolean messages can be exchanged between the prover and the verifier. We prove that the classes BIP _{R +} and PAR _{R +}, the class of languages accepted in parallel polynomial time coincide. In the case of multiplicative machines, we show that BIP _{R x} $\subseteq$ PAR _{R x}). We also separate BIP _R from IP _R in both models by exhibiting a language L which is not in PAR _{R x} but in IP _{R +}. As a consequence we show that additive quantifier elimination can't be solved in PAR _{R x} and that all boolean languages are in IP _{R +}.