Interactive protocols over the reals

We introduce the class IPR+, (resp. IPRx,) as the class of languages that a dmit an interactive protocol on the reals when the verifier is a BSS-machin e with addition (resp. addition and multiplication). Let BIPR+, (resp. BIPR x) be its restriction when only boolean messages can be exchanged between t he prover and the verifier. We prove that the classes BIPR+ and PAR(R+), th e class of languages accepted in parallel polynomial time, coincide. In the case of multiplicative machines, we show that BIPRx subset of or equal to PAR(Rx) We also separate BIPR from IPR in both models by exhibiting a language L wh ich is not in PAR(Rx). but in IPR+. As a consequence we show that additive quantifier elimination cannot be solved in PAR(Rx) and that all boolean lan guages admit an interactive proof with addition and a real constant.