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 +.