On the power of interaction

A hierarchy of probabilistic complexity classes generalizing NP has recently emerged in the work of [B], [GMR], and [GS]. The IP hierarchy is defined through the notion of an interactive proof system, in which an all powerful prover tries to convince a probabilistic polynomial time verifier that a string x is in a language L. The verifier tosses coins and exchanges messages back and forth with the prover before he decides whether to accept x. This proof-system yields "probabilistic" proofs: the verifier may erroneously accept or reject x with small probability. The class IP[f(|x|)] is said to contain L if, there exists an interactive proof system with f(|x|)- message exchanges (interactions) such that with high probability the verifier accepts x if and only if x ε L. Babai [B] showed that all languages recognized by interactive proof systems with bounded number of interactions, can be recognized by interactive proof systems with only two interactions. Namely, for every constant k, IP[k] collapses to Ip[2]. In this paper, we give evidence that interactive proof systems with unbounded number of interactions may be more powerful than interactive proof systems with bounded number of interactions. We show that for any unbounded function f(n) there exists an oracle B such that IPB [f(|x|)] ⊄ PHB. This implies that IPB[f(n)] ≠ IPB[2], since IPB[2] ⊆ Π2B for all oracles B. The techniques employed are extensions of the techniques for proving lower bounds on small depth circuits used in [FSS], [Y] and [H1].