Abstract

The traditional studies of multi-prover interactive proof systems have concentrated on cooperating provers. These are systems in which a verifier interacts with a set of provers who collaborate in their attempt to convince the verifier that a word omega is in a prespecified language L. Results on probabilistically checkable proofs coupled with parallel repetition techniques characterize NP in terms of multi-prover proof systems: languages in NP can be verified through a one-round interaction with two cooperating provers using limited randomness and communication. Less attention has been paid to the study of competition in this complexity-theoretic setting of interactive proof systems. We consider, for example, one-round proof systems where the first prover is trying to convince the verifier to accept and the second prover is trying to convince the verifier to reject. We build into these proof systems a (crucial) asymmetry between the provers, analogous to quantifier alternation. We show that such proof systems capture, with restrictions on communication and randomness, languages in NP. We generalize this model by defining alternating prover proof systems which we show characterize each level of the polynomial hierarchy. Alternating oracle proof systems are also examined. The main contribution of this work is the first exact characterization of the polynomial hierarchy in terms of interactive (prover) proof systems.