# Relativized Arthur-Merlin versus Merlin-Arthur games

## Abstract

Arthur-Merlin games were introduced recently by Babai in order to capture the intuitive notion of efficient, probabilistic proof-systems. Considered as complexity classes, they are extensions of NP. It turned out, that one exchange of messages between the two players is sufficient to simulate a constant number of interactions. Thus at most two new complexity classes survive at the constant levels of this new hierarchy: AM and MA, depending on who starts the communication. It is known that \(MA \subseteq AM\). In this paper we answer an open problem of Babai: we construct an oracle C such that \(AM^C - \Sigma _2^{P,C} \ne \emptyset\). Since \(MA^C \subseteq \Sigma _2^{P,C}\), it follows that for some oracle C, MA^{C}≠AM^{C}. Our prooftechnique is a modification of the technique used by Baker and Selman to show that ∑ _{2} ^{P} and ∏ _{2} ^{P} can be separated by some oracle. This result can be interpreted as an evidence that with one exchange of messages, the proof-system is stronger when Arthur starts the communication.

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