Random debaters and the hardness of approximating stochastic functions

A probabilistically checkable debate system (PCDS) for a language L consists of a probabilistic polynomial-time verifier V and a debate between Player 1, who claims that the input x is in L, and Player 0, who claims that the input x is not in L. It is known that there is a PCDS for L in which V flips O(log n) coins and reads O(1) bits of the debate if and only if L is in PSPACE [A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Chicago J. Theoret. Comput. Sci., 1995, No. 4]. In this paper, we restrict attention to RPCDSs, which are PCDSs in which Player 0 follows a very simple strategy: On each turn, Player 0 chooses uniformly at random from the set of legal moves. We prove the following result. THEOREM. L has an RPCDS in which the verifier flips O(log n) coins and reads O(1) bits of the debate if and only if L is in PSPACE. This new characterization of PSPACE is used to show that certain stochastic PSPACE-hard functions are as hard to approximate closely as they are to compute exactly. Examples of such functions include optimization versions of Dynamic Graph Reliability, Stochastic Satisfiability, Mah-Jongg, Stochastic Generalized Geography, and other ''games against nature'' of the type introduced in [C. Papadimitriou, J. Comput. System Sci., 31 (1985), pp. 288-301].