Random debaters and the hardness of approximating stochastic functions

A random probabilistically checkable debate system (RPCDS) for a language L consists of a probabilistic polynomial-time verifier V and a debate between Player 1, who aims to prove that the input x is in L, and Player 0, who selects a move uniformly at random from the set of legal moves. This model is a natural restriction of the PCDS model (Condon et al., Proc. 25th ACM Symposium on Theory of Computing, p.304-15, 1993,). We show that 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. Using this new characterization of PSPACE, we show that certain stochastic PSPACE-hard functions are as hard to approximate closely as they are to compute exactly. Examples include optimization versions of dynamic graph reliability, stochastic satisfiability, Mah-Jongg, stochastic coloring, stochastic generalized geography, and other “games against nature”