Probabilistically Checkable Debate Systems and Nonapproximability of PSPACE-Hard Functions

We initiate an investigation of probabilistically checkable debate systems (PCDS), a natural generalization of probabilistically checkable proof systems (PCPS). A 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. We show that there is a PCDS for L in which V flips O(log n) random coins and reads O(1) bits of the debate if and only if L is in PSPACE. This characterization of PSPACE is used to show that certain PSPACE-hard functions are as hard to approximate closely as they are to compute exactly.

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