Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions

Abstract

We initiate an investigation of probabilistically check-able debate systems (PCDS'S), a natural generalization of the probabilistically checkable proof systems studied in [1, 2, 3, 8]. 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 z is in L, and player O, 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 as they are to compute exactly. * The full version of this paper has been submitted for journal publication and is available as DIMACS TR 93-1o. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 1 Introduction Suppose that two candidates, B and C, agree to a debate format. Voter V is too busy to catch more than a very small number of bits of the debate. How does V decide which of B or C won the debate? In this paper , we show that if B and C choose the right debate format, V's problem is solved. By listening to a few, randomly chosen, sounds bites of the debate, V can with near certainty figure out who won. Similarly, suppose that B or C is giving a speech to a set of voters VI,. .. . Vn, represented by finite au-tomata. He would like to give the speech that results in acceptance (votes) by the greatest number of Vi 's, We show that not only can he not compute this maximum exactly, but he cannot come within an arbitrary constant factor, unless he has access to an oracle (political consultant) with the full power of P!3PACE. Our work builds on the recent progress that has been made in the theory of probabilistically checkable proof systems (PCPS 's). Results about the language.-recognition power of PCPS'S have …