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

We initiate an investigation of probabilistically checkable 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. t University of Wisconsin, Computer Sciences Department, 1210 West Dayton Street, Madison, WI 57306 USA, condon@cs. wise. edu. Supported in part by NSF grants CCR9100886 and CCR-9257241. tAT&T Ben Laboratories, Room 2C473, 600 Mowt~n Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, jf~research. att . corn. ~AT&T Ben Laboratories, Room 2C324, 600 Momt& Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, lrmd@research. att. coin. ~AT&T Bell Laboratory=, Room 2DI 49, 600 Momtain Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, shor~research. att. corn. * 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. 25th ACM STOC ‘93-5 /93/CA, USA 01993 ACM 0-S9791-591 -7/93 /0005 /0305 . ..$1.50