help us decide where and how to put our efforts into solving We show that MAEXP, the exponential time version of problems in complexity theory. It is still true that virtually the Merlin-Arthur class, does not have polynomial size cir- all of the theorems in computational complexity theory that cuits. This significantly improves the previous known result have reasonable relativizations do relativize (see [For94]). due to Kannan since we furthermore show that our result But we do have a small number of exceptions that arise does not relativize. This is the first separation result in com- from the area of interactive proofs. These results have preplexity theory that does not relativize. As a corollary to our viously always taken the form of collapses such as IP= separation result we also obtain that PEXP, the exponen- PSPACE [LFKN92, Sha92], MIP=NEXP [BFL91] and tial time version of PP is not in P=poly. PCP(O(1);O(logn))=NP [ALM+92]. In this paper we give the first reasonable nonrel-1

We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman-Hartmanis isomorphism conjecture [BH77] holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP^A-complete sets are polynomial-time isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.

We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the first where P A = UP A 6= NP A = coNP A : ffl The construction gives a much simpler proof than Fenner, Fortnow and Kurtz of a relativized world where all NP-complete sets are polynomial-time isomorphic. It is the first such computable oracle. ffl Relative to A we have a collapse of \PhiEXP A ` ZPP A ` P A /poly. We also create a different relativized world where there exists a set L in NP that is NP- complete under reductions that make one query to L but not under traditional many-one reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. 1 Introduction Valiant and Vazirani [VV86] show the sur...

In this paper we demonstrate an oracle relative to which there are one-way functions but every paddable 1-li-degree collapses to an isomorphism type, thus yielding a relativized failure of the Joseph-Young Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but one-way functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of one-way functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomial-time many-to-one reductions and if they are both paddable then they are polynomial-time isomorphic. After surveying all of the then-known NP-complete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NP-complete lan...

Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions. For generators [31] we show a perfect analogue of Rice’s Theorem. Mathematics Subject Classification: 03D15, 68Q15. Keywords: Rice’s Theorem, Counting problems, Promise classes, UP-hard, NP-hard, generators.