Results 1 
4 of
4
The Isomorphism Conjecture Holds Relative to an Oracle
, 1996
"... 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 BermanHartmanis isomorphism conjecture [BH77] holds relative to any spgeneric oracle, i.e., for any symmetric perfect generic set A, ..."
Abstract

Cited by 26 (11 self)
 Add to MetaCart
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 BermanHartmanis isomorphism conjecture [BH77] holds relative to any spgeneric oracle, i.e., for any symmetric perfect generic set A, all NP^Acomplete sets are polynomialtime 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.
NP Might Not Be As Easy As Detecting Unique Solutions
, 1998
"... 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 fi ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
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 NPcomplete sets are polynomialtime 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 manyone 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...
The Isomorphism Conjecture Holds and Oneway Functions Exist Relative to an Oracle
 Journal of Computer and System Sciences
, 1994
"... In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which t ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but oneway 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 oneway functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomialtime manytoone reductions and if they are both paddable then they are polynomialtime isomorphic. After surveying all of the thenknown NPcomplete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NPcomplete lan...
Looking for an analogue of Rice's Theorem in circuit complexity theory
 Mathematical Logic Quarterly
, 1989
"... 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 UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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 UPhard with respect to polynomialtime 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, UPhard, NPhard, generators.