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The Isomorphism Conjecture Fails Relative to a Random Oracle
 J. ACM
, 1996
"... Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kc ..."
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Cited by 40 (4 self)
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Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kcreative setsand defined a class of sets (the K k f 's) that are necessarily kcreative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NPcomplete sets. Clearly, the BermanHartmanis and JosephYoung conjectures cannot both be correct. We introduce a family of strong oneway functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NPcomplete sets, as Joseph and Young conjectured, and the BermanHartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 29 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Properties of NPcomplete sets
 In Proceedings of the 19th IEEE Conference on Computational Complexity
, 2004
"... We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreo ..."
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Cited by 10 (6 self)
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We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p mhard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences for longstanding open questions about whether NPcomplete sets are immune. For example, assuming that pseudorandom generators and secure oneway permutations exist, it follows easily that NPcomplete sets are not pimmune. Assuming only that secure oneway permutations exist, we prove that no NPcomplete set is DTIME(2nɛ)immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomialclose to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for NP are not closed under union. Our hypothesis asserts existence of a UPmachine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ timebounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)biimmune sets, then this hypothesis is true.
Gaplanguages and logtime complexity classes
 Theoretical Computer Science
, 1997
"... This paper shows that classical results about complexity classes involving “delayed diagonalization ” and “gap languages, ” such as Ladner’s Theorem and Schöning’s Theorem and independence results of a kind noted by Schöning and Hartmanis, apply at very low levels of complexity, indeed all the way d ..."
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Cited by 9 (6 self)
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This paper shows that classical results about complexity classes involving “delayed diagonalization ” and “gap languages, ” such as Ladner’s Theorem and Schöning’s Theorem and independence results of a kind noted by Schöning and Hartmanis, apply at very low levels of complexity, indeed all the way down in Sipser’s logtime hierarchy. This paper also investigates refinements of Sipser’s classes and notions of logtime reductions, following on from recent work by Cai, Chen, and others. 1
Scalability and the Isomorphism Problem
 Information Processing Letters
, 1995
"... Scalable sets are defined and their properties studied. It is shown that the set of scalable sets is the isomorphism closure of the set of rankable sets and that every scalable set is Pisomorphic to some rankable set. Scalable sets coincide with Pprintable sets when sparse, and with Ppaddable set ..."
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Cited by 2 (1 self)
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Scalable sets are defined and their properties studied. It is shown that the set of scalable sets is the isomorphism closure of the set of rankable sets and that every scalable set is Pisomorphic to some rankable set. Scalable sets coincide with Pprintable sets when sparse, and with Ppaddable sets when thick. Using scalability as a tool, the Pisomorphism question for polynomialtime computable sets of similar densities is examined. 1 Introduction This paper defines and investigates the new concept of scalability for polynomialtime sets. A set is scalable if there is an efficient method for computing the number of elements in the set (or its complement) which are less than a given element, relative to some polynomialtime computable and invertible order on \Sigma . All scalable sets are polynomialtime computable. Scalability generalizes the previously studied concept of ranking which was based on the same property with respect to a fixed ordering, the lexicographic ordering, ...
Querymonotonic Turing Reductions
"... ... A for which any set that Turing reduces to A will also reduceto A via both queryincreasing and querydecreasing Turing reductions. In particular, this holds for all tight paddable sets, where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bou ..."
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Cited by 1 (1 self)
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... A for which any set that Turing reduces to A will also reduceto A via both queryincreasing and querydecreasing Turing reductions. In particular, this holds for all tight paddable sets, where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bounded tightlyboth from above and from below in the length of the input. We prove that many natural NPcomplete problems such as satisfiability, clique, and vertex cover aretight paddable.
News from the Isomorphism Front
"... this article. First, however, we will need to make a digression, while we discuss some recent progress on the isomorphism conjecture. ..."
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this article. First, however, we will need to make a digression, while we discuss some recent progress on the isomorphism conjecture.