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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 ..."
Abstract

Cited by 30 (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
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, ...
Abstract
, 2005
"... We prove that Psel, the class of all Pselective sets, is EXPimmune, but is not EXP/1immune. That is, we prove that some infinite Pselective set has no infinite EXPtime subset, but we also prove that every infinite Pselective set has some infinite subset in EXP/1. Informally put, the immunity ..."
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We prove that Psel, the class of all Pselective sets, is EXPimmune, but is not EXP/1immune. That is, we prove that some infinite Pselective set has no infinite EXPtime subset, but we also prove that every infinite Pselective set has some infinite subset in EXP/1. Informally put, the immunity of Psel is so fragile that it is pierced by a single bit of information. The above claims follow from broader results that we obtain about the immunity of the Pselective sets. In particular, we prove that for every recursive function f, Psel is DTIME(f)immune. Yet we also prove that Psel is not Π p 2 /1immune. 1