Results 11  20
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24
Every PolynomialTime 1Degree Collapses iff P = PSPACE
, 1996
"... A set A is mreducible (or Karpreducible) to B iff there is a polynomialtime computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1equivalent iff each is mreducible to the other by oneone reductions; ffl pinvertible equivalent iff each is mreducible to the othe ..."
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A set A is mreducible (or Karpreducible) to B iff there is a polynomialtime computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1equivalent iff each is mreducible to the other by oneone reductions; ffl pinvertible equivalent iff each is mreducible to the other by oneone, polynomialtime invertible reductions; and ffl pisomorphic iff there is an mreduction from one set to the other that is oneone, onto, and polynomialtime invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1equivalent sets are pisomorphic. (c) Every two pinvertible equivalent sets are pisomorphic. 2 1. Overview If A is mreducible to B, we usually interpret this to mean that A is computationally no more difficult than B, since a procedure for computing B is easily converted into a procedure for computing A of comparable complexity. In fact, this interpretation is supported by muc...
Collapsing PolynomialTime Degrees
"... For reducibilities r and r 0 such that r is weaker than r 0 , we say that the rdegree of A, i.e., the class of sets which are requivalent to A, collapses to the r 0 degree of A if both degrees coincide. We investigate for the polynomialtime bounded manyone, bounded truthtable, truthtable, and ..."
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For reducibilities r and r 0 such that r is weaker than r 0 , we say that the rdegree of A, i.e., the class of sets which are requivalent to A, collapses to the r 0 degree of A if both degrees coincide. We investigate for the polynomialtime bounded manyone, bounded truthtable, truthtable, and Turing reducibilities whether and under which conditions such collapses can occur. While we show that such collapses do not occur for sets which are hard for exponential time, we have been able to construct a recursive set such that its bounded truthtable degree collapses to its manyone degree. The question whether there is a set such that its Turing degree collapses to its manyone degree is still open; however, we show that such a set  if it exists  must be recursive.
DSPACE(n) ? = NSPACE(n): A degree theoretic characterization
 in Proc. 10th Structure in Complexity Theory Conference
, 1995
"... It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1 ..."
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It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1
OneWay Functions and the Isomorphism Conjecture
, 2009
"... We study the Isomorphism Conjecture proposed by Berman and Hartmanis. It states that all sets complete for NP under polynomialtime manyone reductions are Pisomorphic to each other. From previous research it has been widely believed that all NPcomplete sets are reducible each other by onetoone ..."
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We study the Isomorphism Conjecture proposed by Berman and Hartmanis. It states that all sets complete for NP under polynomialtime manyone reductions are Pisomorphic to each other. From previous research it has been widely believed that all NPcomplete sets are reducible each other by onetoone and lengthincreasing polynomialtime reductions, but we may not hope for the full pisomorphism due to the existence of oneway functions. Here we showed two results on the relation between oneway functions and the Isomorphism Conjecture. Firstly, we imporve the result of Agrawal [Agrawal, CCC’02] to show that if regular oneway functions exist, then all NPcomplete sets are indeed reducible each other by onetoone, lengthincreasing and P/polyreductions. A consequence of this result is the complete description of the structure of manyone complete sets of NP relative to a random oracle: all NPcomplete sets are reducible each other by oneone and lengthincreasing polynomialtime reductions but (as already shown by [Kurtz etal, JACM 95]) they are not Pisomorphic. Neverthless, we also conjecture that (different from the random oracle world) all oneway functions should have some dense easy parts, which we call P/polyeasy cylinders, where they are P/polyinvertible. Then as our second result we show that if regular oneway functions exist and furthermore all oneone, lengthincreasing and P/polycomputable functions have P/polyeasy cylinders, then all manyone complete sets for NP are P/polyisomorphic.
ON THE ISOMORPHISM CONJECTURE FOR 2DFA REDUCTIONS
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCEINTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 1999
"... The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under loglin reductions:  All complete sets for the class C under 2DFA reductions are also complete under oneone, lengthincreasing 2DFA reductions and are firstorder i ..."
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The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under loglin reductions:  All complete sets for the class C under 2DFA reductions are also complete under oneone, lengthincreasing 2DFA reductions and are firstorder isomorphic.  The 2DFAisomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions.
For completeness, sublogarithmic space is no space
"... It is shown that for any class C closed under lineartime reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under firstorder reductions. ..."
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It is shown that for any class C closed under lineartime reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under firstorder reductions.
The isomorphism conjecture for constant depth reductions
 Journal of Computer and System Sciences
"... For any class C closed under TC 0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under uuniform AC 0 reductions are isomorphic under uuniform AC 0computable isomorphisms. 1 ..."
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For any class C closed under TC 0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under uuniform AC 0 reductions are isomorphic under uuniform AC 0computable isomorphisms. 1
OneWay Functions and the BermanHartmanis Conjecture
"... Abstract—The BermanHartmanis conjecture states that all NPcomplete sets are Pisomorphic each other. On this conjecture, we first improve the result of [3] and show that all NPcomplete sets are ≤ p/poly li,11reducible to each other based on the assumption that there exist regular oneway functi ..."
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Abstract—The BermanHartmanis conjecture states that all NPcomplete sets are Pisomorphic each other. On this conjecture, we first improve the result of [3] and show that all NPcomplete sets are ≤ p/poly li,11reducible to each other based on the assumption that there exist regular oneway functions that cannot be inverted by randomized polynomialtime algorithms. Secondly, we show that, besides the above assumption, if all oneway functions have some easy part to invert, then all NPcomplete sets are P/polyisomorphic to each other. Index Terms—averagecase oneway function; Pisomorophism conjecture; P/polyisomorophism in NP; oneway function with easy cylinder I.
Sparse Hard Sets for P
 DIMACS TECHNICAL REPORT
, 1996
"... Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various re ..."
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Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for lowlevel complexity classes. In particular, the Hartmanis' sparseness conjectures for P and NL have been resolved. This article overviews the history of sparse hard set problems and exposes some of the recent results.
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.