Results 1  10
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11
Dimension in Complexity Classes
 SIAM Journal on Computing
, 2000
"... A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Othe ..."
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Cited by 114 (17 self)
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A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 [0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain at most of all strings, has dimension H()  the binary entropy of  in E and in E 2 . 2. For every real number 0 1, the set SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE.
Comparing reductions to NPcomplete sets
 Electronic Colloquium on Computational Complexity
, 2006
"... Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondetermin ..."
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Cited by 12 (4 self)
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Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNPcomplete for NP but not Turingcomplete. (3) Every problem that is manyone complete for NP is complete under lengthincreasing reductions that are computed by polynomialsize circuits. The first item solves one of Lutz and Mayordomo’s “Twelve Problems in ResourceBounded Measure ” (1999). We also show that every manyone complete problem for NE is complete under onetoone, lengthincreasing reductions that are computed by polynomialsize circuits. 1
Randomness is Hard
 SIAM Journal on Computing
, 2000
"... We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomi ..."
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Cited by 6 (3 self)
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We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity, CS introduced by Hartmanis. For all of these measures we dene the set of random strings R CD t , R CND t , and R CS s as the set of strings x such that CD t (x), CND t (x), and CS s (x) is greater than or equal to the length of x, for s and t polynomials. We show the following: MA NP R CD t , where MA is the class of MerlinArthur games dened by Babai. AM NP R CND t , where AM is the class of ArthurMerlin games. PSPACE NP cR CS s . In the last item cR CS s is the set of pairs <x; y> so that x is random given y. These results show that the set of random strings for various resource bounds is hard for...
Autoreducibility, mitoticity and immunity
 Mathematical Foundations of Computer Science: Thirtieth International Symposium, MFCS 2005
, 2005
"... We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are we ..."
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Cited by 6 (4 self)
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We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are weakly manyone mitotic. • PSPACEcomplete sets are weakly Turingmitotic. • If oneway permutations and quick pseudorandom generators exist, then NPcomplete languages are mmitotic. • If there is a tally language in NP ∩ coNP − P, then, for every ɛ> 0, NPcomplete sets are not 2 n(1+ɛ)immune. These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets. 1
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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Cited by 4 (0 self)
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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.
Query Order and NPCompleteness
"... The effect of query order on NPcompleteness is investigated. A sequence ~ D = (D 1 ; : : : ; D k ) of decision problems is defined to be sequentially complete for NP if each D i 2 NP and every problem in NP can be decided in polynomial time with one query to each of D 1 ; : : : ; D k in this orde ..."
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Cited by 3 (1 self)
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The effect of query order on NPcompleteness is investigated. A sequence ~ D = (D 1 ; : : : ; D k ) of decision problems is defined to be sequentially complete for NP if each D i 2 NP and every problem in NP can be decided in polynomial time with one query to each of D 1 ; : : : ; D k in this order . It is shown that, if NP contains a language that is pgeneric in the sense of AmbosSpies, Fleischhack, and Huwig [3], then for every integer k 2, there is a sequence ~ D = (D 1 ; : : : ; D k ) such that ~ D is sequentially complete for NP, but no nontrivial permutation (D i 1 ; : : : ; D i k ) of ~ D is sequentially complete for NP. It follows that such a sequence ~ D exists if there is any strongly positive, pcomputable probability measure such that p (NP) 6= 0. 1 Introduction The success or efficiency of a computation sometimes dependsand sometimes does not dependupon the order in which it is allowed access to the various pieces of information that it may require. Altho...
The Computational Complexity Column
, 1998
"... Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful too ..."
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Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], AmbosSpies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resourcebounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resourcebounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their
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, truth ..."
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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. 1 Introduction and Notation 1.1 Introduction Ladner, Lynch and Selman [12] first compared the strength of the polynomialtime reducibilities. For the most common reducibilities  namely Turing (pT)...
Hard Instances of Hard Problems
"... This paper investigates the instance complexities of problems that are hard or weakly hard for exponential time under polynomial time, manyone reductions. It is shown that almost every instance of almost every problem in exponential time has essentially maximal instance complexity. It follows tha ..."
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This paper investigates the instance complexities of problems that are hard or weakly hard for exponential time under polynomial time, manyone reductions. It is shown that almost every instance of almost every problem in exponential time has essentially maximal instance complexity. It follows that every weakly hard problem has a dense set of such maximally hard instances. This extends the theorem, due to Orponen, Ko, Schöning and Watanabe (1994), that every hard problem for exponential time has a dense set of maximally hard instances. Complementing this, it is shown that every hard problem for exponential time also has a dense set of unusually easy instances.