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18
Almost Everywhere High Nonuniform Complexity
, 1992
"... . We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds ..."
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Cited by 170 (34 self)
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. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR8809238 and CCR9157382 and in ...
The quantitative structure of exponential time
 Complexity theory retrospective II
, 1997
"... ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuit ..."
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Cited by 90 (13 self)
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ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuitsize complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1
Equivalence of Measures of Complexity Classes
"... The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
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Cited by 70 (19 self)
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The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomialtime, truthtable reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the cointoss probability measure given by the sequence ~ fi. (2) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...
Cook versus KarpLevin: Separating Completeness Notions If NP Is Not Small
 Theoretical Computer Science
, 1992
"... Under the hypothesis that NP does not have pmeasure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T complete ("Cook complete "), but not P m complete ("KarpLevin complete"), for NP. This conclusion, widely be ..."
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Cited by 56 (12 self)
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Under the hypothesis that NP does not have pmeasure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T complete ("Cook complete "), but not P m complete ("KarpLevin complete"), for NP. This conclusion, widely believed to be true, is not known to follow from P 6= NP or other traditional complexitytheoretic hypotheses. Evidence is presented that "NP does not have pmeasure 0" is a reasonable hypothesis with many credible consequences. Additional such consequences proven here include the separation of many truthtable reducibilities in NP (e.g., k queries versus k+1 queries), the class separation E 6= NE, and the existence of NP search problems that are not reducible to the corresponding decision problems. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International. 1 Introduction The NPcompleteness of decision problems has...
Prequential Probability: Principles and Properties
, 1997
"... this paper we first illustrate the above considerations for a variety of appealling criteria, and then, in an attempt to understand this behaviour, introduce a new gametheoretic framework for Probability Theory, the `prequential framework', which is particularly suited for the study of such problem ..."
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Cited by 33 (2 self)
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this paper we first illustrate the above considerations for a variety of appealling criteria, and then, in an attempt to understand this behaviour, introduce a new gametheoretic framework for Probability Theory, the `prequential framework', which is particularly suited for the study of such problems.
Scaled dimension and nonuniform complexity
 Journal of Computer and System Sciences
, 2004
"... Resourcebounded dimension is a complexitytheoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resourcebounded measure 0. For example, while it has long been known that the Boolean circuitsize complexity cla ..."
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Cited by 25 (11 self)
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Resourcebounded dimension is a complexitytheoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resourcebounded measure 0. For example, while it has long been known that the Boolean circuitsize complexity class SIZE � α 2n n has measure 0 in ESPACE for all 0 ≤ α ≤ 1, we now know that SIZE � α 2n n has dimension α in ESPACE for all 0 ≤ α ≤ 1. The present paper furthers this program by developing a natural hierarchy of “rescaled” resourcebounded dimensions. For each integer i and each set X of decision problems, we define the ithorder dimension of X in suitable complexity classes. The 0thorder dimension is precisely the dimension of Hausdorff (1919) and Lutz (2000). Higher and lower orders are useful for various sets X. For example, we prove the following for 0 ≤ α ≤ 1 and any polynomial q(n) ≥ n2. 1. The class SIZE(2 αn) and the time and spacebounded Kolmogorov complexity classes KT q (2 αn) and KS q (2 αn) have 1 storder dimension α in ESPACE. 2. The classes SIZE(2nα), KT q (2nα), and KS q (2nα) have 2ndorder dimension α in ESPACE.
Weakly Hard Problems
, 1994
"... A weak completeness phenomenon is investigated in the complexity class E = DTIME(2 linear ). According to standard terminology, a language H is P m hard for E if the set Pm (H), consisting of all languages A P m H , contains the entire class E. A language C is P m complete for E if it ..."
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Cited by 14 (6 self)
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A weak completeness phenomenon is investigated in the complexity class E = DTIME(2 linear ). According to standard terminology, a language H is P m hard for E if the set Pm (H), consisting of all languages A P m H , contains the entire class E. A language C is P m complete for E if it is P m hard for E and is also an element of E. Generalizing this, a language H is weakly P m hard for E if the set Pm (H) does not have measure 0 in E. A language C is weakly P m complete for E if it is weakly P m hard for E and is also an element of E. The main result of this paper is the construction of a language that is weakly P m complete, but not P m complete, for E. The existence of such languages implies that previously known strong lower bounds on the complexity of weakly P m hard problems for E (given by work of Lutz, Mayordomo, and Juedes) are indeed more general than the corresponding bounds for P m hard problems for E. The proof of this result in...
Weakly Complete Problems are Not Rare
 COMPUTATIONAL COMPLEXITY
, 1995
"... Certain natural decision problems are known to be intractable because they are complete for E, the class of all problems decidable in exponential time. Lutz recently conjectured that many other seemingly intractable problems are not complete for E, but are intractable nonetheless because they are we ..."
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Cited by 7 (2 self)
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Certain natural decision problems are known to be intractable because they are complete for E, the class of all problems decidable in exponential time. Lutz recently conjectured that many other seemingly intractable problems are not complete for E, but are intractable nonetheless because they are weakly complete for E. The main result of this paper shows that Lutz's intuition is at least partially correct; many more problems are weakly complete for E than are complete for E. The main result of this paper states that weakly complete problems are not rare in the sense that they form a nonmeasure 0 subset of E. This extends a recent result of Lutz that establishes the existence of problems that are weakly complete, but not complete, for E. The proof of Lutz's original result employs a sophisticated martingale diagonalization argument. Here we simplify and extend Lutz's argument to prove the main result. This simplified martingale diagonalization argument may be applicable to other quest...
Kolmogorov's Contributions to the Foundations of Probability
"... Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical and philosophical foundations of probability in the twentieth century, and his thinking on the topic is still potent today. In this article we first review the three stages of Kolmogorov's work on the foundations of proba ..."
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Cited by 7 (2 self)
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Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical and philosophical foundations of probability in the twentieth century, and his thinking on the topic is still potent today. In this article we first review the three stages of Kolmogorov's work on the foundations of probability: (1) his formulation of measuretheoretic probability, 1933, (2) his frequentist theory of probability, 1963, and (3) his algorithmic theory of randomness, 19651987. We also discuss another approach to the foundations of probability, based on martingales, that Kolmogorov did not consider.
Effective Fractal Dimension in Algorithmic Information Theory
, 2006
"... Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that ..."
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Cited by 7 (7 self)
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Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that