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24
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 173 (36 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
"... ..."
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 (21 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 c ..."
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Cited by 58 (14 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 pr ..."
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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 26 (12 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...
Basic Elements and Problems of Probability Theory
, 1999
"... After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probabil ..."
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Cited by 8 (0 self)
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After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measuretheoretical codification of stochastic processes genuine chance processes can be defined rigorously as socalled regular processes which do not allow a longterm prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory.
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
Why Computational Complexity Requires Stricter Martingales
"... The word "martingale " has related, but different, meanings in probability theory and theoretical computer science. In computational complexity and algorithmic information theory, a martingale is typically a function d on strings such that E(d(wb)w) = d(w) for all strings w, whe ..."
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Cited by 7 (0 self)
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The word &quot;martingale &quot; has related, but different, meanings in probability theory and theoretical computer science. In computational complexity and algorithmic information theory, a martingale is typically a function d on strings such that E(d(wb)w) = d(w) for all strings w, where the conditional expectation is computed over all possible values of the next symbol b. In modern probability theory a martingale is typically a sequence,0,,1,,2,... of random variables such that E(,n+1,0,...,,n) =,n for all n.