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25
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 166 (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 ...
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
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Cited by 45 (4 self)
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An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
ResourceBounded Measure and Randomness
"... We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than ..."
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Cited by 39 (5 self)
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We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than timebounded randomness but which suffice for many of the applications in complexity theory.
Instance Complexity
, 1994
"... We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that ..."
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Cited by 31 (1 self)
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We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that runs in time t, decides x correctly, and makes no mistakes on other strings ("don't know" answers are permitted). We prove that a set A is in P if and only if there exist a polynomial t and a constant c such that ic t (x : A) c for all x
Hausdorff dimension in exponential time
 Computational Complexity, IEEE Computer Society
"... In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including ..."
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Cited by 31 (3 self)
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In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resourcebounded dimension we show that the class of pmcomplete sets for E has dimension 1 in E. Moreover, we show that there are pmlower spans in E of dimension H(β) for any rational β between 0 and 1, where H(β) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz’s concept of weak completeness. Finally we characterize resourcebounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions. 1.
Universal Algorithmic Intelligence: A mathematical topdown approach
 Artificial General Intelligence
, 2005
"... Artificial intelligence; algorithmic probability; sequential decision theory; rational ..."
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Cited by 22 (6 self)
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Artificial intelligence; algorithmic probability; sequential decision theory; rational
On ResourceBounded Instance Complexity
 Theoretical Computer Science A
, 1995
"... The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where "don't know" answers are permitted). The Instance Complexit ..."
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Cited by 20 (9 self)
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The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where "don't know" answers are permitted). The Instance Complexity Conjecture of Ko, Orponen, Schoning, and Watanabe states that for every recursive set A not in P and every polynomial t there is a polynomial t 0 and a constant c such that for infinitely many x, ic t (x : A) C t 0 (x) \Gamma c, where C t 0 (x) is the t 0 time bounded Kolmogorov complexity of x. In this paper the conjecture is proved for all recursive tally sets and for all recursive sets which are NPhard under honest reductions, in particular it holds for all natural NPhard problems. The method of proof also yields the polynomialspace bounded and the exponentialtime bounded versions of the conjecture in full generality. On the other hand, the conjecture itself turns out to be oracl...
ResourceBounded Balanced Genericity, Stochasticity and Weak Randomness
 In Complexity, Logic, and Recursion Theory
, 1996
"... . We introduce balanced t(n)genericity which is a refinement of the genericity concept of AmbosSpies, Fleischhack and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resourcebounded version of Church's stochas ..."
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Cited by 20 (7 self)
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. We introduce balanced t(n)genericity which is a refinement of the genericity concept of AmbosSpies, Fleischhack and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resourcebounded version of Church's stochasticity [6]. By uniformly describing these concepts and weaker notions of stochasticity introduced by Wilber [19] and Ko [11] in terms of prediction functions, we clarify the relations among these resourcebounded stochasticity concepts. Moreover, we give descriptions of these concepts in the framework of Lutz's resourcebounded measure theory [13] based on martingales: We show that t(n)stochasticity coincides with a weak notion of t(n)randomness based on socalled simple martingales but that it is strictly weaker than t(n)randomness in the sense of Lutz. 1 Introduction Over the last years resourcebounded versions of Baire category and Lebesgue measure have been introduced in complexity theor...
A Theory of Universal Artificial Intelligence based on Algorithmic Complexity
, 2000
"... Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ide ..."
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Cited by 18 (10 self)
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Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameterless theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline for a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement and supervised learning, how the AIXI model can formally solve them. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXItl, which is still effectively more intelligent than any other time t and space l bounded agent. The computation time of AIXItl is of the order t·2^l. Other discussed topics are formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches.
Applications of TimeBounded Kolmogorov Complexity in Complexity Theory
 Kolmogorov complexity and computational complexity
, 1992
"... This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functi ..."
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Cited by 17 (4 self)
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This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functions, pseudorandom generators, and hierarchy theorems in circuit complexity.