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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 182 (42 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 ...
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 39 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Dynamics, computation, and the “edge of chaos”: A reexamination
 Complexity:Metaphors, Models, and Reality
, 1994
"... In this paper we review previous work and present new work concerning the relationship between dynamical systems theory and computation. In particular, we review work by Langton [21] and Packard [29] on the relationship between dynamical behavior and computational capability in cellular automata (CA ..."
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Cited by 38 (3 self)
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In this paper we review previous work and present new work concerning the relationship between dynamical systems theory and computation. In particular, we review work by Langton [21] and Packard [29] on the relationship between dynamical behavior and computational capability in cellular automata (CAs). We present results from an experiment similar to the one described by Packard [29], which was cited as evidence for the hypothesis that rules capable of performing complex computations are most likely to be found at a phase transition between ordered and chaotic behavioral regimes for CAs (the “edge of chaos”). Our experiment produced very different results from the original experiment, and we suggest that the interpretation of the original results is not correct. We conclude by discussing general issues related to dynamics, computation, and the “edge of chaos ” in cellular automata. 1
On the relationship between complexity and entropy for Markov chains and regular languages
 Complex Systems
, 1991
"... Abstract. Using the pastfuture mutual information as a measure of complexity, the relation between the complexity and the Shannon entropy is determined analytically for sequences generated by Markov chains and regular languages. It is emphasized that, given an entropy value, there are many possible ..."
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Cited by 30 (2 self)
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Abstract. Using the pastfuture mutual information as a measure of complexity, the relation between the complexity and the Shannon entropy is determined analytically for sequences generated by Markov chains and regular languages. It is emphasized that, given an entropy value, there are many possible complexity values, and vice versa; that is, the relationship between complexity and entropy is not onetoone, but rather manytoone or onetomany. It is also emphasized that there are structures in the complexityversusentropy plots, and these structures depend on the details of a Markov chain or a regular language grammar. 1.
Randomness Space
 AUTOMATA, LANGUAGES AND PROGRAMMING, PROCEEDINGS OF THE 25TH INTERNATIONAL COLLOQUIUM, ICALP’98
, 1998
"... MartinLöf defined infinite random sequences over a finite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After st ..."
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Cited by 24 (4 self)
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MartinLöf defined infinite random sequences over a finite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved which gives conditions under which a function between randomness spaces preserves randomness. This corrects and extends a result by Schnorr. Calude and Jürgensen proved that the randomness notion for real numbers obtained by considering their bary representations is independent from the base b. We use our invariance result to show that this notion is identical with the notion which one obtains by viewing the real number space directly as a randomness space. Furthermore, arithmetic properties of random real numbers are derived, for example that every computable analytic function pres...
Circuit size relative to pseudorandom oracles
 THEORETICAL COMPUTER SCIENCE A 107
, 1993
"... Circuitsize complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspacerandom oracle A, and relative toalmost every oracle A 2 ESPACE. (i) NP A is not contained in SIZE ..."
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Cited by 17 (4 self)
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Circuitsize complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspacerandom oracle A, and relative toalmost every oracle A 2 ESPACE. (i) NP A is not contained in SIZE A (2 n) for any real < 1 3. (ii) E A is not contained in SIZE A ( 2n n). Thus, neither NP A nor E A is contained in P A /Poly. In fact, these separations are shown to hold for almost every n. Since a randomly selected oracle is pspacerandom with probability one, (i) and (ii) immediately imply the corresponding random oracle separations, thus improving a result of Bennett and Gill [9] and answering open questions of Wilson [47].
Computational depth and reducibility
 Theoretical Computer Science
, 1994
"... This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible ..."
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Cited by 14 (2 self)
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This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost
Ontic and epistemic descriptions of chaotic systems
 Computing Anticipatory Systems
, 2000
"... Abstract. Traditional philosophical discourse draws a distinction between ontology and epistemology and generally enforces this distinction by keeping the two subject areas separated and unrelated. In addition, the relationship between the two areas is of central importance to physics and philosophy ..."
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Cited by 13 (8 self)
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Abstract. Traditional philosophical discourse draws a distinction between ontology and epistemology and generally enforces this distinction by keeping the two subject areas separated and unrelated. In addition, the relationship between the two areas is of central importance to physics and philosophy of physics. For instance, all kinds of measurementrelated problems force us to consider both our knowledge of the states and observables of a system (epistemic perspective) and its states and observables independent of such knowledge (ontic perspective). This applies to quantum systems in particular. In this contribution we present an example which shows the importance of distinguishing between ontic and epistemic levels of description even for classical systems. Corresponding conceptions of ontic and epistemic states and their evolution will be introduced and discussed with respect to aspects of stability and information ow. These aspects show whytheontic/epistemic distinction is particularly important for systems exhibiting deterministic chaos. Moreover, this distinction provides some understanding of the relationships between determinism, causation, predictability, randomness, and stochasticity inchaotic systems.
Automatic Bias Learning: An Inquiry into the Inductive Basis of Induction
, 1999
"... This thesis combines an epistemological concern about induction with a computational exploration of inductive mechanisms. It aims to investigate how inductive performance could be improved by using induction to select appropriate generalisation procedures. The thesis revolves around a metalearning ..."
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Cited by 12 (5 self)
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This thesis combines an epistemological concern about induction with a computational exploration of inductive mechanisms. It aims to investigate how inductive performance could be improved by using induction to select appropriate generalisation procedures. The thesis revolves around a metalearning system, called designed to investigate how inductive performances could be improved by using induction to select appropriate generalisation procedures. The performance of is discussed against the background of epistemological issues concerning induction, such as the role of theoretical vocabularies and the value of simplicity.