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42
Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
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Cited by 57 (31 self)
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Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of "highly nonrandom " reals that have been previously studied.
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 34 (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.
Computing over the reals: Foundations for scientific computing
 Notices of the AMS
"... We give a detailed treatment of the “bitmodel ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative BlumShubSmale model. In the final section we d ..."
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Cited by 32 (3 self)
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We give a detailed treatment of the “bitmodel ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative BlumShubSmale model. In the final section we discuss the issue of whether physical systems could defeat the ChurchTuring Thesis. 1
Some ComputabilityTheoretical Aspects of Reals and Randomness
 the Lect. Notes Log. 18, Assoc. for Symbol. Logic
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. ..."
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Cited by 25 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
Theory of representations
 Theoretical Computer Science
, 1985
"... Abstract. An approach for a simple, general, and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented. A standard theory of effectivity on F = {f: N 3 N} has been developed in a previous paper. By representations 6: B * M this theory is extende ..."
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Cited by 20 (3 self)
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Abstract. An approach for a simple, general, and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented. A standard theory of effectivity on F = {f: N 3 N} has been developed in a previous paper. By representations 6: B * M this theory is extended to other sets M. Topological and recursion theoretical properties of representations are studied, where the final topology of a representation plays an essential role. It is shown that for any separable T,space an (up to equivalence) unique admissible representation can be defined which reflects the topological properties correctly. 1.
Computability, noncomputability and undecidability of maximal intervals of IVPs
 Trans. Amer. Math. Soc
"... Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initialvalue problem { dx = f(t, x) dt x(t0) = x0, where E is an open subset of R m+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view o ..."
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Cited by 14 (13 self)
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Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initialvalue problem { dx = f(t, x) dt x(t0) = x0, where E is an open subset of R m+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initialvalue problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable. 1.
Efficiently Approximable RealValued Functions
 Electronic Colloquium on Computational Complexity
, 2000
"... We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a nat ..."
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Cited by 12 (2 self)
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We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a natural complete problem: computing the acceptance probability of a given Boolean circuit; in contrast, no complete problems are known for BPP. We observe that all known complexitytheoretic assumptions under which BPP is easy (i.e., can be efficiently derandomized) imply that APP is easy; on the other hand, we show that BPP may be easy while APP is not, by constructing an appropriate oracle. 1 Introduction The complexity class BPP is traditionally considered a class of languages that can be efficiently decided with the help of randomness. While it does contain some natural problems, the "semantic" nature of its definition (on every input, a BPP machine must have either at least 3=4 or at...
Computability of probability measures and MartinLöf randomness over metric spaces
 Information and Computation
"... In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability ..."
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Cited by 12 (6 self)
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In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measuretheoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption). 1
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...