Results 1  10
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19
DegreeTheoretic Aspects of Computably Enumerable Reals
 in Models and Computability
, 1998
"... A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequ ..."
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A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to #: For example, every representation A of # is Turing reducible to L###: Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below deg T L### necessarily contains a representation of #: 1 Introduction Computability theory essentially studies the relative computability of sets of natural numbers. Since G#odel introduced a method for coding s...
Real number calculations and theorem proving
 Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. ..."
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Cited by 11 (4 self)
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Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 1
Computable Approximations of Reals: An InformationTheoretic Analysis
 Fundamenta Informaticae
, 1997
"... How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, ca ..."
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Cited by 10 (3 self)
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How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, can be approximated (very fast) with a computable convergence rate. Random reals, whose binary expansions contain very much information in their prefixes, can be approximated only very slowly by computable sequences of rationals (this is the case, for example, for Chaitin's \Omega numbers) if they can be computably approximated at all. We show that one can computably approximate any computable real also very slowly, with a convergence rate slower than any computable function. However, there is still a large gap between computable reals and random reals: any computable sequence of rationals which converges (monotonically) to a random real converges slower than any computable sequence of rat...
Monotonically computable real numbers
 Math. Log. Quart
, 2002
"... Key words hmonotone computable real, ωmonotone computable real. ..."
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Cited by 7 (5 self)
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Key words hmonotone computable real, ωmonotone computable real.
On approximating realworld halting problems
 Reischuk (Eds.), Proc. FCT 2005, in: Lectures Notes Comput. Sci
, 2005
"... Abstract. No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most ’ instances and thus solve it at least approximately. Whether and how ..."
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Abstract. No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most ’ instances and thus solve it at least approximately. Whether and how well such approximations are feasible highly depends on the underlying encodings and in particular the Gödelization (programming system) which in practice usually arises from some programming language. We consider BrainF*ck (BF), a simple yet Turingcomplete realworld programming language over an eight letter alphabet, and prove that the natural enumeration of its syntactically correct sources codes induces a both efficient and dense Gödelization in the sense of [Jakoby&Schindelhauer’99]. It follows that any algorithm M approximating the Halting Problem for BF errs on at least a constant fraction εM> 0 of all instances of size n for infinitely many n. Next we improve this result by showing that, in every dense Gödelization, this constant lower bound ε to be independent of M; while, the other hand, the Halting Problem does admit approximation up to arbitrary fraction δ> 0byan appropriate algorithm M δ handling instances of size n for infinitely many n. The last two results complement work by [Lynch’74]. 1
Omega and the time evolution of the Nbody problem
, 2007
"... The series solution of the behavior of a finite number of physical bodies and Chaitin’s Omega number share quasialgorithmic expressions; yet both lack a computable radius of convergence. ..."
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Cited by 4 (4 self)
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The series solution of the behavior of a finite number of physical bodies and Chaitin’s Omega number share quasialgorithmic expressions; yet both lack a computable radius of convergence.
Resolution of the uniform lower bound problem in constructive analysis
, 2007
"... constructive analysis ..."
Periods and elementary real numbers
, 805
"... The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In this paper, we compare the periods with hierarchy of real numbers induced fr ..."
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The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In this paper, we compare the periods with hierarchy of real numbers induced from computational complexities. In particular we prove that periods can be effectively approximated by elementary rational Cauchy sequences. As an application, we exhibit a computable real number which is not a period. 1
Rates of convergence of recursively defined sequences
 CCA 2004
, 2004
"... This paper gives a generalization of a result by Matiyasevich which gives explicit rates of convergence for monotone recursively defined sequences. The generalization is motivated by recent developments in fixed point theory and the search for applications of proof mining to the field. It relaxes th ..."
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This paper gives a generalization of a result by Matiyasevich which gives explicit rates of convergence for monotone recursively defined sequences. The generalization is motivated by recent developments in fixed point theory and the search for applications of proof mining to the field. It relaxes the requirement for monotonicity to the form xn+1 ≤ (1 + an)xn + bn where the parameter sequences have to be bounded in sum, and also provides means to treat computational errors. The paper also gives an example result, an application of proof mining to fixed point theory, that can be achieved by the means discussed in the paper.