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Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
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The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
Relative randomness and real closed fields
 J. Symbolic Logic
, 2005
"... Abstract. We show that for any real number, the class of real numbers less random than it, in the sense of rKreducibility, forms a countable real closed subfield of the real ordered field. This generalizes the wellknown fact that the computable reals form a real closed field. With the same techniq ..."
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Abstract. We show that for any real number, the class of real numbers less random than it, in the sense of rKreducibility, forms a countable real closed subfield of the real ordered field. This generalizes the wellknown fact that the computable reals form a real closed field. With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master’s Thesis, National University of Singapore, in preparation). Lastly, we show that the class of d.c.e. reals is properly contained in the class of reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rKreducibility). §1. Introduction. What does it mean for one real number to be less random than another? In attempts to answer this question, to measure the relative randomness of reals, computability theorists have invented a variety of preorders
Computable Function Representations Using Effective Chebyshev Polynomial
"... Abstract—We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of ..."
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Abstract—We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.
Relative Randomness via RKReducibility
, 2006
"... Its focus is relative randomness as measured by rKreducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rKreducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ nB ↾ n) < d, where K(στ) is the conditional prefixfree descr ..."
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Its focus is relative randomness as measured by rKreducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rKreducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ nB ↾ n) < d, where K(στ) is the conditional prefixfree descriptional complexity of σ given τ. Herein i study the relationship between relative randomness and (standard) absolute randomness and that between relative randomness and computable analysis. i Acknowledgements Foremost, i would like to thank my advisor, Steffen Lempp, for all his words of wisdom and encouragement throughout the long years of the Ph.D. Also, thanks to Frank Stephan who worked with me on some of the questions herein at the Computational Prospects of
Computable versions of the uniform boundedness theorem
 Logic Colloquium 2002
, 2006
"... Abstract. We investigate the computable content of the Uniform Boundedness Theorem and of the closely related BanachSteinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequ ..."
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Abstract. We investigate the computable content of the Uniform Boundedness Theorem and of the closely related BanachSteinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the sequence is “given”. If it is just given with respect to the compact open topology (i.e. if just a sequence of “programs ” is given), then we cannot even compute an upper bound of the uniform bound in general. If, however, the pointwise bounds are available as additional input information, then we can effectively compute an upper bound of the uniform bound. Additionally, we prove an effective version of the contraposition of the Uniform Boundedness Theorem: given a sequence of linear bounded operators which is not uniformly bounded, we can effectively find a witness for the fact that the sequence is not pointwise bounded. As an easy application of this theorem we obtain a computable function whose Fourier series does not converge. §1. Introduction. In this paper we want to study the computational content of some theorems of functional analysis. The Uniform Boundedness Theorem is one of the central theorems of functional analysis and it has first been published in Banach’s thesis [1].
Computable Function Representations Using Effective Chebyshev Polynomial
"... Abstract—We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of ..."
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Abstract—We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.
Computable Function Representations Using Effective Chebyshev Polynomial
"... Abstract—We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of ..."
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Abstract—We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.
A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK
"... Abstract. We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ω1 which measures how complex it is to verify differentiability for that function. We show that for each recursi ..."
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Abstract. We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ω1 which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal α> 0, the set of Turing indices of C[0, 1] functions that are differentiable with rank at most α is Π2α+1complete. This result is expressed in the notation of Ash and Knight. 1.
TOPOLOGICAL CONJUGATIONS ARE NOT CONSTRUCTABLE
, 2013
"... Abstract. We construct two computable topologically conjugate functions for which no conjugacy is computable, or even hyperarithmetic, resolving an open question of Kennedy and Stockman [Pea07]. Author’s Note After this was written up I learned that most of the ideas below are effectivizations of f ..."
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Abstract. We construct two computable topologically conjugate functions for which no conjugacy is computable, or even hyperarithmetic, resolving an open question of Kennedy and Stockman [Pea07]. Author’s Note After this was written up I learned that most of the ideas below are effectivizations of folk theorems which were known to descriptive set theorists. Hjorth has given a nice account of these in section 4.2 of his book Classification and orbit equivalence relations [Hjo00]. The nonlogician who wants to understand the basic idea could still use Sections 2 and 3 as an accessible expository construction of a pair of computable topologically conjugate functions with no computable conjugacy. If one already has some familiarity with computable analysis and is interested in the hyperarithmetic aspect, I would recommend reading Hjorth’s book instead, because the presentation there suggests a cleaner way to encode linear orders into homeomorphisms.