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50
The Realizability Approach to Computable Analysis and Topology
, 2000
"... policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. ..."
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Cited by 52 (20 self)
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policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
A DomainTheoretic Approach to Computability on the Real Line
, 1997
"... In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and ..."
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Cited by 49 (11 self)
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In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of StoltenbergHansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turingmachine based approach which dates back to Grzegorczyk and Lacombe, is used by PourEl & Richards in their found...
Weihrauch degrees, omniscience principles and weak computability
, 2009
"... In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multivalued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partia ..."
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Cited by 32 (6 self)
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In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multivalued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semilattice with the disjoint union of multivalued functions as greatest lower bound operation. We show that parallelization is a closure operator for this semilattice and the parallelized Weihrauch degrees even form a lattice with the product of multivalued functions as greatest lower bound operation. We show that the Medvedev lattice and hence the Turing upper semilattice can both be embedded into the parallelized Weihrauch lattice in a natural way. The importance of Weihrauch degrees is based on the fact that multivalued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means
Effective Choice and Boundedness Principles in Computable Analysis
, 2009
"... In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which a ..."
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In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles on closed sets which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. Wellknown omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 27 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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Cited by 20 (0 self)
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Conformal mapping in linear time
, 2006
"... Abstract. Given any ɛ> 0 and any planar region Ω bounded by a simple ngon P we construct a (1 + ɛ)quasiconformal map between Ω and the unit disk in time C(ɛ)n. One can take C(ɛ) = C + C log 1 ɛ log log ..."
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Cited by 19 (13 self)
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Abstract. Given any ɛ> 0 and any planar region Ω bounded by a simple ngon P we construct a (1 + ɛ)quasiconformal map between Ω and the unit disk in time C(ɛ)n. One can take C(ɛ) = C + C log 1 ɛ log log
Formal Topology and Constructive Mathematics: the Gelfand and StoneYosida Representation Theorems
 Journal of Universal Computer Science
, 2005
"... Abstract. We present a constructive proof of the StoneYosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for falgebras. In turn, this theorem implies the Gelfand representation theorem for C*alge ..."
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Cited by 16 (6 self)
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Abstract. We present a constructive proof of the StoneYosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for falgebras. In turn, this theorem implies the Gelfand representation theorem for C*algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.
Is the Mandelbrot set computable?
 MATH. LOGIC QUART
, 2005
"... We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components sa ..."
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We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two–sided distance function of the Mandelbrot set is computable if the hyperbolicity conjecture is true. We formulate the question whether the distance function of the Mandelbrot set is computable also in terms of the escape time.
Constructive algebraic integration theory without choice”, in Mathematics, Algorithms and Proofs
 Dagstuhl Seminar Proceedings, 05021, Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
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Cited by 9 (5 self)
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Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.