K. Weihrauch, A simple introduction to computable analysis. Informatik--Berichte 171--7/1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....j = k, and the direct sense of the lemma. 2 3.1 BSS undecidability of Mort R (2; 2) Talking about the decidability or undecidability of MortR (2) requires one to talk about machines that manipulate real numbers. One approach is to use the machine model studied in recursive analysis (e.g. see [26]) However, this model does not meet our needs because one cannot decide whether a real number is equal to zero in this model [26] Another approach is to use the Turing machine model for real numbers proposed by Blum, Shub, and Smale in [4, 5] Roughly speaking, a BSS machine is an extended ....

....of MortR (2) requires one to talk about machines that manipulate real numbers. One approach is to use the machine model studied in recursive analysis (e.g. see [26] However, this model does not meet our needs because one cannot decide whether a real number is equal to zero in this model [26]. Another approach is to use the Turing machine model for real numbers proposed by Blum, Shub, and Smale in [4, 5] Roughly speaking, a BSS machine is an extended Random Access Machine [10] that treats real numbers as basic entities; namely a BSS machine contains an unbounded number of real ....

K. Weihrauch. A simple introduction to computable analysis. Technical Report 171-2/1995.

.... domains, e.g. 29, 30, 16, 7, 8, 11, 35 37] are generalisation of algebraic domains, e.g. 2, 27, 32, 33] The continuous domain (more precisely, the interval domain) for the reals was rst proposed by Dana Scott [29] and later was applied to mathematics, physics and real number computation in [7, 8, 36, 37, 27] and other publications. In this section we propose continuous domains, named as function domains to construct a computational model of operators and real valued functionals de ned on the set of continuous real valued functions. In Section 4 we give semantic characterisations of computable ....

.... domains, e.g. 29, 30, 16, 7, 8, 11, 35 37] are generalisation of algebraic domains, e.g. 2, 27, 32, 33] The continuous domain (more precisely, the interval domain) for the reals was rst proposed by Dana Scott [29] and later was applied to mathematics, physics and real number computation in [7, 8, 36, 37, 27] and others. In this section we propose continuous domains, named as function domains to construct a computational model of operators and real valued functionals de ned on the set of continuous real valued functions. 3.1 Interval Domain for the reals By the interval domain for the reals we mean ....

K. Weihrauch, A simple introduction to computable analysis, Informatik Berichte 171, FernUniversitat, Hagen, 1995, 2-nd edition.

.... domains, e.g. 31, 32, 14, 6, 7, 10, 38 40] are generalisation of algebraic domains, e.g. 2, 29, 34, 35] The continuous domain (more precisely, the interval domain) for the reals was rst proposed by Dana Scott [31] and later was applied to mathematics, physics and real number computation in [6, 7, 39, 40, 29] and others. In this article we propose continuous domains, named as function domains to construct a computational model of operators and real valued functionals de ned on the set of continuous real valued functions. In Section 2, we recall basic de nitions and tools from [7] and introduce some ....

K. Weihrauch, A simple introduction to computable analysis, Informatik Berichte 171, FernUniversitat, Hagen, 1995, 2-nd edition.

....or for very efficient implementations of log(x) This ability of computing limits seems to be a unique feature of the iRRAM. The limited scope of algebraic or symbolic computations on real numbers is left as soon as these operators are used. From this point on, Type 2 Theory of Effectivity (TTE) [Ko91,We95,We97] is the best fitting theoretical model. This implies e.g. that computable functions must essentially be continuous, that it is no longer possible to check whether two real numbers are equal, etc. A certain relaxation of the law that computability implies continuity has been discussed e.g. in ....

....computing extends the computational power of the iRRAM in a way that the scope of algebraic of symbolic computations is left, also the computational model of Blum, Shub, and Smale[BSS89] is no longer applicable. The correct corresponding theoretical model is Type 2 Theory of Effectivity (TTE) [Ko91,We95,We97], although this model works essentially on a Turing machine level. Example definitions of computations on topological structures like R on an abstract level (i.e. without using representations) can be found in [TZ99] but without multi valued functions and without an internal limit operator) or ....

K. Weihrauch, A Simple Introduction to Computable Analysis, Informatik Berichte 171 - 2/1995, FernUniversitt Hagen

....Th. Mller LG Mathematische Informatik Modellierung University of Trier D 54286 Trier http: www.informatik.uni trier.de mueller email: mueller uni trier.de 1 Introduction The theory of computability and complexity on the field of real numbers is already well developed, see e.g. [Ko91, We95]. On the other hand, the reality of computing with reals is still tied to the old concept of single and double precision approximations. Improvements like [Br78] did not find widespread usage, whereas newer approaches like [Kl93] are not satisfying at least from the viewpoint of type 2 ....

Weihrauch, K., A Simple Introduction to Computable Analysis, Informatik Berichte 171 - 2/1995, FernUniversitt Hagen, technical report 8

....generalized models of computation. Finally, the several theories of effective analysis and type 2 computability should be mentioned in this context, even if their paradigm of computation by approximation differs considerably from our point of view. Related surveys and discussions can be found in [3, 72, 73, 36]. In some sense, our model is a modification of Friedman s generalized Turing algorithms. Moreover, for structures of finite signatures, it is equivalent to a uniform version of Friedman s effective definitional schemes. In contrast to almost all former approaches however, we explicitly consider ....

K. Weihrauch, A simple introduction to computable analysis. Informatik--Berichte 171--2/1995, FernUniv. Hagen, 1995 56

....right. This kind of machine with in nite input and output is called a Type 2 machine. We can also de ne the notion of a computable real number as a special case with no input. This notion of computability dates back to Turing[Turing 1936] and is the basis of the e ective analysis [Weihrauch 1985, Weihrauch 1995] It looks intuitively natural in that, for example, it says is computable because there is a program which outputs the sequence 3:14159: in nitely. The wide use of stream based programming also shows the naturality of this kind of computation. Though this notion of computability depends on ....

.... 10 , f0; 1g 110 , and f0; 1g 010 , and the interpretation of a nite sequence like 01 becomes a closed interval [ 1 = 4 ; 1 = 2 ] instead of an open interval ( 1 = 4 ; 1 = 2 ) As an application of our representation, we give a simple proof of Theorem 3. 4 of [Weihrauch 1995], which says that there is no e ective enumeration of computable real numbers. Theorem 9 When (x i ) i2 is a computable sequence, then a computable number x with x 6= x i for all i 2 can be determined. Proof: Let s i be the Gray code representation of x i and suppose that there is a ....

K. Weihrauch. (1995) A simple introduction to computable analysis. Technical Report 171-7/1995, FernUniversiat. 21

....on the infinite input words given by (the code of) the objects approximating the arguments and produces as output the sequence of (codes of the) objects approximating the function value. An analysis of these approaches led Weihrauch to the development of his Type Two Theory of E#ectivity (TTE) [14, 22, 23, 24]. Here infinite entities like the reals are represented by infinite words over a finite alphabet. If such a word is computable (as a sequence of letters) then also the corresponding entity is considered as computable. A map between spaces of such # The paper mainly contains results from the ....

....the set of all words in # # that have a prefix from A. In order to compute with infinite words Turing machines with a finite number of readonly input tapes and a write only output tape are used so that on the output tape the head can move only in one direction. They are called type two machines [23]. If M is such a machine, say with k input tapes and input output alphabets # 1 , # k , #, then the partial map #M : # # 1 # # k # # # computed by M is defined as follows: #M (p 1 , p k ) p, if M with p 1 , p k written on its input tapes computes forever, writing ....

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K. Weihrauch, A simple introduction to computable analysis, Informatik Berichte 171, FernUniversitat Hagen, Hagen, 1995.

....due to the presence of roundoff errors. Moreover, they are inappropriate for problems whose solution is sensitive to small variations on the input. As a consequence, exact real number computation has been advocated as an alternative solution (see e.g. 4, 5, 25] on the practical side and e.g. [2, 20, 21, 24, 26, 27, 28] on the foundational side) However, work on exact real number computation has focused on representations of real numbers and has neglected the issue of data types for real numbers. In particular, programming languages for exact real number computation with an explicit distinction between ....

K. Weihrauch. A simple introduction to computable analysis. Technical Report 171 -- 7/1995, FernUniversitat, 1995.

....and the universality result follows. In order to obtain our definability results, we consider a domain equation like structure on the real numbers data type. The Real PCF notion of computability induces classical notions of computability on real numbers and real valued functions of real variables [3, 11, 12, 16, 19, 20], but this material is not included in this extended abstract due lack of space. Also, we only consider the unit interval type of Real PCF, although we indicate how the type for the whole real line can be handled. Contents 1 The real numbers domains 2 2 Effectively given coherent domains 5 3 ....

....real line. We thus sometimes notationally identify real numbers and singleton intervals. The reason for introducing partial numbers is similar to the reason for introducing partial functions N k N in recursion theory. For instance, the set of computable real numbers is countable but not r.e. [20], whereas the set of computable partial real numbers is r.e. because the set of computable elements of any domain is r.e. 6, 15] In particular, no programming language with a real numbers data type can define all computable real numbers without having some divergent programs of real number ....

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K. Weihrauch. A simple introduction to computable analysis. Technical Report 171 -- 7/1995, FernUniversitat, 1995.

....(exact) real number computation see Section 2.1 below. After this discovery, he and other mathematicians, logicians and computer scientists proposed many essentially equivalent concrete representations of real numbers that are suitable for constructive or mechanical computation see Weihrauch [26] for some classical examples. In this paper we consider a variation of Brouwer s solution, proposed by Wiedmer [29] in 1980 and further investigated by Boehm et al. 4] Di Gianantonio [8,10] and Weihrauch [26] among others, which consists of the use of signed digit numerals. One learns from Di ....

....numbers that are suitable for constructive or mechanical computation see Weihrauch [26] for some classical examples. In this paper we consider a variation of Brouwer s solution, proposed by Wiedmer [29] in 1980 and further investigated by Boehm et al. 4] Di Gianantonio [8,10] and Weihrauch [26], among others, which consists of the use of signed digit numerals. One learns from Di Gianantonio [9] that Leslie [14] considered signed digit numerals in 1817 from a philosophical point of view, and from Sunderhauf [23] that Cauchy [6] observed in 1840 that signed digit numerals simplify ....

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K. Weihrauch. A simple introduction to computable analysis. Technical Report 171--7/1995, FernUniversitat, 1995.

....due to the presence of round off errors. Moreover, they are inappropriate for problems whose solution is sensitive to small variations on the input. As a consequence, exact real number computation has been advocated as an alternative solution (see e.g. 7, 8, 42] on the practical side and e.g. [5, 31, 32, 36, 43, 44, 45] on the foundational side) However, work on exact real number computation has focused on representations of real numbers and has neglected the issue of data types for real numbers. In particular, programming languages for exact real number computation with an explicit distinction between ....

K. Weihrauch. A simple introduction to computable analysis. Technical Report 171 -- 7/1995, FernUniversitat, 1995.

....difference among these approaches lies in the way real numbers are represented [Gia93] In the notion of computable real functions given by Grzegorczyk s in [Grz57] is used the concept of computable operators on the set of natural numbers sequences. This notion was investigated and generalized by [Wei95]. In his approach an approximation of the output with arbitrary precision is computed from a suitable approximation of the input [Bra95] Another approach was developed by Blum, Shub and Smale [BSS89] Here real numbers are viewed as entities and the computable functions are generated from a ....

....of effective procedure is intuitive and not limited to countable sets, while the notion of computability on Turing machines and partial recursive functions limits the computable functions to countable ones. The theory of computability on countable sets is called type 1 computability by Weirhauch [Wei95] (here called ChurchTuring computability) whereas the computability on sets with the cardinality of continuum is called by him type 2 computability. The computability on sets with cardinality of the continuum has been more directioned to computability on the real numbers. Very different ....

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Klaus Weihrauch. A simple Introduction to Computable Analysis. Technical Report 171-2/1995, Fern Universitat, 1995.

....to prove that several set valued operators are not computably invariant. 1 Introduction In the model of computability of Computable Analysis, as introduced by Grzegorczyk and Lacombe (cf. 9, 17] and further developed by Pour El Richards, Friedman Ko, Kreitz Weihrauch and others (cf. [20, 11, 26]) each computable function is continuous. Consequently, a lot of operations fail to be computable, simply because they are discontinuous. For instance, the operator of differentiation d : C[0; 1] C[0; 1] f 7 f 0 is discontinuous w.r.t. the usual topology of uniform convergence on C[0; 1] ....

....each computable metric space is separable. Sometimes, we will say by abuse of notation that X or (X; d) is a computable metric space if ff resp. d is fixed or out of consideration. One can precisely define computable operations in computable metric spaces via Cauchy representations (cf. [25, 26]) or equivalently recursive operations via recursion operators (cf. 4, 6] We will omit these definitions and refer the reader to the references. As a further notion we will need computable Banach spaces which additionally have computable vector space operations. Definition 2.2 (Computable ....

K. Weihrauch, A Simple Introduction to Computable Analysis, Informatik Berichte 171, FernUniversit at Hagen (1995)

....due to the presence of round off errors. Moreover, they are inappropriate for problems whose solution is sensitive to small variations on the input. As a consequence, exact real number computation has been advocated as an alternative solution (see e.g. 4, 5, 20] on the practical side and e.g. [2, 15, 16, 19, 21, 22, 23] on the foundational side) However, work on exact real number computation has focused on representations of real numbers and has neglected the issue of data types for real numbers. In particular, programming languages for exact real number computation with an explicit distinction between ....

Klaus Weihrauch. A simple introduction to computable analysis. Technical Report 171 -- 7/1995, FernUniversitat, 1995.

....TTE, Type 2 Theory of Effectivity, to measure theory. TTE has been introduced by Kreitz and Weihrauch [KW84, KW85] as a general framework for studying effectivity, i.e. continuity, computability and computational complexity, in Analysis. For details the reader is referred to the introduction [Wei95b] and a recent short survey [Wei95a] containing most of the notations we shall use in this paper. More details can be found in [KW85, Wei87] Since this paper is a first attempt, we consider only the space of probability measures on the Borel subsets of the real unit interval. By f : A Gamma B ....

....notations, i.e. surjections : Sigma Gamma S, and representations, i.e. surjections ffi : Sigma Gamma M . Continuity and computability concepts are transferred from Sigma and Sigma via notations and representations, respectively, to the named sets straightforwardly, see [KW85, Wei87, Wei95b, Wei95a]. Mainly notations or representations which are compatible with some relevant structure on the set under consideration are of practical interest. We do not discuss this for notations (see [RW80, Wei87] and Appendix C in [Wei95b] but we will introduce effective notations explicitly whenever ....

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K. Weihrauch. A simple introduction to computable analysis. In Informatik--Berichte, Band 171, Fernuniversitat Hagen, July 1995. 2nd ed. 14

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K. Weihrauch, A simple introduction to computable analysis. Informatik--Berichte 171--7/1995.

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K. Weihrauch. A simple introduction to computable analysis. Technical Report 171--2/1995.

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Klaus Weihrauch. A simple introduction to computable analysis. Technical report, Fern Universit at, Hagen, 1995.

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Klaus Weihrauch. A simple introduction to computable analysis. Technical Report 171 -- 7/1995, FernUniversitat, 1995.

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