H. J. Boehm and R. Cartwright. Exact real arithmetic: formulating real numbers as functions. In Research topics in functional programming, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computations of the weak parallel or operation on the Sierpinski domain to computations of the addition operation on the interval domain. 1 Introduction Suppose we extend a functional programming language with an abstract data type real meant to represent in nite precision real numbers [3]. What is a sensible semantics for such a type or how does one explain to users what their programs involving real actually mean It is probably not sensible to interpret real as just the set of real numbers as this would not assign meaning to nonterminating computations ( nor to partial ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43-64. Addison-Wesley, 1990.

....in the signed digit system. The signed binary system is also the basis of the first extension of PCF with a real number data type in the work of Di Gianantonio [4,5] In the late 1980 s two other frameworks for exact real number computation were proposed. In the Boehm and Cartwright s approach [3,2], developed and implemented recently by Valerie Menissier, a computable real number is approximated by B adic numbers of the form k=B n where B is the base, n is a natural number and k is an integer. For any basic function in analysis a feasible algorithm has been presented in order to produce ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

.... Computable Real Analysis, Random Access Machines, Limits, Interval Arithmetic, Multi valued functions, C 1 Introduction In the last decade, several implementations of exact real arithmetic based on different theoretical approaches or programming languages have been discussed, see e.g. [BoCa90] (essentially based on decimal representation) EdPo97] linear fractional transformations) or [GL00] using multiple precision arithmetic) All these approaches lack the possibility of full imperative programming that would facilitate the implementation of the usual numerical algorithms. In ....

H. Boehm & R. Cartwright, Exact Real Arithmetic: Formulating real numbers as functions. In T. D., editor, Research Topics in Functional Programming, 43-64 (Addison-Wesley, 1990)

....there must be more than one representation for every real number. In the literature, there are broadly speaking three frameworks for exact real computer arithmetic: i) Infinite sequences of linear maps proposed by Avizienis [1] and appeared in the work of Watanuki et al. [23] Boehm an Cartwright [2], Di Gianantonio [5] Escardo [4] Nielsen et al. [18] and Menissier Morain [16] ii) Continued fraction expansions proposed by Gosper [7] developed by Peyton Jones [10] and Vuillemin [21] and advanced more recently by Kornerup et al. [15, 13, 12, 14] iii) Infinite composition of linear ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....abstract. Related work This paper has its origins in the first author s work on exact real number computation [10, 11] In this approach, real numbers are represented by concrete computational structures such as streams, allowing computations to be performed to any desired degree of accuracy [35, 6, 4, 5, 33]. Of particular relevance to our work is the issue of obtaining an abstract data type of real numbers, in which the underlying computational representation is hidden [5, 8, 10, 11] In the programming language Real PCF [10] the abstract data type is based on simple real number constructors and ....

.... computational structures such as streams, allowing computations to be performed to any desired degree of accuracy [35, 6, 4, 5, 33] Of particular relevance to our work is the issue of obtaining an abstract data type of real numbers, in which the underlying computational representation is hidden [5, 8, 10, 11]. In the programming language Real PCF [10] the abstract data type is based on simple real number constructors and destructors. Mathematically, the constructors are unary midpoint operations x 7 0 x and x 7 x 1 on the unit interval [0; 1] where x y = x y) 2 is the binary midpoint ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....numbers by allowing unbounded integers to represent the numerator and the denominator but it cannot handle basic operations such as the square root or exponential function. In the late 1980 s two frameworks for exact real number computation were proposed. In the approach of Boehm and Cartwright [21, 20], developed and implemented recently by Valerie Menissier Morain [90] a computable real number is approximated by B adic numbers of the form k B n where B is the base, n is a natural number and k is an integer. For any basic function in analysis, a feasible algorithm has been presented in ....

H. J. Boehm and R. Cartwright, Exact real arithmetic: Formulating real numbers as functions, Research topics in functional programming (D. Turner, editor), Addison-Wesley, 1990, pp. 43--64.

....to a product with contractivity 1 25 (in the limit) ln 2 = 1 2 4 6 1 Y n=1 n 2n 1 4n 7n 3 (I 0 ) 13 Historical Remarks and Pointers to Literature In the late 1980 s, two frameworks for exact real number computation were proposed. In the approach of Boehm and Cartwright [3, 4], a computable real number is approximated by rational numbers of the form K=r n where r is the base and K is a (usually big) integer. This approach was further developed and implemented by Valerie Menissier Morain [19] For any basic function in analysis a feasible algorithm has been presented ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43-64. Addison-Wesley, 1990.

....of the first argument is not sufficient to decide if it is smaller or bigger than zero, the function B[ cond] gives as output the most precise approximation of the second and third argument. Remark. The function cond is a parallel test function and cannot be implemented sequentially. In [8] there is a proof showing that sequential primitives are not sufficient to define real numbers as an abstract data type. The constant 9 is similar to the corresponding constant introduced in LCF by Plotkin [21] In LCF the interpretation of 9 is the function H : Nat Bool) Bool defined by: ....

H.-J. Boehm, R. Cartwright, "Exact Real Arithmetic: Formulating Real Numbers as Functions" in "Research Topics in Functional Programming" David Turner editor, Addison-Wesley, 1990, pp. 43-64

....point. Necessary concepts like functions as nonconstant objects, i.e. the possibility of constructing a function during a computation, would be well suited, e.g. for working with real functions which could be implemented with objects of type (N Q) N Q) For an existing implementation see [BoCa90]. 3. To use traditional imperative or object oriented programming languages: Despite the problems of implementing an appropriate semantics, we chose to use C as a base four our prototype, especially as there is a freely available GNU compiler offering high level concepts as overloading of ....

Boehm, H. & Cartwright, R., Exact Real Arithmetic: Formulating real numbers as functions. In T. D., editor, Research Topics in Functional Programming, 43-64 (AddisonWesley, 1990)

....Schemes over Continuous Algebras (Extended Abstract) Wilson R. de Oliveira, Jr. Departamento de Inform atica UFPE CCEN Caixa Postal 7851 CEP 50732 970 Recife PE, Brazil wrdodi.ufpe.br http: www.di. ufpe.br wrdo Abstract Motivated by the real numbers representation as lazy lists [4] a notion of continuous algebras was introduced in [9, 7] which generalizes both the ordered [15] and metric approaches [25] Here we investigate the notion of program scheme acting on this algebras with a view to study notions of computability and complexity of real functions. A byproduct of ....

....to continuous data types disappear. As we shall see in Section 2 the space of the lazy reals is not a domain (algebraic nor continuous cpos) of any kind and can not be seen as an ultrametric space. The reals as data type is an important issue in the analysis and semantics of numeric programs [4]. In what follows a variant of the redundant balanced radix notation [17] is employed. In this representation we allow digits to be negative as well as positive. For a more complete study of this representation in contrast to others see [4] What it is important for the work here is its ....

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Hans Boehm and Robert Cartwright. Exact real arithmetic: Formulating reals numbers as functions. Technical Report Rice COMP TR88-66, Department of Computer Science, Rice University, April 1988.

....j 1 y j 1 2 [k1 k2 ] y j 1 6 [k1 4k2 k3 ] y j 1 6 [k1 2k2 2k3 k4 ] 4 Exact Real Arithmetic Using LFTs By an exact real arithmetic framework we mean any set of algorithms for computing elementary function up to any required accuracy. One example is Boehm and Cartwright s framework [1, 2], developed and implemented by M enissier Morain, 17] which uses B adic numbers. Another example, which uses linear fractional transformations, is based on the work of Gosper, Vuillemin, Kornerup and Nielsen, and has been developed and implemented on the special base interval [0; 1] by Edalat ....

Boehm, H. J., Cartwright, R.: Exact Real Arithmetic: Formulating Real Numbers as Functions. In Turner, D., editor, Research Topics in Functional Programming, Addison{Wesley (1990) 43-64.

....by LFT s [8, 17, 11] After a general introduction, we specialize to the version used by the group of Edalat and Potts at Imperial College [14, 12, 13, 16, 6] 2. 1 From Digit Streams to Linear Fractional Transformations There are many ways to represent real numbers as infinite objects [3, 2, 4, 5]. Here, we are only concerned with representations as infinite streams of digits . These streams are evaluated incrementally; at any given time, only a finite prefix of the stream is known. There are several different stream representations which can be grouped into two large families: variations ....

....as infinite streams of digits . These streams are evaluated incrementally; at any given time, only a finite prefix of the stream is known. There are several different stream representations which can be grouped into two large families: variations of the familiar decimal representation [1, 3, 2, 5, 7, 11, 10], and continued fraction expansions [8, 17, 9] For the first family, consider the usual decimal representation. 1 A number such as 0:142 Delta Delta Delta can be unraveled from left to right as follows: 0:142 Delta Delta Delta = 1 10 (1 0:42 Delta Delta Delta) 0:42 Delta Delta ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....= n(a div n) a mod n and the fact that a mod n 2 ( Gamma n 2 ; n 2 ] 2. 2 Real numbers as sequences The representation of real numbers as limits of Cauchy sequences has a long tradition in constructive analysis [2] Boehm et al. suggested this approach to exact real number computation in [4, 3], where they call it real numbers as functions . In the original definition, a real number x 2 R is represented by a 2 function f : Q Q with jf(q) Gamma xj q for all q 0. For the implementation, this was changed to a function f : Z Z with 8n 2 Z: fi fi fi fi f(n) B n Gamma x fi fi ....

H.-J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Addison--Wesley, 1990.

....arithmetic by LFT s [8, 16, 11] After a general introduction, we specialise to the version used by the group of Edalat and Potts at Imperial College [14, 12, 13, 15, 6] 2. 1 From Digit Streams to Linear Fractional Transformations There are many ways to represent real numbers as infinite objects [3, 2, 4, 5]. Here, we are only concerned with representations as infinite streams of digits . These streams are evaluated incrementally; at any given time, only a finite prefix of the stream is known. There are several different stream representations which can be grouped into two large families: variations ....

....as infinite streams of digits . These streams are evaluated incrementally; at any given time, only a finite prefix of the stream is known. There are several different stream representations which can be grouped into two large families: variations of the familiar decimal representation [1, 3, 2, 5, 7, 11, 10], and continued fraction expansions [8, 16, 9] For the first family, consider the usual decimal representation. 1 A number such as 0:142 Delta Delta Delta can be unravelled from left to right as follows: 0:142 Delta Delta Delta = 1 10 (1 0:42 Delta Delta Delta) 0:42 Delta Delta ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....put the other way round: instead of storing D 2 k 1 , D 2 km as list of matrices or as list [k 1 ; km ] of digits, store it as the number K together with the length m. This brings the digit matrix approach in close relationship with Boehm and Cartwright s functional approach [2,3]. 6.4 Mass Emission Mass absorption may bring down the cost of real number computation, but for a real gain, also mass emission is needed. For, no matter how quick M 0 = M Delta D 2 k 1 Delta Delta Delta D 2 km can be computed, the result has entries of bit size Omega Gamma m) ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....do not produce correct results in practice, 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 ....

H. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In T. D., editor, Research Topics in Functional Programming, pages 43--64. AddisonWesley, 1990.

....arrow of the pushout diagram is the centre of the binary system. We omit a definition by cases scheme for the real line. An explicit definition of h in the case X = I is given by h(x) g 0 (2x) if x 1=2 g 1 (2x Gamma 1) if x 1=2 Equality of real numbers is not decidable in general [1, 2, 14]. Therefore, in order to compute h, we need to start the computations of g 0 (2x) g(2x Gamma 1) and the comparison x versus 1=2 simultaneously. If x 6= 1=2, as soon as we discover which of the cases x 1=2 or x 1=2 hold, we can abort the computation of g 0 (2x) or the computation of g 1 ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....there must be more that one representation for every real number. In the literature, there are broadly speaking three frameworks for exact real computer arithmetic: i) Infinite sequences of linear maps proposed by Avizienis [1] and appeared in the work of Watanuki et al. [27] Boehm an Cartwright [2], Nielsen et al. [20] Menissier Morain [18] Di Gianantonio [7] and Escard o [4] The last two authors studied extensions of PCF with a real number data type. ii) Continued fraction expansions proposed by Gosper [8] developed by Peyton Jones [12] and Vuillemin [25] and advanced more recently by ....

....its computation can be eventually made in the interval [0; 1] The Language for Positive Reals can be extended to a language for all reals using the above framework. Comparison of real numbers may be implemented using the quasi relational comparison operator ffl described by Boehm and Cartwright [2]. Finally, we note that this framework could also be extended to cater for interval inputs and interval outputs. ....

H. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....Sequential case analysis on the reals Sequentiality issues arise when one considers partial elements. For this part of the paper we assume some familiarity with domain theory [1,18] and the programming language PCF [16,11] As opposed to other case analysis operators considered in the literature [3,7], the above operator admits a sequential implementation. One of the simplest examples of a parallel operation is the so called parallel or [20,16] defined by the following table: false true true false false true true true true true Notice that this is part of Kleene s three valued ....

....8 : x if p = true, y if p = false, x u y if p = It is not clear to the author whether the pseudo parallel conditional can also be generalized to a large class of domains. In this paper we consider a generalization to a domain of signed digit numerals (Section 4) Boehm and Cartwright [3] observed that the parallel conditional is useful in connection with a computable partial inequality test (x y) 8 : true if x y, false if x y, otherwise. For instance, one can implement min by min(x; y) pif x y then x else y; because when the test diverges the parallel ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....do not produce correct results in practice, 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 ....

H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43--64. Addison-Wesley, 1990.

....only possible way to compute on real numbers. Also a form of exact computation is feasible. In this case the results of the computations can be obtained with arbitrary precision. The exact computation on real numbers has been studied by several authors. Here we just mention Boehm and Cartwright ([2, 3]) Ko ( 9] Martin Lof ( 12] Vuillemin ( 16] and Weihrauch ( 17] In [2] and [6] an approach to real number computation via lazy functional languages is considered. In this approach a real number is represented by an infinite lazy sequence of digits. A real function is implemented by a lazy ....

H.-J. Boehm and R. Cartwright. Exact real arithmetic: formulating real numbers as functions. In David Turner, editor, Research topics in functional programming, pages 43--64. Addison-Wesley, 1990.

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H. J. Boehm and R. Cartwright. Exact real arithmetic: formulating real numbers as functions. In Research topics in functional programming, 1990.

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Boehm, H. and Cartwright, R., Exact Real Arithmetic: Formulating Real Numbers as Functions, in D.A. Turner (ed.), Research Topics in Functional Programming, pp. 43-64, Addison-Wesley (1990).

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H. J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In Turner. D., editor, Research Topics in Functional Programming, pages 43-64. Addison-Wesley, 1990.

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H.-J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43--64. Adison-- Weseley, 1990.

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