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PCF extended with real numbers
, 1996
"... We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (singlepoint intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be ..."
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Cited by 47 (14 self)
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We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (singlepoint intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be considered as “continuous words”. Concatenation of continuous words corresponds to refinement of partial information. The usual basic operations cons, head and tail used to explicitly or recursively define functions on words generalize to partial real numbers. We use this fact to give an operational semantics to the above referred extension of PCF. We prove that the operational semantics is sound and complete with respect to the denotational semantics. A program of real number type evaluates to a headnormal form iff its value is different from ⊥; if its value is different from ⊥ then it successively evaluates to headnormal forms giving better and better partial results converging to its value.
Lazy Functional Algorithms for Exact Real Functionals
 Lec. Not. Comput. Sci
, 1998
"... . We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a ..."
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Cited by 23 (0 self)
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. We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a practical application of a method, due to Berger, for computing quantifiers over streams. Correctness proofs for the algorithms make essential use of domain theory. 1 Introduction In exact real number computation, infinite representations of reals are employed to avoid the usual rounding errors that are inherent in floating point computation [46, 17]. For certain real number computations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floatingpoint algorithms can lead to completely erroneous results [1, 14]. In such situations, exact real number computation provides guaranteed correctness, although at the (probably...
Computation and Hypercomputation
 MINDS AND MACHINES
, 2003
"... Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification o ..."
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Cited by 15 (4 self)
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Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification of any experiment capable of refuting hypercomputation. We consider the implications of relativistic algorithms capable of solving the (Turing) Halting Problem. We also reject as a fallacy the argument that hypercomputation has no relevance because noncomputable values are indistinguishable from sufficiently close computable approximations. In addition to
Integration in real PCF
 Information and Computation
, 1996
"... Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number ..."
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Cited by 7 (3 self)
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Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number of steps. Based on a domaintheoretic approach to integration, we show how to define integration in Real PCF. We propose two approaches to integration in Real PCF. One consists in adding integration as primitive. The other consists in adding a primitive for function maximization and then recursively defining integration from maximization. In both cases we have a computational adequacy theorem for the corresponding extension of Real PCF. Moreover, based on previous work on Real PCF definability, we show that Real PCF extended with the maximization operator is universal. 1
The constructive reals as a Java Library
 J. Log. Algebr. Program
, 2004
"... We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the ..."
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Cited by 4 (0 self)
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We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the implementation could be easily understood, and to allow simple informal correctness arguments. We hope to demonstrate that even such a basic implementation of constructive real arithmetic can be useful in a number of contexts, including in a desk calculator utility distributed with the package. A secondary goal was to demonstrate that some secondorder functions on the reals, such as restricted inverse and derivative operations, can be implemented with su#cient performance to be useful.
Streaming RepresentationChangers
 LNCS
, 2004
"... Unfolds generate data structures, and folds consume them. ..."
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Cited by 3 (0 self)
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Unfolds generate data structures, and folds consume them.
Continuity and Limits of Programs
, 2003
"... In demonstrations of proofs [3] of correctness for programs that are designed to compute mathematical functions we attempted to show that programs have the right properties, namely the same properties as the mathematical functions they are alleged to compute. In the cited paper we were forced to han ..."
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In demonstrations of proofs [3] of correctness for programs that are designed to compute mathematical functions we attempted to show that programs have the right properties, namely the same properties as the mathematical functions they are alleged to compute. In the cited paper we were forced to handwave (our excuse was the need for brevity) in at least two places, trying to sidestep sticky issues. Here we try to address these issues by clarifying two concepts: (a) computational continuity and (b) equality in a domain where floatingpoint computations can be done to variable (presumably high) precision. We introduce notations prepresentable and pnegligible where p denotes precision, and show how this helps in our applications. 1