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39
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 (15 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.
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 43 (8 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...
Beyond The Universal Turing Machine
, 1998
"... We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a phi ..."
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Cited by 29 (1 self)
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We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.
Some ComputabilityTheoretical Aspects of Reals and Randomness
 the Lect. Notes Log. 18, Assoc. for Symbol. Logic
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. ..."
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Cited by 24 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
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 19 (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
DegreeTheoretic Aspects of Computably Enumerable Reals
 in Models and Computability
, 1998
"... A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequ ..."
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Cited by 13 (0 self)
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A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to #: For example, every representation A of # is Turing reducible to L###: Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below deg T L### necessarily contains a representation of #: 1 Introduction Computability theory essentially studies the relative computability of sets of natural numbers. Since G#odel introduced a method for coding s...
Induction And Recursion On The Real Line
"... We characterize the real line by properties similar to the socalled Peano axioms for natural numbers. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, s ..."
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Cited by 11 (9 self)
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We characterize the real line by properties similar to the socalled Peano axioms for natural numbers. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, sine, arc sine, etc. from simpler ones. In order to obtain such a characterization, we introduce a notion of infinitely iterated composition of morphisms in categories, and we state a fixed point theorem and an infinite composition theorem for uniform spaces. 1 Introduction We characterize the real line by properties similar to the socalled Peano axioms for natural numbers [10, 11, 19]. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, sine, arc sine, etc. from simpler ones. 1.1 Programme We begin by characterizing the unit interval I = [0; 1]. F...
The Broad Conception Of Computation
 American Behavioral Scientist
, 1997
"... A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, ..."
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Cited by 11 (2 self)
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A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine  a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'ChurchTuring thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of nonclassical computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a numb...
Presentations of computably enumerable reals
 Theoretical Computer Science
, 2002
"... Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings. ..."
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Cited by 10 (5 self)
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Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings.
Number Computability and Domain Theory
 Information and Computation
, 1996
"... We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. Thi ..."
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Cited by 8 (0 self)
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We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the literature using different methods. Domain theory is a useful tool to study higher order computability on real numbers. An interesting connection between Scotttopology and the standard topologies on the real line and on the space of continuous functions on reals is stated. An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space. 1