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11
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 49 (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.
Is wave propagation computable or can wave computers beat the Turing machine?
 PROC. LONDON MATH SOC
, 2002
"... By the ChurchTuring Thesis a numerical function is computable by a physical device if and only if it is computable by a Turing machine. The `if'part is plausible since every (sufficiently small) Turing machine can be simulated by a computer program which operates correctly as long as sufficie ..."
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Cited by 24 (5 self)
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By the ChurchTuring Thesis a numerical function is computable by a physical device if and only if it is computable by a Turing machine. The `if'part is plausible since every (sufficiently small) Turing machine can be simulated by a computer program which operates correctly as long as sufficient time and storage
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Cited by 15 (4 self)
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
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 8 (4 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
Degrees of unsolvability of continuous functions
 Journal of Symbolic Logic
"... Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce ..."
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Cited by 5 (0 self)
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Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees nontotal. Several fundamental results are proved: a continuous function with nontotal degree has no least degree representation, settling a question asked by PourEl and Lempp; every noncomputable f ∈ C[0, 1] computes a noncomputable subset of N; there is a nontotal degree between Turing degrees a <T b iff b is a PA degree relative to a; S ⊆ 2N is a Scott set iff it is the collection of fcomputable subsets of N for some f ∈ C[0, 1] of nontotal degree; and there are computably incomparable f, g ∈ C[0, 1] which compute exactly the same subsets of N. Proofs draw from classical analysis and constructive analysis as well as from computability theory. §1. Introduction. The computable real numbers were introduced in Alan Turing’s famous 1936 paper, “On computable numbers, with an application to the Entscheidungsproblem ” [40]. Originally, they were defined to be the reals
A critical look at design, verification, and validation of large scale simulations
"... Note to the Reader. I see six constituencies in CSE: computer, mathematical, and physical scientists; engineers; and technical and nontechnical managers. I have adopted a conversational tone to make this article as widely accessible as possible. Ihave also provided a Bibliography. ..."
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Cited by 1 (0 self)
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Note to the Reader. I see six constituencies in CSE: computer, mathematical, and physical scientists; engineers; and technical and nontechnical managers. I have adopted a conversational tone to make this article as widely accessible as possible. Ihave also provided a Bibliography.
Abstract
"... 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 ..."
Abstract
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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 maximization of functions and then recursively defining integration from maximization. In both cases we have an 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, which implies that it is also fully abstract. 1.
Real PCF extended with ∃ is universal (Extended Abstract ∗)
, 1996
"... Real PCF is an extension of the programming language PCF with a data type for the real line, introduced elsewhere. We show that Real PCF extended with ∃ is universal, in the sense that all computable elements of all types of its universe of discourse are definable. We also show that ∃ is not necessa ..."
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Real PCF is an extension of the programming language PCF with a data type for the real line, introduced elsewhere. We show that Real PCF extended with ∃ is universal, in the sense that all computable elements of all types of its universe of discourse are definable. We also show that ∃ is not necessary to define first order computable functions in Real PCF. In order to obtain our definability results, we consider a domainequationlike structure on the real numbers data type.