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Parallel Functions In Recursive Program Schemes
"... Abstract: We consider models of sequential programs (recursive program schemes) and analyze their extension with parallel functions. For this purpose, we introduce a special class of parallel functions (called invariant functions) that don’t depend on interpretation of domain on which they are defi ..."
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Abstract: We consider models of sequential programs (recursive program schemes) and analyze their extension with parallel functions. For this purpose, we introduce a special class of parallel functions (called invariant functions) that don’t depend on interpretation of domain on which they are defined. Expressive power of extended classes of recursive schemes is analyzed in terms of sequential reducibility between the used parallel functions. It is shown that the obtained hierarchy of schemes is infinite but not dense. KeyWords: Program models, semantics, parallel functions, expressive power.
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1
On the calculating power of Laplace’s demon
"... Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameter ..."
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Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameterized by the choice of an appropriate notion of computability. We present some simple examples to show that these choices can indeed make a difference to whether a given physical theory is ‘deterministic’ or not. Keywords: Laplace’s demon, physical determinism, philosophy of mathematics, notions of computability. 1
A Denotional Semantics for . . .
, 2007
"... We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by ..."
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We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by Boehm and Cartwright, is to use a nondeterministic test. For any two rational numbers p < q and any real number x, at least one of the relations p < x or x < q can be determined to hold; thus, an operator rtest is used, whose evaluation never diverges when x is a real number: 1. rtestp,q(x) evaluates to true or to false, 2. rtestp,q(x) may evaluate to true iff x < q and 3. rtestp,q(x) may evaluate to false iff p < x. Since a program can in general produce different results in different runs, Escardó and MarcialRomero took the view in previous work that programs of realnumber type denote sets of real numbers, and the question arose as to which power domains would be suitable for modelling the behaviour of rtest. It was shown that, among the known power domains,
Total Correctness for Sequential Real Programs 1
"... Real PCF is a higher type language for exact real number computation introduced by Martín Escardó [1]. It is universal, in the sense that it can define all computable functions over the reals, at all types. But, in place of the usual, sequential if operator, it includes a parallel conditional satisf ..."
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Real PCF is a higher type language for exact real number computation introduced by Martín Escardó [1]. It is universal, in the sense that it can define all computable functions over the reals, at all types. But, in place of the usual, sequential if operator, it includes a parallel conditional satisfying pif true then xelsey = x, pif falsethenxelse y = y, pif ⊥ thenxelsex = x, whose use is unavoidable, at least in a deterministic setting with a domainbased model (cf. [2]). In our approach, a realnumber computation is an infinite sequence of shrinking rational intervals. This infinite character of real numbers makes the tests x = y and x ≤ y undecidable. In particular, such tests cannot be used to control the execution flow of realnumber programs. An alternative solution to the use of a parallel operator is the use of a nondeterministic test: for any two numbers p < q and any number x, at least one of the relations p < x or x < q can be determined to hold (Boehm and Cartwright [3]). One can use a construct rtestp,q, for p < q rational, such that, for any real number x, 1. rtestp,q(x) evaluates to true or to false, 2. rtestp,q(x) may evaluate to true iff x < q, and 3. rtestp,q(x) may evaluate to false iff p < x. It is important here that evaluation never diverges.
Semidecidability of may, must . . . in a highertype setting
, 2009
"... We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic pro ..."
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We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semidecidability of similar tests under the simultaneous presence of nondeterministic and probabilistic choice. Keywords: Nondeterministic and probabilistic computation, highertype computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains.
Equivalent Transformations for Invariant Parallel Functions
"... Abstract: Monotonic parallel functions were extensively studied in research on semantics of programming languages. While most of this research concentrated on expressive power of parallel functions, this paper focuses on development of a rich catalog of equivalent transformations associated with in ..."
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Abstract: Monotonic parallel functions were extensively studied in research on semantics of programming languages. While most of this research concentrated on expressive power of parallel functions, this paper focuses on development of a rich catalog of equivalent transformations associated with invariant parallel functions. Such functions are independent of interpretation of their definition domain, and they can be naturally used as additional control means to enrich models of sequential programs (such as recursive program schemes). It is shown how the offered transformations can be applied for a variety of goals, such as regularization of terms and reducing the strength of the used operations.
Functional first order definability of LRTp
"... Abstract. The language LRTp is a nondeterministic language for exact real number computation. It has been shown that all computable first order relations in the sense of Brattka are definable in the language. If we restrict the language to singlevalued total relations (e.g. functions), all polynom ..."
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Abstract. The language LRTp is a nondeterministic language for exact real number computation. It has been shown that all computable first order relations in the sense of Brattka are definable in the language. If we restrict the language to singlevalued total relations (e.g. functions), all polynomials are definable in the language. In this paper we show that the nondeterministic version of the limit operator, which allows to define all computable first order relations, when restricted to singlevalued total inputs, produces singlevalued total outputs. This implies that not only the polynomials are definable in the language but also all computable first order functions.
Inteligencia Artificial. Revista Iberoamericana de Inteligencia Artificial
"... How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System ..."
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How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System