Conference Paper

Semantics of a Sequential Language for Exact Real-Number Computation.

Birmingham Univ., England, UK
DOI: 10.1109/LICS.2004.1319637 Conference: 19th IEEE Symposium on Logic in Computer Science (LICS 2004), 14-17 July 2004, Turku, Finland, Proceedings
Source: DBLP

ABSTRACT We study a programming language with a built-in ground type for real numbers. In order for the language to be sufficiently expressive but still sequential, we consider a construction proposed by Boehm and Cartwright. The non-deterministic nature of the construction suggests the use of powerdomains in order to obtain a denotational semantics for the language. We show that the construction cannot be modelled by the Plotkin or Smyth powerdomains, but that the Hoare powerdomain gives a computationally adequate semantics. As is well known, Hoare semantics can be used in order to establish partial correctness only. Since computations on the reals are infinite, one cannot decompose total correctness into the conjunction of partial correctness and termination as it is traditionally done. We instead introduce a suitable operational notion of strong convergence and show that total correctness can be proved by establishing partial correctness (using denotational methods) and strong convergence (using operational methods). We illustrate the technique with a representative example.

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Available from: Martin Escardo, Dec 16, 2013
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    • "Most of the recent work on exact real number computation describes algorithms for functions on certain exact representations of the reals (for example streams of signed digits [18] [19] or linear fractional transformations [17]) and proves their correctness using a certain proof method (for example coinduction [16] [11] [8] [30]). Our work has a similar aim, and builds on the work cited above, but there are two important differences. "
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    ABSTRACT: Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps.
    Logical Methods in Computer Science 01/2011; DOI:10.2168/LMCS-7(1:8)2011 · 0.44 Impact Factor
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    • "A proof of this fact and of correctness of the program is provided in Section 6, using the techniques developed below. For further examples see [22] "
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    ABSTRACT: We study a programming language with a built-in ground type for real numbers. In order for the language to be sufficiently expressive but still sequential, we consider a construction proposed by Boehm and Cartwright. The non-deterministic nature of the construction suggests the use of powerdomains in order to obtain a denotational semantics for the language. We show that the construction cannot be modelled by the Plotkin or Smyth powerdomains, but that the Hoare powerdomain gives a computationally adequate semantics. As is well known, Hoare semantics can be used in order to establish partial correctness only. Since computations on the reals are infinite, one cannot decompose total correctness into the conjunction of partial correctness and termination as it is traditionally done. We instead introduce a suitable operational notion of strong convergence and show that total correctness can be proved by establishing partial correctness (using denotational methods) and strong convergence (using operational methods). We illustrate the technique with a representative example.
    19th IEEE Symposium on Logic in Computer Science (LICS 2004), 14-17 July 2004, Turku, Finland, Proceedings; 01/2004
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