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A Declarative Specification of Tree-Based Symbolic Arithmetic Computations

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Practical Aspects of Declarative Languages (PADL 2012)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7149))

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Abstract

We use Prolog as a flexible meta-language to provide executable specifications of some interesting mathematical objects and their operations. In the process, isomorphisms are unraveled between natural numbers and rooted ordered trees representing hereditarily finite sequences and rooted ordered binary trees representing Gödel’s System T types. Our isomorphisms result in an interesting “paradigm shift”: we provide recursive definitions that perform the equivalent of arbitrary-length integer computations directly on rooted ordered trees. Besides the theoretically interesting fact of “breaking the arithmetic/symbolic barrier”, our arithmetic operations performed with symbolic objects like trees or types turn out to be genuinely efficient – we derive implementations with asymptotic performance comparable to ordinary bitstring implementations of arbitrary-length integer arithmetic. The Prolog code of the paper, organized as a literate program, is available at http://logic.cse.unt.edu/tarau/research/2012/padl12.pl

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, University of North Texas, USA

    Paul Tarau

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  1. Paul Tarau
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Editors and Affiliations

  1. Microsoft Research Ltd, 7 J J Thomson Ave, CB3 0FB, Cambridge, United Kingdom

    Claudio Russo

  2. Dept. of Computer and Information Science, Brooklyn College, 2900 Bedfora Ave, 11210-2889, Brooklyn, NY, USA

    Neng-Fa Zhou

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Tarau, P. (2012). A Declarative Specification of Tree-Based Symbolic Arithmetic Computations. In: Russo, C., Zhou, NF. (eds) Practical Aspects of Declarative Languages. PADL 2012. Lecture Notes in Computer Science, vol 7149. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27694-1_20

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  • DOI: https://doi.org/10.1007/978-3-642-27694-1_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27693-4

  • Online ISBN: 978-3-642-27694-1

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