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The Functions Provable by First Order Abstraction

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Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2001)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2250))

Abstract

Function provability in higher-order logic is a versatile and powerful framework for conceptual classification as well as verification and derivation of declarative programs. Here we show that the functions provable in second-order logic with first-order set-abstraction are precisely the elementary functions. This holds regardless of whether the logic is classical, intuitionistic, or minimal.

The notion of provability here is not purely logical, as it incorporates a trivial theory of data, with axioms stating that each data object has a detectable main constructor which can be destructed. We show that this is necessary, by proving that without such rudimentary axioms the provable functions are merely the functions broadly-represented in the simply typed lambda calculus, a collection that does not even include integer subtraction.

Research supported by NSF grant CS-0105651

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Author information

Authors and Affiliations

  1. Computer Science Department, Indiana University, 47405, Bloomington, IN

    Daniel Leivant

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  1. Daniel Leivant
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Editor information

Editors and Affiliations

  1. Department of Software, Technical University of Catalonia, Jordi Girona 1, 08034, Barcelona, Spain

    Robert Nieuwenhuis

  2. Department of Computer Science, University of Manchester, Kilburn Building, Oxford Road, M13 9PL, Manchester, UK

    Andrei Voronkov

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© 2001 Springer-Verlag Berlin Heidelberg

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Leivant, D. (2001). The Functions Provable by First Order Abstraction. In: Nieuwenhuis, R., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2001. Lecture Notes in Computer Science(), vol 2250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45653-8_23

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  • DOI: https://doi.org/10.1007/3-540-45653-8_23

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42957-9

  • Online ISBN: 978-3-540-45653-7

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