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The hereditary partial effective functionals and recursion theory in higher types1

  • G. Longo (a1) and E. Moggi (a1)
  • DOI:
  • Published online: 01 March 2014

A type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings).

By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (HEO) are easily characterized.

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Research partially supported by Min. P.I. (fondi 60%) and, in part, by Consiglio Nazionale delle Ricerche (Comitato per la Matematica).

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[1]Yu. L. Ershov , The theory of A-spaces, Algebra and Logic, vol. 12 (1973), pp. 209232.

[7]J. Myhill and J. Shepherdson , Effective operations on partial recursive functions, Zeitschrift für Mathematische Logik and Grundlagen der Mathematik, vol. 1 (1955), pp. 310317.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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