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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Recursive functionals and quantifiers of finite types revisited. V
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by S. C. Kleene PDF
Trans. Amer. Math. Soc. 325 (1991), 593-630 Request permission

Abstract:

This is the last in a sequence of papers that redoes the theory of recursion in finite types. A key feature of the theory is that a computation can succeed (or finish) even if some of its subcomputations do not, if these turn out to be irrelevant to the total computation. I give a detailed description of computations involving oracles for type $3$ functionals. The computation may be viewed formally as a transfinite sequence of symbolic expressions, but I also describe a semantics in which each expression is given a concrete realization.
References
  • S. C. Kleene, Recursive functionals and quantifiers of finite types revisited. I, Generalized recursion theory, II (Proc. Second Sympos., Univ. Oslo, Oslo, 1977) Studies in Logic and the Foundations of Mathematics, vol. 94, North-Holland, Amsterdam-New York, 1978, pp. 185–222. MR 516936
  • S. C. Kleene, Recursive functionals and quantifiers of finite types revisited. II, The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Studies in Logic and the Foundations of Mathematics, vol. 101, North-Holland, Amsterdam-New York, 1980, pp. 1–29. MR 591873
  • S. C. Kleene, Recursive functionals and quantifiers of finite types revisited. III, Patras Logic Symposion (Patras, 1980) Studies in Logic and the Foundations of Mathematics, vol. 109, North-Holland, Amsterdam-New York, 1982, pp. 1–40. MR 694251
  • Sympos. Pure Math, (from the American Mathematical Society’s 1982 Summer Research Institute on Recursion Theory at Cornell University, June 27-July 16) (A. Nerode, ed.), vol. 42, Amer. Math. Soc., Providence, R.I., pp. 119-138.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 325 (1991), 593-630
  • MSC: Primary 03D65
  • DOI: https://doi.org/10.1090/S0002-9947-1991-0974519-8
  • MathSciNet review: 974519