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Recursive functionals and quantifiers of finite types revisited. V


Author: S. C. Kleene
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
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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.


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  • S. C. Kleene, Recursive functionals and quantifiers of finite types revisited. I, Generalized recursion theory, II (Proc. Second Sympos., Univ. Oslo, Oslo, 1977) Stud. Logic Foundations Math., 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), Stud. Logic Foundations Math., 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) Stud. Logic Foundations Math., 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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Article copyright: © Copyright 1991 American Mathematical Society