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Some modifications of iterated Kleene calculability. (English. Russian original) Zbl 0636.03042

Algebra Logic 25, 184-199 (1986); translation from Algebra Logika 25, No. 3, 292-314 (1986).
N. V. Belyakin [Mat. Sbornik, Nov. Ser. 101(143), 21-43 (1976; Zbl 0342.02031); Algebra Logika 22, No.1, 3-25 (1983; Zbl 0538.03040)] considered a mechanism for constructing recursive hierarchies, iterated Kleene computability (IKC); along an ordinal enumeration \(\nu\) we generate oracles of increasing strength, Kleene relativized to a functional G of type 2, i.e., these oracles
1) calculate the functional G;
2) uniformly in the \(\nu\)-indices, solve the graphs of the preceding oracles (answer the “problems of the past”).
In this article we consider variations of conditions 1 and 2; in Sec. 2 we study a method of iterating an arbitrary monotonic or extending operator; in Sec. 3 we consider iterated Kleene computability with respect to multivalued functionals of type 2, simultaneously considered for three weaker variants of condition 2. We show that for almost all variations of condition 2, we do not obtain a great difference from IKC, whereas changing condition 1 can have a great effect on the properties of hierarchies.

MSC:

03D55 Hierarchies of computability and definability
03D65 Higher-type and set recursion theory
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References:

[1] N. V. Belyakin, ”Iterated Kleene calculability and the superjump,” Mat. Sb.,101, No. 1, 21–43 (1976). · Zbl 0342.02031
[2] N. V. Belyakin, ”A method of modeling second-degree classical arithmetic,” Algebra Logika,22, No. 1, 3–25 (1983). · Zbl 0538.03040
[3] N. V. Belyakin, ”Generalized calculations and second degree arithmetic,” Algebra Logika,9, No. 4, 375–407 (1970). · Zbl 0239.20059
[4] N. V. Belyakin and L. N. Pobedin, ”Dialog aspects in the foundations of mathematics,” All-Union Conf. on Applied Logic, Lecture Theses, Novosibirsk (1985), pp. 17–18.
[5] E. G. Hikiforova, ”A method of constructing enumerations, Mathematical logic founcations of the MOZ problem,” Vych. Sistemy,107, 80–95 (1985).
[6] E. G. Nikiforova, ”On a class of recursive hierarchies,” All-Union Conf. on Applied Logic, Lecture These, Novosibirsk (1985), pp. 164–165.
[7] S. C. Kleene, ”Recursive functionals and quantifiers of finite types,” Trans. Am. Math. Soc.,91, No. 1, 1–52 (1959). · Zbl 0088.01301
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