Abstract
Infinite hierarchies of operators in constructive metric spaces (CMS's) are constructed, based on the convergence of an approximate representation of the operators. Under fairly general restrictions on the CMS (these restrictions are satisfied by, e.g., a CMS of constructive real numbers and a CMS of general recursive functions) it is shown that these hierarchies are nondegenerate. The hierarchies constructed are used for studying the complexity of operators on general recursive functions. Operators of superposition and bounded and unbounded minimization are considered, along with diverse recursion operators.
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Literature cited
S. V. Pakhomov, “Approximability of operators in constructive metric spaces,”Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. im. V. A. Steklova Akad. Nauk SSSR,60, 171–182 (1976).
N. A. Shanin, “Constructive real numbers and constructive function spaces,” Tr. Mat. Inst. Akad. Nauk SSSR,67, 15–294 (1962).
A. Grzegorczyk, “Some classes of recursive functions,” Rozpr. Mat.,4, 1–46 (1953).
S. V. Pakhomov, “Some properties of the graphs of functions in the Grzegorczyk hierarchy,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. im. V. A. Steklova Akad.Nauk SSSR,12, 105–107 (1972).
S. V. Pakhomov, “A simple syntactical definition of all the classes in the Grzegorczyk hierarchy,” Zap. Nauchn. Sem. Lening. Otd. Mat. Inst. im. V. A. Steklova Akad. Nauk SSSR,40, 127–130 (1974).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 183–193, 1976. Results announced May 15, 1975.
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Pakhomov, S.V. Hierarchies of operators in constructive metric spaces. J Math Sci 14, 1547–1554 (1980). https://doi.org/10.1007/BF01693985
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DOI: https://doi.org/10.1007/BF01693985