Abstract
The Δ-calculus is an applicative system having the axioms of universality and approximation and the so-called s-m-n-axiom. It is proved that this calculus is interpreted in the theory of partially continuous operators on the space C[0, 1] consisting of continuous real functions defined on the segment [0, 1] . The problems of realizability, continuity, and computability are considered. The realizability of functionals and operators is understood to be the possibility of their representation by macrotransducers over labeled trees.
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L. P. Lisovik, “Algorithmic questions for real functions,” Kibernetika, No. 1, 12–17 (1987).
L. P. Lisovik, “Macrotransducers on labeled trees and partially continuous operators,” Kibern. Sist. Anal., No. 1, 16–24 (1992).
L. P. Lisovik, “Axiomatic theory of partially continuous functions and the Peano curve,” Kibern. Sist. Anal., No. 3, 58–68 (1993).
L. P. Lisovik, “Logical properties of partially continuous functions,” in: Mathematical Logic and Algorithmic Problems, Tr. IM SOAN SSSR, 12, Novosibirsk (1989), pp. 39–72.
G. S. Tseitin, “Algorithmic operators in constructive complete separable metric spaces,” Dokl. Akad. Nauk SSSR, 128, 49–52 (1959).
B. A. Kushner, Lectures on Constructive Mathematical Analysis [in Russian], Nauka, Moscow (1975).
P. Martin-Loef, An Outline of Constructive Mathematics [Russian translation], Mir, Moscow (1975).
L. P. Lisovik, “Classes of functions defined by transducers,” Kibernetika, No. 6, 1–8 (1990).
L. P. Lisovik, “Applications of finite transducers for specifying mappings and fractal sets,” Dokl. Ross. Akad. Nauk, 358 No. 1, 19–21 (1998).
H. Jr. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).
M. J. Beeson, Foundations of Constructive Mathematics, Springer-Verlag, Berlin (1985).
Y. Moschovakis, “Recursive metric spaces,” Fund. Math., 55 215–238 (1964).
L. P. Lisovik, “Another representation of finite transducers over real numbers,” Kibern. Sist. Anal., No. 2, 23–31 (1997).
A. I. Mal'tsev, Algorithms and Recursive Functions [in Russian], Nauka, Moscow (1965).
L. P. Lisovik, “Problems of interpolation of integer functions with infinite number of nodes,” Vychisl. Prikl. Mat., 55 26–34 (1991).
G. Polya andG. Szegoe, Aufgaben und Lehrsatze aus der Analysis [Russian translation], Nauka, Moscow (1978).
A. N. Kolmogorov, “Representation of continuous multivariable functions in the form of superpositions of continuous functions of one variable and addition,” Dokl. Akad. Nauk SSSR, 114 953–956 (1957).
L. A. Lyusternik andV. I. Sobolev, A Brief Course in Functional Analysis [in Russian], Vysshaya Shkola, Moscow (1982).
L. P. Lisovik, “Using macrotransducers to specify partial continuous operators in metric spaces,” Kibern. Sist. Anal., No. 5, 18–29 (1996).
B. A. Kushner, “Markov's constructive analysis: A participant's view,” Theor. Comput. Sci., 219 267–285 (1999).
A. N. Kolmogorov, “Representation of continuous multivariable functions by superpositions of continuous functions of a smaller number of variables,” Dokl. Akad. Nauk SSSR, 108 179–182 (1956).
V. I. Arnol'd, “Functions of three variables,” Dokl. Akad. Nauk SSSR, 114 679–681 (1957).
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Lisovik, L.P. Macrotransducers and Applicative Systems for Partially Continuous Functions. Cybernetics and Systems Analysis 37, 151–160 (2001). https://doi.org/10.1023/A:1016741416419
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DOI: https://doi.org/10.1023/A:1016741416419