Algebra of Countably Functions and Theorems of Completeness
Pure and Applied Mathematics Journal
Volume 8, Issue 1, February 2019, Pages: 1-9
Received: Feb. 15, 2019; Accepted: Mar. 18, 2019; Published: Apr. 9, 2019
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Maydim Malkov, Department of Mathematics, Research Center for Artificial Intelligence, Moscow, Russia
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An algebraic approach to the theory of countable functions is given. Compositions (superpositions) of functions are used instead of recursions. Arithmetic and analytic algorithms are defined. All closed sets are founded. Mathematically precise definitions of logic algorithms with quantifiers of existence and universality are given. Logic algorithm for fast-growing function is built as example. Classification of functions is given. There are non-computable functions. These functions are fictitious (useless) and their set is continual. The set of computable functions is countable. Incompleteness of disjunction and negation, conjunction and negation, of Pierce, Sheffer and diagonal Webb functions is proved. The completeness of the set of one-place functions and any all-valued essential function (Slupecki theorem) is proved for computable functions. Existence of generators of all computable functions is proved too.
Discrete Functions, Complete Sets of Functions, Algebra Countably-valued Functions, Logic Programming, Theory of Algorithms
To cite this article
Maydim Malkov, Algebra of Countably Functions and Theorems of Completeness, Pure and Applied Mathematics Journal. Vol. 8, No. 1, 2019, pp. 1-9. doi: 10.11648/j.pamj.20190801.11
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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