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 Proceedings of the YSU, Physical and Mathematical Sciences, 2015, Issue 1, Pages 52–60 (Mi uzeru15)

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On non-classical theory of computability

S. A. Nigiyan

Yerevan State University

Abstract: Definition of arithmetical functions with indeterminate values of arguments is given. Notions of computability, strong computability and $\lambda$-definability for such functions are introduced. Monotonicity and computability of every $\lambda$-definable arithmetical function with indeterminate values of arguments is proved. It is proved that every computable, naturally extended arithmetical function with indeterminate values of arguments is $\lambda$-definable. It is also proved that there exist strong computable, monotonic arithmetical functions with indeterminate values of arguments, which are not $\lambda$-definable. The $\delta$-redex problem for strong computable, monotonic arithmetical functions with indeterminate values of arguments is defined. It is proved that there exist strong computable, $\lambda$-definable arithmetical functions with indeterminate values of arguments, for which the $\delta$-redex problem is unsolvable.

Keywords: arithmetical function, indeterminate value of argument, computability, strong computability, $\lambda$-definability, $\beta$-redex, $\delta$-redex.

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MSC: Primary 68Q01; Secondary 68Q05
Accepted:17.12.2014
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Citation: S. A. Nigiyan, “On non-classical theory of computability”, Proceedings of the YSU, Physical and Mathematical Sciences, 2015, no. 1, 52–60

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\Bibitem{Nig15} \by S.~A.~Nigiyan \paper On non-classical theory of computability \jour Proceedings of the YSU, Physical and Mathematical Sciences \yr 2015 \issue 1 \pages 52--60 \mathnet{http://mi.mathnet.ru/uzeru15} 

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This publication is cited in the following articles:
1. T. V. Khondkaryan, “On typed and untyped lambda-terms”, Uch. zapiski EGU, ser. Fizika i Matematika, 2015, no. 2, 45–52
2. S. A. Nigiyan, “On $\lambda$-definability of arithmetical functions with indeterminate values of arguments”, Uch. zapiski EGU, ser. Fizika i Matematika, 2016, no. 2, 39–47
3. S. A. Nigiyan, T. V. Khondkaryan, “On canonical notion of $\delta$-reduction and on translation of typed $\lambda$-terms into untyped $\lambda$-terms”, Uch. zapiski EGU, ser. Fizika i Matematika, 51:1 (2017), 46–52
4. S. A. Nigiyan, T. V. Khondkaryan, “On translation of typed functional programs into untyped functional programs”, Uch. zapiski EGU, ser. Fizika i Matematika, 51:2 (2017), 177–186
5. D. A. Grigoryan, “On incomparability of interpretation algorithms of typed functional programs with respect to undefined value”, Uch. zapiski EGU, ser. Fizika i Matematika, 52:2 (2018), 109–118
6. D. A. Grigoryan, “On main canonical notion of $\delta$-reduction”, Uch. zapiski EGU, ser. Fizika i Matematika, 52:3 (2018), 191–199
7. L. Budaghyan, D. A. Grigoryan, L. H. Torosyan, “A necessary and sufficient condition for the uniqueness of $\beta\delta$-normal form of typed $\lambda$-terms”, Uch. zapiski EGU, ser. Fizika i Matematika, 53:1 (2019), 28–36
8. D. A. Grigoryan, “On the uniqueness of $\beta\delta$-normal form of typed $\lambda$-terms for the canonical notion of $\delta$-reduction”, Uch. zapiski EGU, ser. Fizika i Matematika, 53:1 (2019), 37–46
9. S. A. Nigiyan, “$\lambda$-definability of built-in McCarthy functions as functions with indeterminate values of arguments”, Uch. zapiski EGU, ser. Fizika i Matematika, 53:3 (2019), 191–202
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