Representations of Natural Numbers and Computability of Various Functions

Wrocławski, Michał

doi:10.1007/978-3-030-22996-2_26

Cited by 3 publications

(1 citation statement)

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“…A computable copy of a structure is thus a notation in which all the basic relations are computable (meaning that their images within the copy are computable). This is essentially Shapiro's idea (in philosophy of mathematics), as studied, though in a very restricted sense, in [21] and further in [22] and [23].…”

Section: Related Workmentioning

confidence: 96%

Degrees of relations on canonically ordered natural numbers and integers

Wrocławski

Bazhenov

Kalociński

2023

Preprint

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We investigate the degree spectra of computable relations on canonically ordered natural numbers (ω,<) and integers (ζ,<). As for (ω,<), we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all Δ2 degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to [1], we obtain a more general solution to the problem regarding possible degree spectra on (ω,<), answering the question whether there are infinitely many such spectra. As for (ζ,<), we prove the following dichotomy result: given an arbitrary computable relation R on (ζ,<), its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for (ω,<) obtained in [2], and provide initial insight to Wright's question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of [1]. MSC Classification: 03C57

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Section: Related Workmentioning

confidence: 96%

Degrees of relations on canonically ordered natural numbers and integers

Wrocławski

Bazhenov

Kalociński

2023

Preprint

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What is the Church-Turing Thesis?

Boker¹,

Dershowitz

2022

Axiomatic Thinking II

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Generalization of Shapiro’s theorem to higher arities and noninjective notations

Kalociński

Wrocławski

2022

Arch. Math. Logic

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In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination.

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Representations of Natural Numbers and Computability of Various Functions

Cited by 3 publications

References 5 publications

Degrees of relations on canonically ordered natural numbers and integers

Degrees of relations on canonically ordered natural numbers and integers

What is the Church-Turing Thesis?

Generalization of Shapiro’s theorem to higher arities and noninjective notations

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