One of the basic differences between the primitive recursive functions on the natural numbers and the primitive recursive ordinal functions (PR) is the nearly complete absence of constant functions in PR. Since ω is ...
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One of the basic differences between the primitive recursive functions on the natural numbers and the primitive recursive ordinal functions (PR) is the nearly complete absence of constant functions in PR. Since ω is closed under all of the functions in PR, if α is any infinite ordinal, then λξ·α is not in PR. It is easily seen, however, that if one adds to the initial functions of PR the constant function λξ·ω, then all of the ordinals up to ω#, the next largest PR-closed ordinal, have their constant functions in this class. Since, however, such collections of functions are always countable, it is also the case that if one adds to the initial functions of PR the function λξ. α for uncountable α, then there are ordinals β < α whose constant functions are not in this collection. Because of this, the following objects are of interest:Definition. For all α,(i)PR(α) is the collection of functions obtained by adding to the initial primitive recursive ordinal functions, the function λξ· α;(ii) PRsp(α), the primitive recursive spectrum of α, is the set {β < α ∣ λξβ ∈ PR(α);(iii) Λ (α)= μρ(ρ∉ PRsp(α)).
Let ƒ be a real number. It is well known [7] that the set of rational numbers which are less than ƒ is a recursive set if and only if ƒ is representable as the limit of a recursive, recursively convergent sequence of ...
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Let ƒ be a real number. It is well known [7] that the set of rational numbers which are less than ƒ is a recursive set if and only if ƒ is representable as the limit of a recursive, recursively convergent sequence of rational numbers. In this paper we replace the condition that the set of rational numbers less than ƒ is recursive by the condition that this set is at various points in the Kleene hierarchy, and we replace the recursive, recursively convergent limit by a variety of other recursive limiting processes.
Using a recent result of ***, an extention of Ackermann-Peter hierarchy of unary primitive recursive functions to string-functions is obtained. The resulting hierarchy classifies the string-functions according to thei...
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Pure Lisp style recursive function programs are represented in a new way by sentences and schemata of first order logic. This permits easy and natural proofs of extensional properties of such programs by methods that ...
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In this paper we summarize our approach and findings in a point-free setting for recursive topology and recursive analysis. recursive analysis has received intensive attention recently and our approach, while differin...
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This article presents the design of a library for attaching and checking executable contracts to code written in the Elixir programming language. In addition to classical contract constructs such as preconditions and ...
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This article presents the design of a library for attaching and checking executable contracts to code written in the Elixir programming language. In addition to classical contract constructs such as preconditions and postconditions, the library allows specifying exceptional behaviour (i.e., which exceptions are thrown and under which conditions), detecting non-termination issues in recursive functions by specifying a strictly decreasing order in function arguments, and associating timers with function calls to detect slow computations. The library also focuses on language-specific features, enabling the association of contracts with the reception of messages sent by processes and the attachment of constraints to variable names (useful due to variable shadowing in Elixir). Moreover, stateful contracts (i.e., with a model state) permit specifying the behaviour of stateful APIs whose operations can be linearized. Using the stateful contracts, a monitor can be employed to check that the observed state can be explained in terms of possible linearizations.
We furnish a core-logical development of the G & ouml;del numbering framework that allows metamathematicians to attain limitative results about arithmetical truth without incorporating a genuine truth predicate in...
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We furnish a core-logical development of the G & ouml;del numbering framework that allows metamathematicians to attain limitative results about arithmetical truth without incorporating a genuine truth predicate into the language in a way that would lead to semantic closure. We show how Tarski's celebrated theorem on the arithmetical undefinability of arithmetical truth can be established using only core logic in both the object language and the metalanguage. We do so at a high level of abstraction, by augmenting the usual first-order language of arithmetic with a primitive predicate Tr and then showing how it cannot be a truth predicate for the augmented language. McGee established an important result about consistent theories that are in the language of arithmetic augmented by such a "truth predicate" Tr and that use G & ouml;del numbering to refer to expressions of the augmented language. Given the nature of his sought result, he was forced to use classical reasoning at the meta level. He did so, however, on the additional and tacit presupposition that the arithmetical theories in question (in the object language) would be closed under classical logic. That left open the dialectical possibility that a constructivist (or intuitionist) could claim not to be discomfited by the results, even if they were to "give a pass" on the unavoidably classical reasoning at the meta level. In this study we "constructivize" McGee's result, by presuming only core logic for the object language. This shows that the perplexity induced by McGee's result will confront the constructivist (or intuitionist) as well.
As the reliance of businesses and organizations on online operations continues to grow, the importance of addressing software security vulnerabilities becomes increasingly critical. This paper delves into the phenomen...
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A capacitor bank of appropriate capacitance value helps in self-excitation of Standalone Asynchronous Generator. İn order to determine the appropriate value of the capacitor to be connected in real time systems, simul...
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Corresponding to the definition of mu-recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applica...
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Corresponding to the definition of mu-recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applications of some specified operators. We prove that our class of recursive relations essentially coincides with our class of densely computable relations, defined via Turing machines. In the special case of the real numbers our subclass of recursive functions coincides with the classical class of computable real-valued functions, defined via Turing machines by Grzegorczyk, Lacombe and others.
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