The Experts below are selected from a list of 15 Experts worldwide ranked by ideXlab platform
Herbert B. Enderton - One of the best experts on this subject based on the ideXlab platform.
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4 – Recursive Enumerability
Computability Theory, 2020Co-Authors: Herbert B. EndertonAbstract:Publisher Summary The class of general recursive Partial Functions is exactly the same as the class of register-machine Computable Partial Functions. The fact that two different approaches yield the same class of Functions is evidence that one has here a “natural” class. The members of this class are called Computable Partial Functions. It covers both the total and nontotal Functions; it can be omitted in cases where one knows that the Function is total. Church's thesis is the assertion that the concept of being a Computable Partial Function is the correct formalization of the informal idea of being an effectively calculable Partial Function. The class of Computable Partial Functions includes all of the primitive recursive Functions. A Partial Function is a Computable Partial Function if and only if its graph is a recursively enumerable relation.
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The Computability Concept
Computability Theory, 2020Co-Authors: Herbert B. EndertonAbstract:This chapter concerns computability theory, also known as recursion theory, the area of mathematics dealing with the concept of an effective procedure—a procedure that can be carried out by specific rules. Effective procedures show how limiting the concept of decidability is. One can utilize the concepts of countable and uncountable sets. Computability theory arose before the development of digital computers. It is relevant to certain considerations in mathematical logic. This chapter describes several equivalent ways of formulating the concept in precise terms. The mathematical concept of a Computable Partial Function is the correct formalization of the informal concept of an effectively calculable Partial Function. It provides a general overview of a number of different ways of formalizing the concept of effective calculability. The idea behind the concept of effective calculable Functions is that one should be able to give explicit instructions—a program—for calculating such a Function. The precise concept of a Computable Partial Function is an accurate formalization of the informal concept of an effectively calculable Function.
Fredrik Dahlgren - One of the best experts on this subject based on the ideXlab platform.
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Computability and continuity in metric Partial algebras equipped with computability structures
Mathematical Logic Quarterly, 2004Co-Authors: Fredrik DahlgrenAbstract:In this paper we give an axiomatisation of the concept of a computability structure with Partial sequences on a many-sorted metric Partial algebra, thus extending the axiomatisation given by Pour-El and Richards in [9] for Banach spaces. We show that every Banach-Mazur Computable Partial Function from an effectively separable Computable metric Partial Σ-algebra A to a Computable metric Partial Σ-algebra B must be continuous, and conversely, that every effectively continuous Partial Function with semidecidable domain and which preserves the computability of a computably enumerable dense set must be Computable. Finally, as an application of these results we give an alternative proof of the first main theorem for Banach spaces first proved by Pour-El and Richards. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Itamar Pitowsky - One of the best experts on this subject based on the ideXlab platform.
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CiE - From Logic to Physics: How the Meaning of Computation Changed over Time
Lecture Notes in Computer Science, 2007Co-Authors: Itamar PitowskyAbstract:The common formulation of the Church-Turing thesis runs as follows: Every Computable Partial Function is Computable by a Turing machine Where by Partial Function I mean a Function from a subset of natural numbers to natural numbers. As most textbooks relate, the thesis makes a connection between an intuitive notion (Computable Function) and a formal one (Turing machine). The claim is that the definition of a Turing machine captures the pre-analytic intuition that underlies the concept computation. Formulated in this way the Church-Turing thesis cannot be proved in the same sense that a mathematical proposition is provable. However, it can be refuted by an example of a Function which is not Turing Computable, but is nevertheless calculable by some procedure that is intuitively acceptable.
Detlef Plump - One of the best experts on this subject based on the ideXlab platform.
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FoSSaCS - Computational Completeness of Programming Languages Based on Graph Transformation
Lecture Notes in Computer Science, 2001Co-Authors: Annegret Habel, Detlef PlumpAbstract:We identify a set of programming constructs ensuring that a programming language based on graph transformation is computationally complete. These constructs are (1) nondeterministic application of a set of graph transformation rules, (2) sequential composition and (3) iteration. This language is minimal in that omitting either sequential composition or iteration results in a computationally incomplete language. By computational completeness we refer to the ability to compute every Computable Partial Function on labelled graphs. Our completeness proof is based on graph transformation programs which encode arbitrary graphs as strings, simulate Turing machines on these strings, and decode the resulting strings back into graphs.
Annegret Habel - One of the best experts on this subject based on the ideXlab platform.
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FoSSaCS - Computational Completeness of Programming Languages Based on Graph Transformation
Lecture Notes in Computer Science, 2001Co-Authors: Annegret Habel, Detlef PlumpAbstract:We identify a set of programming constructs ensuring that a programming language based on graph transformation is computationally complete. These constructs are (1) nondeterministic application of a set of graph transformation rules, (2) sequential composition and (3) iteration. This language is minimal in that omitting either sequential composition or iteration results in a computationally incomplete language. By computational completeness we refer to the ability to compute every Computable Partial Function on labelled graphs. Our completeness proof is based on graph transformation programs which encode arbitrary graphs as strings, simulate Turing machines on these strings, and decode the resulting strings back into graphs.