Computability and Models: Perspectives East and West (University Series in Mathematics)

leondumoulin.nl: Computability and Models: Perspectives East and West (University Series in Mathematics): S. Barry Cooper, Sergei S. Goncharov.
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Review: Computability and Models: Perspectives East and West | EMS

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Assuming the conjecture that probabilistic polynomial time BPP equals deterministic polynomial time P , the word 'probabilistic' is optional in the complexity-theoretic Church—Turing thesis. A similar thesis, called the invariance thesis , was introduced by Cees F. Slot and Peter van Emde Boas. In other words, there would be efficient quantum algorithms that perform tasks that do not have efficient probabilistic algorithms.

This would not however invalidate the original Church—Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church—Turing thesis for efficiency reasons. Consequently, the quantum complexity-theoretic Church—Turing thesis states: Eugene Eberbach and Peter Wegner claim that the Church—Turing thesis is sometimes interpreted too broadly, stating "the broader assertion that algorithms precisely capture what can be computed is invalid".

Philosophers have interpreted the Church—Turing thesis as having implications for the philosophy of mind. Jack Copeland states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain.

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When applied to physics, the thesis has several possible meanings:. There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. One can formally define functions that are not computable.

A well-known example of such a function is the Busy Beaver function. This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input.

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Finding an upper bound on the busy beaver function is equivalent to solving the halting problem , a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church—Turing thesis states that this function cannot be effectively computed by any method. Several computational models allow for the computation of Church-Turing non-computable functions. These are known as hypercomputers. Mark Burgin argues that super-recursive algorithms such as inductive Turing machines disprove the Church—Turing thesis.

This interpretation of the Church—Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above. The argument that super-recursive algorithms are indeed algorithms in the sense of the Church—Turing thesis has not found broad acceptance within the computability research community. From Wikipedia, the free encyclopedia. For the axiom CT in constructive mathematics, see Church's thesis constructive mathematics. History of the Church—Turing thesis.

This section relies largely or entirely upon a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. November Learn how and when to remove this template message. Merriam Webster's New Collegiate Dictionary 9th ed.


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Merriam-Webster's Online Dictionary 11th ed. What is effectively calculable is computable. He calls this "Church's Thesis". Church uses the words "effective calculability" on page ff. Church in Davis Archived from the original PDF on Editor's footnote to Post Finite Combinatory Process. With respect to his proposed Gandy machine he later adds LC.

Also a review of this collection: Smith, Peter July 11, Archived PDF from the original on July 27, Retrieved July 27, Bulletin of Symbolic Logic. Archived from the original PDF on November 24, An introduction to quantum computing. Handbook of Theoretical Computer Science A. On tape versus core: Jack November 10, In Zalta, Edward N.

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Stanford Encyclopedia of Philosophy. Classical and Contemporary Readings. The Blackwell guide to the philosophy of computing and information. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Monographs in Computer Science. Barwise, Jon ; Keisler, H. Bernstein, E; Vazirani, U.

Blass, Andreas ; Gurevich, Yuri October Monographs in computer science. Church, Alonzo April American Journal of Mathematics. Date and year link Church, Alonzo June Could a Computer Write Hamlet? Incomputability, Emergence, and the Turing Universe Authors: Slides of talk at the University of Sofia, Bulgaria, June 26, Post's Programme and the Algorithmic Structure of Information: Slides of talk given at the University of Sofia, Bulgaria, June 24, How Computationally Relevant is Definability?

The Interactive Structure of Information: Post's Programme Revisited Authors: Computability in Europe - a very brief introduction Authors: Also - suggestions received of missing faces from the slide of people from computability in Europe before Genericity and Enumeration Reducibility Authors: Heidelberg version , Kazan version Status: Slides of a talk given for Turing A celebration of his life and achievements , held at the University of Manchester, June 5th, , and run jointly by the British Logic Colloquium and the British Society for the History of Mathematics.

Nonrigidity as Paradigm Shift Author: The Machine as Data: In Synthese , See published version at http: The final publication is available at link. Harris, Kyung Il Lee and S.

To appear in the Mathematical Logic Quarterly. Embodying Computation at Higher Types Authors: Barry Cooper and Jan van Leeuwen Formats: Alan Turing and Enigmatic Statistics Authors: Incomputability after Alan Turing Authors: In Computation, Physics and Beyond eds. In A Computable Universe: Understanding and Exploring Nature as Computation H. The Incomputable Reality Authors: Nature , 23 February , p. In Logic and Philosophy Today A. In Computability in Context: Computation and Logic in the Real World S. International Journal of Unconventional Computing , 6 , The Computational Content of Supervenience Authors: In Theory and Applications of Models of Computation: Theoretical Computer Science , , Extending and Interpreting Post's Programme Authors: Annals of Pure and Applied Logic , , Emergence as a Computability-Theoretic Phenomenon Authors: Applied Mathematics and Computation , , Wah , Wiley-Interscience, , pp.

Barry Cooper, Iskander Sh. Kalimullin and Mariya I. Barry Cooper, Matthew C. Salts and Guohua Wu Formats: Mariya Ivanova Soskova and S. Journal of Symbolic Logic , 73 , The Strongest Nonsplitting Theorem Authors: Post's Programme for the Ershov Hierarchy Authors: Bahareh Afshari, George Barmpalias, S. Barry Cooper and Frank Stephan Formats: In Journal of Logic and Computation , 17 , - Immunity properties and the n-c. Bahareh Afshari, George Barmpalias and S. Barry Cooper, Angsheng Li, eds.

Definability as hypercomputational effect Authors: Computation and Computability Authors: If CiE did not exist, it would be necessary to invent It Authors: Introduction to CiE Computability and emergence Authors: Logics for the XXIst Century eds.


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