Alexander Meduna, Petr Horáček, Martin Tomko · Institution of Engineering and Technology eBooks · 2019
This theory makes use of Turing machines to demonstrate the mathematical limits of computation. Indeed, any computation beyond the power of Turing machines is also beyond the computer power. The Turing machine represents a relatively simple language -defining model. Consider Turing machines as computers of functions over nonnegative integers and demonstrates the existence of functions whose computation cannot be specified by any procedure.
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J. Buescu, D. S. Graça, N. Zhong · 2014
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Tao Jiang, Ming Li, Bala Ravikumar et al. · Chapman & Hall/CRC applied algorithms and data structures series · 1998
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Gian‐Carlo Rota · Advances in Mathematics · 1989
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Chris Hankin · 1994
Abstract A classical application of the A-calculus was in the study of computability. This is the topic of this chapter. Since most computations involve repetitive execution of “code”, fixed points (for encoding recursion) play a fundamental role; we start by reviewing the Fixed Point Theorem and (re-)introducing the concept of a fixed point combinator. Next, rather than add delta rules for constants (see Chapter 3), we explore how constants can be handled in the pure calculus. We introduce several ways of encoding numerals and functions on them. In the third section, we introduce the notion of A-definability- this allows us to relate the A-calculus to other formalisms such as Kleene’s Recursive Functions and Turing Machines. The final section of this chapter discusses the issue of decidability in the A-calculus and we present some undecidable problems.
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Toniann Pitassi Sheila, Sheila Mcilraith, Timothy Brecht · 2007
this article, we are interested in characterizing as precisely as possible, those problems that can and cannot be solved by a computer. We are not concerned here with the efficiency or practicality of the solutions; we are merely interested in any computer solution to the problem, even one that is unrealistically slow or that operates on an imaginary computer, even one exploiting unbounded time and memory space. The equally important problem of characterizing the problems that can and cannot be computed efficiently is addressed elsewhere in this encyclopedia. See Computational Complexity Theory.
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J. R. Shoenfield · Lecture notes in logic · 1993
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Tao Jiang, Ming Li, Bala Ravikumar et al. · Chapman & Hall/CRC applied algorithms and data structures series · 2009
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Studies in logic and the foundations of mathematics · 2000
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Princeton University Press eBooks · 2018
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Douglas Bridges · Graduate texts in mathematics · 1994
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Jack Pope · 2026
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J. R. Shoenfield · 2018
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Klaus Weihrauch · 1987
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Cutland, Nigel · 1980
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Nigel J. Cutland · Cambridge University Press eBooks · 1980
What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gödel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest.
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Löfgren, Lars · Lund University Publications (Lund University) · 1987
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The MIT Press eBooks · 2019
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George Tourlakis · 2022
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Ashwin Lall · Mathematical Foundations of Computer Science · 2024
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