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Author: Oliver Aberth
Publisher: Academic Press
ISBN: 9780120417520
Category : Mathematics
Languages : en
Pages : 212
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Book Description
Computable Calculus treats the fundamental topic of calculus in a novel way that is more in tune with today's computer age. Comprising 11 chapters and an accompanying CD-ROM, the book presents mathematical analysis that has been created to deal with constructively defined concepts. The book's "show your work" approach makes it easier to understand the pitfalls of various computations and, more importantly, how to avoid these pitfalls. The accompanying CD-ROM has self-contained programs that interact with the text, providing for easy grasp of the new concepts and enabling readers to write their own demonstration programs. Contains software on CD ROM: The accompanying software demonstrates, through simulation and exercises, how each concept of calculus can be associated with a program for the 'ideal computer' Using this software readers will be able to write their own demonstration programs
Author: Oliver Aberth
Publisher: Academic Press
ISBN: 9780120417520
Category : Mathematics
Languages : en
Pages : 212
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Book Description
Computable Calculus treats the fundamental topic of calculus in a novel way that is more in tune with today's computer age. Comprising 11 chapters and an accompanying CD-ROM, the book presents mathematical analysis that has been created to deal with constructively defined concepts. The book's "show your work" approach makes it easier to understand the pitfalls of various computations and, more importantly, how to avoid these pitfalls. The accompanying CD-ROM has self-contained programs that interact with the text, providing for easy grasp of the new concepts and enabling readers to write their own demonstration programs. Contains software on CD ROM: The accompanying software demonstrates, through simulation and exercises, how each concept of calculus can be associated with a program for the 'ideal computer' Using this software readers will be able to write their own demonstration programs
Author: Hector Zenil
Publisher: World Scientific
ISBN: 9814374296
Category : Computers
Languages : en
Pages : 855
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Book Description
This volume discusses the foundations of computation in relation to nature. It focuses on two main questions: What is computation? and How does nature compute?
Author: John Longley
Publisher: Springer
ISBN: 3662479923
Category : Computers
Languages : en
Pages : 587
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Book Description
This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers
Author: Rod G. Downey
Publisher: Springer Nature
ISBN: 3031537440
Category : Computable functions
Languages : en
Pages : 361
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Book Description
The ideas and techniques comprised in the mathematical framework for understanding computation should form part of the standard background of a graduate in mathematics or computer science, as the issues of computability and complexity permeate modern science. This textbook/reference offers a straightforward and thorough grounding in the theory of computability and computational complexity. Among topics covered are basic naive set theory, regular languages and automata, models of computation, partial recursive functions, undecidability proofs, classical computability theory including the arithmetical hierarchy and the priority method, the basics of computational complexity and hierarchy theorems. Topics and features: · Explores Conway's undecidability proof of the "3x+1" problem using reductions from Register Machines and "Fractran" · Offers an accessible account of the undecidability of the exponential version of Hilbert's 10th problem due to Jones and Matijacevič · Provides basic material on computable structure, such as computable linear orderings · Addresses parameterized complexity theory, including applications to algorithmic lower bounds and kernelization lower bounds · Delivers a short account of generic-case complexity and of smoothed analysis · Includes bonus material on structural complexity theory and priority arguments in computability theory This comprehensive textbook will be ideal for advanced undergraduates or beginning graduates, preparing them well for more advanced studies or applications in science. Additionally, it could serve such needs for mathematicians or for scientists working in computational areas, such as biology.
Author: Rod Adams
Publisher: Docent Press
ISBN: 0983700400
Category : Mathematics
Languages : en
Pages : 312
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Book Description
Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.
Author: K. Vela Velupillai
Publisher: Routledge
ISBN: 1134253362
Category : Business & Economics
Languages : en
Pages : 514
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Book Description
Computable Foundations for Economics is a unified collection of essays, some of which are published here for the first time and all of which have been updated for this book, on an approach to economic theory from the point of view of algorithmic mathematics. By algorithmic mathematics the author means computability theory and constructive mathematics. This is in contrast to orthodox mathematical economics and game theory, which are formalised with the mathematics of real analysis, underpinned by what is called the ZFC formalism, i.e., set theory with the axiom of choice. This reliance on ordinary real analysis and the ZFC system makes economic theory in its current mathematical mode completely non-algorithmic, which means it is numerically meaningless. The book provides a systematic attempt to dissect and expose the non-algorithmic content of orthodox mathematical economics and game theory and suggests a reformalization on the basis of a strictly rigorous algorithmic mathematics. This removes the current schizophrenia in mathematical economics and game theory, where theory is entirely divorced from algorithmic applicability – for experimental and computational exercises. The chapters demonstrate the uncomputability and non-constructivity of core areas of general equilibrium theory, game theory and recursive macroeconomics. The book also provides a fresh look at the kind of behavioural economics that lies behind Herbert Simon’s work, and resurrects a role for the noble classical traditions of induction and verification, viewed and formalised, now, algorithmically. It will therefore be of particular interest to postgraduate students and researchers in algorithmic economics, game theory and classical behavioural economics.
Author: Hans Hermes
Publisher: Springer Science & Business Media
ISBN: 3642461786
Category : Mathematics
Languages : en
Pages : 260
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Book Description
Once we have accepted a precise replacement of the concept of algo rithm, it becomes possible to attempt the problem whether there exist well-defined collections of problems which cannot be handled by algo rithms, and if that is the case, to give concrete cases of this kind. Many such investigations were carried out during the last few decades. The undecidability of arithmetic and other mathematical theories was shown, further the unsolvability of the word problem of group theory. Many mathematicians consider these results and the theory on which they are based to be the most characteristic achievements of mathe matics in the first half of the twentieth century. If we grant the legitimacy of the suggested precise replacements of the concept of algorithm and related concepts, then we can say that the mathematicians have shown by strictly mathematical methods that there exist mathematical problems which cannot be dealt with by the methods of calculating mathematics. In view of the important role which mathematics plays today in our conception of the world this fact is of great philosophical interest. Post speaks of a natural law about the "limitations of the mathematicizing power of Homo Sapiens". Here we also find a starting point for the discussion of the question, what the actual creative activity of the mathematician consists in. In this book we shall give an introduction to the theory of algorithms.
Author: Neil D. Jones
Publisher: MIT Press
ISBN: 9780262100649
Category : Computers
Languages : en
Pages : 494
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Book Description
Computability and complexity theory should be of central concern to practitioners as well as theorists. Unfortunately, however, the field is known for its impenetrability. Neil Jones's goal as an educator and author is to build a bridge between computability and complexity theory and other areas of computer science, especially programming. In a shift away from the Turing machine- and G�del number-oriented classical approaches, Jones uses concepts familiar from programming languages to make computability and complexity more accessible to computer scientists and more applicable to practical programming problems. According to Jones, the fields of computability and complexity theory, as well as programming languages and semantics, have a great deal to offer each other. Computability and complexity theory have a breadth, depth, and generality not often seen in programming languages. The programming language community, meanwhile, has a firm grasp of algorithm design, presentation, and implementation. In addition, programming languages sometimes provide computational models that are more realistic in certain crucial aspects than traditional models. New results in the book include a proof that constant time factors do matter for its programming-oriented model of computation. (In contrast, Turing machines have a counterintuitive "constant speedup" property: that almost any program can be made to run faster, by any amount. Its proof involves techniques irrelevant to practice.) Further results include simple characterizations in programming terms of the central complexity classes PTIME and LOGSPACE, and a new approach to complete problems for NLOGSPACE, PTIME, NPTIME, and PSPACE, uniformly based on Boolean programs. Foundations of Computing series
Author: Benedikt Löwe
Publisher: Springer Science & Business Media
ISBN: 3642218741
Category : Computers
Languages : en
Pages : 330
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Book Description
This book constitutes the refereed proceedings of the 7th Conference on Computability in Europe, CiE 2011, held in Sofia, Bulgaria, in June/July 2011. The 22 revised papers presented together with 11 invited lectures were carefully reviewed and selected with an acceptance rate of under 40%. The papers cover the topics computability in analysis, algebra, and geometry; classical computability theory; natural computing; relations between the physical world and formal models of computability; theory of transfinite computations; and computational linguistics.
Author: Nigel Cutland
Publisher: Cambridge University Press
ISBN: 9780521294652
Category : Computers
Languages : en
Pages : 268
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Book Description
What can computers do in principle? What are their inherent theoretical limitations? 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 - a function whose values can be calculated in an automatic way.
