On the extraction of computational content from noneffective convergence proofs in analysis PDF Download

Author: Helmut Schwichtenberg
Publisher: Cambridge University Press
ISBN: 1139504169
Category : Mathematics
Languages : en
Pages : 480

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Book Description
Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Author: Pavol Safarik
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832537651
Category : Computers
Languages : en
Pages : 164

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Book Description
The Fields medalist, Terence Tao, recently emphasized the importance of ''hard'' (or finitary) analysis and connected the finitisation to the methods we will employ in this thesis: ... The main advantage of working in a finitary setting ... is that the underlying dynamical system becomes extremely explicit. ... In proof theory, this finitisation is known as Gödel functional interpretation ... For convergence theorems Tao calls the finitary formulation metastability and the corresponding explicit content its rate(s). In the case of the mean ergodic theorem such a rate can be used to obtain even an effective bound on the number of fluctuations. We introduce effective learnability and three other natural kinds of such finitary information and analyze the corresponding proof-theoretic conditions. Effective learnability not only provides means to know when to expect a bound on the number of fluctuations but also explains a very common pattern in the realizers for strong ergodic theorems. Moreover, we will see how a most natural example for a non-learnable convergence theorem closely relates to a notable exception to this pattern, the strong nonlinear ergodic theorem due to Wittmann. Finally, we show how can computational content be extracted in the context of non-standard analysis.

Author: Sanjeev Arora
Publisher: Cambridge University Press
ISBN: 0521424267
Category : Computers
Languages : en
Pages : 609

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Book Description
New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

Author: Richard H. Hammack
Publisher:
ISBN: 9780989472111
Category : Mathematics
Languages : en
Pages : 314

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Book Description
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

Author: Helmut Schwichtenberg
Publisher: Springer Science & Business Media
ISBN: 3642590489
Category : Computers
Languages : en
Pages : 405

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Book Description
The Marktoberdorf Summer School 1995 'Logic of Computation' was the 16th in a series of Advanced Study Institutes under the sponsorship of the NATO Scientific Affairs Division held in Marktoberdorf. Its scientific goal was to survey recent progress on the impact of logical methods in software development. The courses dealt with many different aspects of this interplay, where major progress has been made. Of particular importance were the following. • The proofs-as-programs paradigm, which makes it possible to extract verified programs directly from proofs. Here a higher order logic or type theoretic setup of the underlying language has developed into a standard. • Extensions of logic programming, e.g. by allowing more general formulas and/or higher order languages. • Proof theoretic methods, which provide tools to deal with questions of feasibility of computations and also to develop a general mathematical understanding of complexity questions. • Rewrite systems and unification, again in a higher order context. Closely related is the now well-established Grabner basis theory, which recently has found interesting applications. • Category theoretic and more generally algebraic methods and techniques to analyze the semantics of programming languages. All these issues were covered by a team of leading researchers. Their courses were grouped under the following headings.

Author: Andrew M. Pitts
Publisher: Cambridge University Press
ISBN: 0521580579
Category : Computers
Languages : en
Pages : 375

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Book Description
The aim of this volume is to present modern developments in semantics and logics of computation in a way that is accessible to graduate students. The book is based on a summer school at the Isaac Newton Institute and consists of a sequence of linked lecture course by international authorities in the area. The whole set have been edited to form a coherent introduction to these topics, most of which have not been presented pedagogically before.

Author: John MacCormick
Publisher: Princeton University Press
ISBN: 0691170665
Category : Computers
Languages : en
Pages : 404

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Book Description
An accessible and rigorous textbook for introducing undergraduates to computer science theory What Can Be Computed? is a uniquely accessible yet rigorous introduction to the most profound ideas at the heart of computer science. Crafted specifically for undergraduates who are studying the subject for the first time, and requiring minimal prerequisites, the book focuses on the essential fundamentals of computer science theory and features a practical approach that uses real computer programs (Python and Java) and encourages active experimentation. It is also ideal for self-study and reference. The book covers the standard topics in the theory of computation, including Turing machines and finite automata, universal computation, nondeterminism, Turing and Karp reductions, undecidability, time-complexity classes such as P and NP, and NP-completeness, including the Cook-Levin Theorem. But the book also provides a broader view of computer science and its historical development, with discussions of Turing's original 1936 computing machines, the connections between undecidability and Gödel's incompleteness theorem, and Karp's famous set of twenty-one NP-complete problems. Throughout, the book recasts traditional computer science concepts by considering how computer programs are used to solve real problems. Standard theorems are stated and proven with full mathematical rigor, but motivation and understanding are enhanced by considering concrete implementations. The book's examples and other content allow readers to view demonstrations of—and to experiment with—a wide selection of the topics it covers. The result is an ideal text for an introduction to the theory of computation. An accessible and rigorous introduction to the essential fundamentals of computer science theory, written specifically for undergraduates taking introduction to the theory of computation Features a practical, interactive approach using real computer programs (Python in the text, with forthcoming Java alternatives online) to enhance motivation and understanding Gives equal emphasis to computability and complexity Includes special topics that demonstrate the profound nature of key ideas in the theory of computation Lecture slides and Python programs are available at whatcanbecomputed.com

Author: Patricia Martinková
Publisher: CRC Press
ISBN: 1000899179
Category : Psychology
Languages : en
Pages : 348

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Book Description
This book covers the computational aspects of psychometric methods involved in developing measurement instruments and analyzing measurement data in social sciences. It covers the main topics of psychometrics such as validity, reliability, item analysis, item response theory models, and computerized adaptive testing. The computational aspects comprise the statistical theory and models, comparison of estimation methods and algorithms, as well as an implementation with practical data examples in R and also in an interactive ShinyItemAnalysis application. Key Features: Statistical models and estimation methods involved in psychometric research Includes reproducible R code and examples with real datasets Interactive implementation in ShinyItemAnalysis application The book is targeted toward a wide range of researchers in the field of educational, psychological, and health-related measurements. It is also intended for those developing measurement instruments and for those collecting and analyzing data from behavioral measurements, who are searching for a deeper understanding of underlying models and further development of their analytical skills.

Author: Matthias Baaz
Publisher: Cambridge University Press
ISBN: 1139498436
Category : Mathematics
Languages : en
Pages : 541

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Book Description
This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Author: Helmut Schwichtenberg
Publisher: Springer Science & Business Media
ISBN: 9401004137
Category : Mathematics
Languages : en
Pages : 419

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Book Description
As society comes to rely increasingly on software for its welfare and prosperity there is an urgent need to create systems in which it can trust. Experience has shown that confidence can only come from a more profound understanding of the issues, which in turn can come only if it is based on logically sound foundations. This volume contains contributions from leading researchers in the critical disciplines of computing and information science, mathematics, logic, and complexity. All contributions are self-contained, aiming at comprehensibility as well as comprehensiveness. The volume also contains introductory hints to technical issues, concise surveys, introductions, and various fresh results and new perspectives.