Proof And Computation
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Proofs And Computations
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Author : Helmut Schwichtenberg
language : en
Publisher: Cambridge University Press
Release Date : 2011-12-15
Proofs And Computations written by Helmut Schwichtenberg and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-15 with Mathematics categories.
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.
Computation Proof Machine
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Author : Gilles Dowek
language : en
Publisher: Cambridge University Press
Release Date : 2015-05-05
Computation Proof Machine written by Gilles Dowek and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-05 with Computers categories.
Computation, calculation, algorithms - all have played an important role in mathematical progress from the beginning - but behind the scenes, their contribution was obscured in the enduring mathematical literature. To understand the future of mathematics, this fascinating book returns to its past, tracing the hidden history that follows the thread of computation.
Proofs And Algorithms
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Author : Gilles Dowek
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-01-11
Proofs And Algorithms written by Gilles Dowek and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-01-11 with Computers categories.
Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel’s incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.
Proof And Computation
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Author : Helmut Schwichtenberg
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Proof And Computation written by Helmut Schwichtenberg and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Computers categories.
Logical concepts and methods are of growing importance in many areas of computer science. The proofs-as-programs paradigm and the wide acceptance of Prolog show this clearly. The logical notion of a formal proof in various constructive systems can be viewed as a very explicit way to describe a computation procedure. Also conversely, the development of logical systems has been influenced by accumulating knowledge on rewriting and unification techniques. This volume contains a series of lectures by leading researchers giving a presentation of new ideas on the impact of the concept of a formal proof on computation theory. The subjects covered are: specification and abstract data types, proving techniques, constructive methods, linear logic, and concurrency and logic.
Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification
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Author : Klaus Mainzer
language : en
Publisher: World Scientific
Release Date : 2021-07-27
Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification written by Klaus Mainzer and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-27 with Mathematics categories.
This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.
Arithmetic Proof Theory And Computational Complexity
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Author : Peter Clote
language : en
Publisher:
Release Date : 2023
Arithmetic Proof Theory And Computational Complexity written by Peter Clote and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with Computational complexity categories.
This book principally concerns the area of "Logical Complexity Theory", the study of bounded arithmetic, propositional proof systems, length of proof, etc and relations to computational complexity theory. This includes an open problem list of 7 fundamental and 39 technical questions together with a bibliography of references.
Mathematics And Computation
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Author : Avi Wigderson
language : en
Publisher: Princeton University Press
Release Date : 2019-10-29
Mathematics And Computation written by Avi Wigderson and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-29 with Computers categories.
From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography
Computational Complexity
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Author : Sanjeev Arora
language : en
Publisher: Cambridge University Press
Release Date : 2009-04-20
Computational Complexity written by Sanjeev Arora and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-04-20 with Computers categories.
New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.
Mathesis Universalis Computability And Proof
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Author : Stefania Centrone
language : en
Publisher: Springer Nature
Release Date : 2019-10-25
Mathesis Universalis Computability And Proof written by Stefania Centrone and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-25 with Philosophy categories.
In a fragment entitled Elementa Nova Matheseos Universalis (1683?) Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects” (“objets quelconques”). It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” (“Gründe”) of others, and the latter are “consequences” (“Folgen”) of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.
Proof Computation And Agency
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Author : Johan van Benthem
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-04-02
Proof Computation And Agency written by Johan van Benthem and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-04-02 with Philosophy categories.
Proof, Computation and Agency: Logic at the Crossroads provides an overview of modern logic and its relationship with other disciplines. As a highlight, several articles pursue an inspiring paradigm called 'social software', which studies patterns of social interaction using techniques from logic and computer science. The book also demonstrates how logic can join forces with game theory and social choice theory. A second main line is the logic-language-cognition connection, where the articles collected here bring several fresh perspectives. Finally, the book takes up Indian logic and its connections with epistemology and the philosophy of science, showing how these topics run naturally into each other.