Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification
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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.
Temporal Logic From Philosophy And Proof Theory To Artificial Intelligence And Quantum Computing
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Author : Klaus Mainzer
language : en
Publisher: World Scientific
Release Date : 2023-05-12
Temporal Logic From Philosophy And Proof Theory To Artificial Intelligence And Quantum Computing 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 2023-05-12 with Mathematics categories.
Calculi of temporal logic are widely used in modern computer science. The temporal organization of information flows in the different architectures of laptops, the Internet, or supercomputers would not be possible without appropriate temporal calculi. In the age of digitalization and High-Tech applications, people are often not aware that temporal logic is deeply rooted in the philosophy of modalities. A deep understanding of these roots opens avenues to the modern calculi of temporal logic which have emerged by extension of modal logic with temporal operators. Computationally, temporal operators can be introduced in different formalisms with increasing complexity such as Basic Modal Logic (BML), Linear-Time Temporal Logic (LTL), Computation Tree Logic (CTL), and Full Computation Tree Logic (CTL*). Proof-theoretically, these formalisms of temporal logic can be interpreted by the sequent calculus of Gentzen, the tableau-based calculus, automata-based calculus, game-based calculus, and dialogue-based calculus with different advantages for different purposes, especially in computer science.The book culminates in an outlook on trendsetting applications of temporal logics in future technologies such as artificial intelligence and quantum technology. However, it will not be sufficient, as in traditional temporal logic, to start from the everyday understanding of time. Since the 20th century, physics has fundamentally changed the modern understanding of time, which now also determines technology. In temporal logic, we are only just beginning to grasp these differences in proof theory which needs interdisciplinary cooperation of proof theory, computer science, physics, technology, and philosophy.
Grenzen K Nstlicher Intelligenz
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Author : Markus Maier
language : de
Publisher: Kohlhammer Verlag
Release Date : 2025-02-19
Grenzen K Nstlicher Intelligenz written by Markus Maier and has been published by Kohlhammer Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-02-19 with Philosophy categories.
Die aktuellen Entwicklungen Künstlicher Intelligenz vermitteln den Eindruck grenzenloser Möglichkeiten. Sie strahlen dabei in alle gesellschaftlichen Bereiche aus und bestimmen unsere lebensweltlichen Praxen maßgeblich mit. Wie bei allen großen technologischen Transformationen stellt sich dabei die Frage nach deren zukünftigen Potentialen, umgekehrt aber auch nach ihren prinzipiellen Grenzen. In diesem Spannungsverhältnis kann philosophische Reflexion durch begriffliche Klarheit für Orientierung sorgen und einen (auf-)klärenden Beitrag leisten, welcher dabei hilft ernstzunehmende technische Möglichkeiten von unvernünftigen Spekulationen sinnvoll unterscheiden zu können. Der vorliegende Band trägt damit zu einem übergreifenden Diskurs über die Beurteilung von Möglichkeiten und Grenzen Künstlicher Intelligenz bei.
Philosophisches Handbuch K Nstliche Intelligenz
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Author : Klaus Mainzer
language : de
Publisher: Springer-Verlag
Release Date : 2024-09-14
Philosophisches Handbuch K Nstliche Intelligenz written by Klaus Mainzer and has been published by Springer-Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-09-14 with Science categories.
Das Handbuch schlägt die Brücke von der Grundlagenforschung zum Orientierungswissen. Es greift damit die Bildungs- und Ausbildungsziele der bundesweiten MINT-Initiative auf, die Mathematik (M), Informatik (I), Naturwissenschaft (N) und Technik (T) als fachübergreifendes Schlüsselwissen für technisch-wissenschaftlich gestützte Gesellschaften versteht. Additives Wissen und Ausbildung in getrennten Disziplinen der Mathematik, Informatik, Naturwissenschaft und Technik reichen aber nicht aus. In der Künstlichen Intelligenz wachsen diese Disziplinen mit den Human- und Sozialwissenschaften zusammen. Zunächst sollen die Grundlagen der KI-Forschung methodisch und begrifflich geklärt werden. Philosophie wird als Grundlagenforschung verstanden, die logisch und methodisch die Prinzipien von Wissenschaft und Technik untersucht. Daher handelt es sich um ein „Philosophisches Handbuch“ (in diesem Fall der KI) und nicht um eine Bindestrich-Philosophie, also ein Handbuch der Philosophie einer Einzelwissenschaft. Denken und Wissen selber und das Selbstverständnis der Menschen verändern sich durch KI grundlegend.
Mathematical Innovation
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Author : Mr. A. Durai Ganesh
language : en
Publisher: QUILL TECH PUBLICATIONS
Release Date : 2025-06-16
Mathematical Innovation written by Mr. A. Durai Ganesh and has been published by QUILL TECH PUBLICATIONS this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-06-16 with categories.
Mathematical Innovation is a comprehensive and forward-looking exploration of how mathematics drives progress across science, technology, and modern industry. This book presents a rich collection of contemporary theories, applied methodologies, and creative problem-solving approaches that showcase the evolving role of mathematics in solving real-world challenges. Covering both pure and applied mathematics, it bridges classical concepts with emerging fields such as artificial intelligence, data science, optimization, and complex systems. Designed for students, educators, researchers, and professionals, the book highlights interdisciplinary connections and demonstrates how mathematical thinking fuels innovation across diverse domains. Through engaging explanations, illustrative examples, and real-world applications, Mathematical Innovation invites readers to see mathematics not just as a subject, but as a dynamic, essential tool for understanding and shaping the future.
Homotopy Type Theory Univalent Foundations Of Mathematics
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Author :
language : en
Publisher: Univalent Foundations
Release Date :
Homotopy Type Theory Univalent Foundations Of Mathematics written by and has been published by Univalent Foundations this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
A Course In Constructive Algebra
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Author : Ray Mines
language : en
Publisher: Springer Science & Business Media
Release Date : 1987-12-18
A Course In Constructive Algebra written by Ray Mines 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 1987-12-18 with Mathematics categories.
The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.
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.
Program Proof
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Author : Samuel Mimram
language : en
Publisher:
Release Date : 2020-07-03
Program Proof written by Samuel Mimram and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-03 with categories.
This course provides a first introduction to the Curry-Howard correspondence between programs and proofs, from a theoretical programmer's perspective: we want to understand the theory behind logic and programming languages, but also to write concrete programs (in OCaml) and proofs (in Agda). After an introduction to functional programming languages, we present propositional logic, λ-calculus, the Curry-Howard correspondence, first-order logic, Agda, dependent types and homotopy type theory.
Mathematics For Computation M4c
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Author : Marco Benini
language : en
Publisher: World Scientific Publishing Company
Release Date : 2023
Mathematics For Computation M4c written by Marco Benini and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with Computational complexity categories.
The overall topic of the volume, Mathematics for Computation (M4C), is mathematics taking crucially into account the aspect of computation, investigating the interaction of mathematics with computation, bridging the gap between mathematics and computation wherever desirable and possible, and otherwise explaining why not. Recently, abstract mathematics has proved to have more computational content than ever expected. Indeed, the axiomatic method, originally intended to do away with concrete computations, seems to suit surprisingly well the programs-from-proofs paradigm, with abstraction helping not only clarity but also efficiency. Unlike computational mathematics, which rather focusses on objects of computational nature such as algorithms, the scope of M4C generally encompasses all the mathematics, including abstract concepts such as functions. The purpose of M4C actually is a strongly theory-based and therefore, is a more reliable and sustainable approach to actual computation, up to the systematic development of verified software. While M4C is situated within mathematical logic and the related area of theoretical computer science, in principle it involves all branches of mathematics, especially those which prompt computational considerations. In traditional terms, the topics of M4C include proof theory, constructive mathematics, complexity theory, reverse mathematics, type theory, category theory and domain theory. The aim of this volume is to provide a point of reference by presenting up-to-date contributions by some of the most active scholars in each field. A variety of approaches and techniques are represented to give as wide a view as possible and promote cross-fertilization between different styles and traditions.