Download or read book Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification written by Klaus Mainzer and published by World Scientific. This book was released on 2021-07-27 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Temporal Logic From Philosophy And Proof Theory To Artificial Intelligence And Quantum Computing written by Klaus Mainzer and published by World Scientific. This book was released on 2023-05-12 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Proof And Computation Digitization In Mathematics Computer Science And Philosophy written by Mainzer Klaus and published by World Scientific. This book was released on 2018-05-30 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Predicative Foundations, Constructive Mathematics and Type Theory, Computation in Higher Types, Extraction of Programs from Proofs, and Algorithmic Aspects in Financial Mathematics. 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. Contents: Proof and Computation (K Mainzer) Constructive Convex Programming (J Berger and G Svindland) Exploring Predicativity (L Crosilla) Constructive Functional Analysis: An Introduction (H Ishihara) Program Extraction (K Miyamoto) The Data Structures of the Lambda Terms (M Sato) Provable (and Unprovable) Computability (S Wainer) Introduction to Minlog (F Wiesnet) Readership: Graduate students, researchers, and professionals in Mathematics and Computer Science. Keywords: Proof Theory;Computability Theory;Program Extraction;Constructive Analysis;PredicativityReview: Key Features: This book gathers recent contributions of distinguished experts It makes emerging fields accessible to a wider audience, appealing to a broad readership with diverse backgrounds It fills a gap between (under-)graduate level textbooks and state-of-the-art research papers
Download or read book Proofs and Computations written by Helmut Schwichtenberg and published by Cambridge University Press. This book was released on 2011-12-15 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Concepts of Proof in Mathematics Philosophy and Computer Science written by Dieter Probst and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-07-25 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.
Download or read book Homotopy Type Theory Univalent Foundations of Mathematics written by and published by Univalent Foundations. This book was released on with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Computational Logic and Set Theory written by Jacob T. Schwartz and published by Springer Science & Business Media. This book was released on 2011-07-16 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the ÆtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Topics and features: describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents an unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn’s lemma.
Download or read book Proof Technology in Mathematics Research and Teaching written by Gila Hanna and published by . This book was released on 2019 with total page 379 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.
Download or read book Proof Theory written by Wolfram Pohlers and published by . This book was released on 2014-01-15 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Selected Papers in Proof Theory written by Grigorii E. Mints and published by North-Holland. This book was released on 1992-01-01 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection includes papers devoted to the structural theory of proofs, which was born in the framework of Hilbert's program and is applied now in connection with various projects using the effective contents of formalized proofs. The main tool and unifying topic here is normalization, i.e. putting proofs into a normal form. The book presupposes some familiarity with the definition and elementary properties of Gentzen-type systems but little more. The first three papers introduce various normalization procedures different from popular ones. The next group deals with unwinding proofs, that is the extraction of an explicit realization from the proof of existential theorems. Normalization (or more precisely, normal form theorems) is applied to the solution of some problems in the following two papers. A separate group is formed by three papers dealing with applications of the theory of proofs to algebra, more specifically to coherence theorems in category theory. The last paper of the volume is a survey of proof theory and elementary model theory for modal logic up to the year 1974.
Download or read book Computation and Proof Theory written by Michael M. Richter and published by Springer. This book was released on 1984 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proof Theory written by Vincent F. Hendricks and published by . This book was released on 2014-01-15 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Program Proof written by Samuel Mimram and published by . This book was released on 2020-07-03 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book QED at Large written by Talia Ringer and published by . This book was released on 2019-09-03 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: Development of formal proofs of correctness of programs can increase actual and perceived reliability and facilitate better understanding of program specifications and their underlying assumptions. Tools supporting such development have been available for over 40 years but have only recently seen wide practical use. Projects based on construction of machine-checked formal proofs are now reaching an unprecedented scale, comparable to large software projects, which leads to new challenges in proof development and maintenance. Despite its increasing importance, the field of proof engineering is seldom considered in its own right; related theories, techniques, and tools span many fields and venues. QED at Large covers the timeline and research literature concerning proof development for program verification, including theories, languages, and tools. It emphasizes challenges and breakthroughs at each stage in history and highlights challenges that are currently present due to the increasing scale of proof developments. This monograph is intended for use by researchers and students who are new to the field. It provides the reader with an insightful overview of the work that has led to modern-day techniques for formally verifying software. In times of increasing automation, this underpins many software systems so future trends are also highlighted.
Download or read book Mathematics without Apologies written by Michael Harris and published by Princeton University Press. This book was released on 2017-05-30 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: An insightful reflection on the mathematical soul What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.
Download or read book Certified Programming with Dependent Types written by Adam Chlipala and published by MIT Press. This book was released on 2013-12-06 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: A handbook to the Coq software for writing and checking mathematical proofs, with a practical engineering focus. The technology of mechanized program verification can play a supporting role in many kinds of research projects in computer science, and related tools for formal proof-checking are seeing increasing adoption in mathematics and engineering. This book provides an introduction to the Coq software for writing and checking mathematical proofs. It takes a practical engineering focus throughout, emphasizing techniques that will help users to build, understand, and maintain large Coq developments and minimize the cost of code change over time. Two topics, rarely discussed elsewhere, are covered in detail: effective dependently typed programming (making productive use of a feature at the heart of the Coq system) and construction of domain-specific proof tactics. Almost every subject covered is also relevant to interactive computer theorem proving in general, not just program verification, demonstrated through examples of verified programs applied in many different sorts of formalizations. The book develops a unique automated proof style and applies it throughout; even experienced Coq users may benefit from reading about basic Coq concepts from this novel perspective. The book also offers a library of tactics, or programs that find proofs, designed for use with examples in the book. Readers will acquire the necessary skills to reimplement these tactics in other settings by the end of the book. All of the code appearing in the book is freely available online.
Download or read book Foundations of Constructive Analysis written by Errett Bishop and published by Ishi Press. This book was released on 2012-07 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, Foundations of Constructive Analysis, founded the field of constructive analysis because it proved most of the important theorems in real analysis by constructive methods. The author, Errett Albert Bishop, born July 10, 1928, was an American mathematician known for his work on analysis. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. From 1965 until his death, he was professor at the University of California at San Diego.
