Download or read book Twenty Five Years of Constructive Type Theory written by Giovanni Sambin and published by Clarendon Press. This book was released on 1998-10-15 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: Per Martin-Löf's work on the development of constructive type theory has been of huge significance in the fields of logic and the foundations of mathematics. It is also of broader philosophical significance, and has important applications in areas such as computing science and linguistics. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Löf over the last twenty-five years. As well as celebrating the anniversary of the birth of the subject it covers many of the diverse fields which are now influenced by type theory. It is an invaluable record of areas of current activity, but also contains contributions from N. G. de Bruijn and William Tait, both important figures in the early development of the subject. Also published for the first time is one of Per Martin-Löf's earliest papers.

Download or read book Intuitionistic Type Theory written by Per Martin-Löf and published by . This book was released on 1984 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 Epistemology versus Ontology written by P. Dybjer and published by Springer Science & Business Media. This book was released on 2012-07-10 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?

Download or read book Proof Theory in Computer Science written by Reinhard Kahle and published by Springer. This book was released on 2003-06-30 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proof theory has long been established as a basic discipline of mathematical logic. It has recently become increasingly relevant to computer science. The - ductive apparatus provided by proof theory has proved useful for metatheoretical purposes as well as for practical applications. Thus it seemed to us most natural to bring researchers together to assess both the role proof theory already plays in computer science and the role it might play in the future. The form of a Dagstuhl seminar is most suitable for purposes like this, as Schloß Dagstuhl provides a very convenient and stimulating environment to - scuss new ideas and developments. To accompany the conference with a proc- dings volume appeared to us equally appropriate. Such a volume not only ?xes basic results of the subject and makes them available to a broader audience, but also signals to the scienti?c community that Proof Theory in Computer Science (PTCS) is a major research branch within the wider ?eld of logic in computer science.

Download or read book Typed Lambda Calculi and Applications written by Jean-Yves Girard and published by Springer. This book was released on 2003-07-31 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 4th International Conference on Typed Lambda Calculi and Applications, TLCA'99, held in L'Aquila, Italy in April 1999. The 25 revised full papers presented were carefully reviewed and selected from a total of 50 submissions. Also included are two invited demonstrations. The volume reports research results on various aspects of typed lambda calculi. Among the topics addressed are noncommutative logics, type theory, algebraic data types, logical calculi, abstract data types, and subtyping.

Download or read book Foundations of Software Science and Computation Structures written by Christel Baier and published by Springer. This book was released on 2018-04-14 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 21st International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2018, which took place in Thessaloniki, Greece, in April 2018, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2018.The 31 papers presented in this volume were carefully reviewed and selected from 103 submissions. The papers are organized in topical sections named: semantics; linearity; concurrency; lambda-calculi and types; category theory and quantum control; quantitative models; logics and equational theories; and graphs and automata.

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 The Realism Antirealism Debate in the Age of Alternative Logics written by Shahid Rahman and published by Springer Science & Business Media. This book was released on 2011-09-22 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: The relation between logic and knowledge has been at the heart of a lively debate since the 1960s. On the one hand, the epistemic approaches based their formal arguments in the mathematics of Brouwer and intuitionistic logic. Following Michael Dummett, they started to call themselves `antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to unify both traditions by means of what is now known as `explicit epistemic logic'. Under this view, epistemic contents are introduced into the object language as operators yielding propositions from propositions, rather than as metalogical constraints on the notion of inference. The Realism-Antirealism debate has thus had three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume believe that in the age of Alternative Logics, where manifold developments in logic happen at a breathtaking pace, this debate should be revisited. Contributors to this volume happily took on this challenge and responded with new approaches to the debate from both the explicit and the implicit epistemic point of view.

Download or read book Types for Proofs and Programs written by Thorsten Altenkirch and published by Springer. This book was released on 2003-06-29 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the strictly refereed post-workshop proceedings of the International Workshop on Types for Proofs and Programs, TYPES '98, held under the auspices of the ESPRIT Working Group 21900. The 14 revised full papers presented went through a thorough process of reviewing and revision and were selected from a total of 25 candidate papers. All current aspects of type theory and type systems and their relation to proof theory are addressed.

Download or read book Logic Colloquium 03 written by Viggo Stoltenberg-Hansen and published by Cambridge University Press. This book was released on 2017-03-30 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the twenty-fourth publication in the Lecture Notes in Logic series, contains the proceedings of the European Summer Meeting of the Association for Symbolic Logic, held in Helsinki, Finland, in August 2003. These articles include an extended tutorial on generalizing finite model theory, as well as seventeen original research articles spanning all areas of mathematical logic, including proof theory, set theory, model theory, computability theory and philosophy.

Download or read book Proof and System Reliability written by Helmut Schwichtenberg and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Mathesis Universalis Computability and Proof written by Stefania Centrone and published by Springer Nature. This book was released on 2019-10-25 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Thirty Five Years of Automating Mathematics written by F.D. Kamareddine and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: THIRTY FIVE YEARS OF AUTOMATING MATHEMATICS: DEDICATED TO 35 YEARS OF DE BRUIJN'S AUTOMATH N. G. de Bruijn was a well established mathematician before deciding in 1967 at the age of 49 to work on a new direction related to Automating Mathematics. By then, his contributions in mathematics were numerous and extremely influential. His book on advanced asymptotic methods, North Holland 1958, was a classic and was subsequently turned into a book in the well known Dover book series. His work on combinatorics yielded influential notions and theorems of which we mention the de Bruijn-sequences of 1946 and the de Bruijn-Erdos theorem of 1948. De Bruijn's contributions to mathematics also included his work on generalized function theory, analytic number theory, optimal control, quasicrystals, the mathematical analysis of games and much more. In the 1960s de Bruijn became fascinated by the new computer technology and as a result, decided to start the new AUTOMATH project where he could check, with the help of the computer, the correctness of books of mathematics. In each area that de Bruijn approached, he shed a new light and was known for his originality and for making deep intellectual contributions. And when it came to automating mathematics, he again did it his way and introduced the highly influential AUTOMATH. In the past decade he has also been working on theories of the human brain.

Download or read book Gentzen s Centenary written by Reinhard Kahle and published by Springer. This book was released on 2015-11-02 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

Download or read book From Sets and Types to Topology and Analysis written by Laura Crosilla and published by Oxford University Press on Demand. This book was released on 2005-10-06 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bridging the foundations and practice of constructive mathematics, this text focusses on the contrast between the theoretical developments - which have been most useful for computer science - and more specific efforts on constructive analysis, algebra and topology.

Download or read book Modern Perspectives in Type Theoretical Semantics written by Stergios Chatzikyriakidis and published by Springer. This book was released on 2017-02-07 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collective volume that reports the state of the art in the applications of type theory to linguistic semantics. The volume fills a 20 year gap from the last published book on the issue and aspires to bring researchers closer to cutting edge alternatives in formal semantics research. It consists of unpublished work by some key researchers on various issues related to the type theoretical study of formal semantics and further exemplifies the advantages of using modern type theoretical approaches to linguistic semantics. Themes that are covered include modern developments of type theories in formal semantics, foundational issues in linguistic semantics like anaphora, modality and plurals, innovational interdisciplinary research like the introduction of probability theory to type theories as well as computational implementations of type theoretical approaches. This volume will be of great interest to formal semanticists that are looking for alternative ways to study linguistic semantics, but will also be of interest to theoretical computer scientists and mathematicians that are interested in the applications of type theory.