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Extensional Constructs In Intensional Type Theory
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Book Synopsis Extensional Constructs in Intensional Type Theory by : Martin Hofmann
Download or read book Extensional Constructs in Intensional Type Theory written by Martin Hofmann and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: Extensional Constructs in Intensional Type Theory presents a novel approach to the treatment of equality in Martin-Loef type theory (a basis for important work in mechanised mathematics and program verification). Martin Hofmann attempts to reconcile the two different ways that type theories deal with identity types. The book will be of interest particularly to researchers with mainly theoretical interests and implementors of type theory based proof assistants, and also fourth year undergraduates who will find it useful as part of an advanced course on type theory.
Book Synopsis Extensional Concepts in Intensional Type Theory by : Martin Hofmann
Download or read book Extensional Concepts in Intensional Type Theory written by Martin Hofmann and published by . This book was released on 1995 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Theories of dependent types have been proposed as a foundation of constructive mathematics and as a framework in which to construct certified programs. In these applications an important role is played by identity types which internalise equality and therefore are essential for accommodating proofs and programs in the same formal system. This thesis attempts to reconcile the two different ways that type theories deal with identity types. In extensional type theory the propositional equality induced by the identity types is identified with definitional equality, i.e. conversion. This renders type-checking and well-formedness of propositions undecidable and leads to non-termination in the presence of universes. In intensional type theory propositional equality is coarser than definitional equality, the latter being confined to definitional expansion and normalisation. Then type-checking and well-formedness are decidable, and this variant is therefore adopted by most implementations. However, the identity type in intensional type theory is not powerful enough for formalisation of mathematics and program development. Notably, it does not identify pointwise equal functions (functional extensionality) and provides no means of redefining equality on a type as a given relation, i.e. quotient types. We call such capabilities extensional concepts. Other extensional concepts of interest are uniqueness of proofs and more specifically of equality proofs, subset types, and propositional extensionality--the identification of equivalent propositions. In this work we investigate to what extent these extensional concepts may be added to intensional type theory without sacrificing decidability and existence of canonical forms. The method we use is the translation of identity types into equivalence relations defined by induction on the type structure. In this way type theory with extensional concepts can be understood as a high-level language for working with equivalence relations instead of equality. Such translations of type theory into itself turn out to be best described using categorical models of type theory. We thus begin with a thorough treatment of categorical models with particular emphasis on the interpretation of type-theoretic syntax in such models. We then show how pairs of types and predicates can be organised into a model of type theory in which subset types are available and in which any two proofs of a proposition are equal. This model has applications in the areas of program extraction from proofs and modules for functional programs. For us its main purpose is to clarify the idea of syntactic translations via categorical model constructions. The main result of the thesis consists of the construction of two models in which functional extensionality and quotient types are available. In the first one types are modelled by types together with proposition-valued partial equivalence relations. This model is rather simple and in addition provides subset types and propositional extensionality. However, it does not furnish proper dependent types such as vectors or matrices. We try to overcome this disadvantage by using another model based on families of type-valued equivalence relations which is however much more complicated and validates certain conversion rules only up to propositional equality. We illustrate the use of these models by several small examples taken from both formalised mathematics and program development. We also establish various syntactic properties of propositional equality including a proof of the undecidability of typing in extensional type theory and a correspondence between derivations in extensional type theory and terms in intensional type theory with extensional concepts added. Furthermore we settle affirmatively the hitherto open question of the independence of unicity of equality proofs in intensional type theory which implies that the addition of pattern matching to intensional type theory does not yield a conservative extension.
Book Synopsis Types for Proofs and Programs by : Stefano Berardi
Download or read book Types for Proofs and Programs written by Stefano Berardi and published by Springer. This book was released on 2009-06-07 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: These proceedings contain a selection of refereed papers presented at or - lated to the Annual Workshop of the TYPES project (EU coordination action 510996), which was held during March 26–29, 2008 in Turin, Italy. The topic of this workshop, and of all previous workshops of the same project, was f- mal reasoning and computer programming based on type theory: languages and computerized tools for reasoning, and applications in several domains such as analysis of programming languages, certi?ed software, mobile code, formali- tion of mathematics, mathematics education. The workshop was attended by more than 100 researchers and included more than 40 presentations. We also had three invited lectures, from A. Asperti (University of Bologna), G. Dowek (LIX, Ecole polytechnique, France) and J. W. Klop (Vrije Universiteit, A- terdam, The Netherlands). From 27 submitted papers, 19 were selected after a reviewing process. Each submitted paper was reviewed by three referees; the ?nal decisions were made by the editors. This workshop is the last of a series of meetings of the TYPES working group funded by the European Union (IST project 29001, ESPRIT Working Group 21900, ESPRIT BRA 6435).
Book Synopsis Twenty Five Years of Constructive Type Theory by : Giovanni Sambin
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.
Book Synopsis Simple Type Theory by : William M. Farmer
Download or read book Simple Type Theory written by William M. Farmer and published by Springer Nature. This book was released on 2023-02-02 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique textbook, in contrast to a standard logic text, provides the reader with a logic that actually can be used in practice to express and reason about mathematical ideas. The book is an introduction to simple type theory, a classical higher-order version of predicate logic that extends first-order logic. It presents a practice-oriented logic called Alonzo that is based on Alonzo Church's formulation of simple type theory known as Church's type theory. Unlike traditional predicate logics, Alonzo admits undefined expressions. The book illustrates, using Alonzo, how simple type theory is suited ideally for reasoning about mathematical structures and constructing libraries of mathematical knowledge. Topics and features: Offers the first book-length introduction to simple type theory as a predicate logic Provides the reader with a logic that is close to mathematical practice Presents the tools needed to build libraries of mathematical knowledge Employs two semantics, one for mathematics and one for logic Emphasizes the model-theoretic view of predicate logic Includes several important topics, such as definite description and theory morphisms, not usually found in standard logic textbooks Aimed at students of computing and mathematics at the graduate or upper-undergraduate level, this book is also well-suited for mathematicians, computing professionals, engineers, and scientists who need a practical logic for expressing and reasoning about mathematical ideas. William M. Farmer is a Professor in the Department of Computing and Software at McMaster University in Hamilton, Ontario, Canada.
Book Synopsis Computation and Logic in the Real World by : Barry S. Cooper
Download or read book Computation and Logic in the Real World written by Barry S. Cooper and published by Springer. This book was released on 2007-07-25 with total page 843 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Third International Conference on Computability in Europe, CiE 2007, held in Sienna, Italy, in June 2007. The 50 revised full papers presented together with 36 invited papers were carefully reviewed and selected from 167 submissions.
Book Synopsis Intelligent Computer Mathematics by : James H. Davenport
Download or read book Intelligent Computer Mathematics written by James H. Davenport and published by Springer Science & Business Media. This book was released on 2011-07-18 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the joint refereed proceedings of three international events, namely the 18th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, Calculemus 2011, the 10th International Conference on Mathematical Knowledge Management, MKM 2011, and a new track on Systems and Projects descriptions that span both the Calculemus and MKM topics, all held in Bertinoro, Italy, in July 2011. All 51 submissions passed through a rigorous review process. A total of 15 papers were submitted to Calculemus, of which 9 were accepted. Systems and Projects track 2011 there have been 12 papers selected out of 14 submissions while MKM 2011 received 22 submissions, of which 9 were accepted for presentation and publication. The events focused on the use of AI techniques within symbolic computation and the application of symbolic computation to AI problem solving; the combination of computer algebra systems and automated deduction systems; and mathematical knowledge management, respectively.
Book Synopsis Theoretical Aspects of Computing - ICTAC 2015 by : Martin Leucker
Download or read book Theoretical Aspects of Computing - ICTAC 2015 written by Martin Leucker and published by Springer. This book was released on 2015-10-08 with total page 628 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 12th International Colloquium on Theoretical Aspects of Computing, ICTAC 2015, held in Cali, Colombia, in October 2015. The 25 revised full papers presented together with 7 invited talks, 3 tool papers, and 2 short papers were carefully reviewed and selected from 93 submissions. The papers cover various topics such as algebra and category theory; automata and formal languages; concurrency; constraints, logic and semantic; software architecture and component-based design; and verification.
Book Synopsis Foundations of Software Science and Computation Structures by : Naoki Kobayashi
Download or read book Foundations of Software Science and Computation Structures written by Naoki Kobayashi and published by Springer Nature. This book was released on with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Intuitionistic Type Theory by : Per Martin-Löf
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:
Book Synopsis Logicism, Intuitionism, and Formalism by : Sten Lindström
Download or read book Logicism, Intuitionism, and Formalism written by Sten Lindström and published by Springer Science & Business Media. This book was released on 2008-11-25 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: This anthology reviews the programmes in the foundations of mathematics from the classical period and assesses their possible relevance for contemporary philosophy of mathematics. A special section is concerned with constructive mathematics.
Book Synopsis Dictionary of World Philosophy by : A. Pablo Iannone
Download or read book Dictionary of World Philosophy written by A. Pablo Iannone and published by Routledge. This book was released on 2013-04-15 with total page 894 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Dictionary of World Philosophy covers the diverse and challenging terminology, concepts, schools and traditions of the vast field of world philosophy. Providing an extremely comprehensive resource and an essential point of reference in a complex and expanding field of study the Dictionary covers all major subfields of the discipline. Key features: * Cross-references are used to highlight interconnections and the cross-cultural diffusion and adaptation of terms which has taken place over time * The user is led from specific terms to master entries which provide valuable historical and cultural context * Each master entry is followed by at least two suggestions for further reading on the subject, creating a substantial bibliography of world philosophy * References extend beyond philosophy to related areas such as cognitive science, computer science, language and physics Subdisciplines covered include:* aesthetics * ethics * sociopolitical philosophy * the philosophy of law * epistemology * logic * the philosophy of science * the philosophy of mind * the philosophy of culture and history * metaphysics * the philosophy of religion Entries are drawn from West Africa, Arabic, Chinese, Indian, Japanese, Jewish, Korean, Latin American, Maori and Native American philosophy including the important and so far largely neglected instance of Pre-Hispanic thought: Nahua philosophy.
Book Synopsis Reflections on the Foundations of Mathematics by : Stefania Centrone
Download or read book Reflections on the Foundations of Mathematics written by Stefania Centrone and published by Springer Nature. This book was released on 2019-11-11 with total page 511 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.
Book Synopsis Handbook of Constructive Mathematics by : Douglas Bridges
Download or read book Handbook of Constructive Mathematics written by Douglas Bridges and published by Cambridge University Press. This book was released on 2023-03-31 with total page 864 pages. Available in PDF, EPUB and Kindle. Book excerpt: Constructive mathematics – mathematics in which 'there exists' always means 'we can construct' – is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.
Book Synopsis Programming Languages and Systems by : Stephanie Weirich
Download or read book Programming Languages and Systems written by Stephanie Weirich and published by Springer Nature. This book was released on with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Games and Full Abstraction for a Functional Metalanguage with Recursive Types by : Guy McCusker
Download or read book Games and Full Abstraction for a Functional Metalanguage with Recursive Types written by Guy McCusker and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a minor revision of the thesis submitted in August 1996; no major changes have been made. However, I would like to take this opportunity to mention that since the thesis was written, discoveries have been made which would allow a substantial simplification and strengthening of the results in Chapters 3 and 6. In particular, it is now possible to model sums correctly in the category I as well as in £, which means that the definability results of Chapter 6 can be stated and proved at the intensional level, making them simpler and much closer in spirit to the original proofs of Abramsky, Jagadeesan, Malacaria, Hyland, Ong and Nickau [10,61,79]. This also leads quite straightforwardly to an understanding of call-by-value languages. Details of these improvements can be found in [14,73]. It is also worth mentioning that progress has been made on some of the topics suggested for future research in Chapter 7. In particular, fully abstract models have been found for various kinds of languages with local variables [8,13-16], and a fully complete games model of the polymorphic language System F has been constructed by Hughes [59]. Guy McCusker February 1998 Acknowledgements First of all, I must thank my supervisor, Samson Abramsky. It was he who first introduced me to game semantics and suggested avenues of research in the area; this book would certainly not exist were it not for him.
Book Synopsis Theoretical Aspects of Computing – ICTAC 2017 by : Dang Van Hung
Download or read book Theoretical Aspects of Computing – ICTAC 2017 written by Dang Van Hung and published by Springer. This book was released on 2017-09-28 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 14th International Colloquium on Theoretical Aspects of Computing, ICTAC 2017, held in Hanoi, Vietnam, in October 2017. The 17 revised full papers presented together with three invited talks were carefully reviewed and selected from 40 submissions. The papers are organized in topical sections on logics; software components and concurrency; automata; SMT solvers and algorithms; and security.
