Temporal Type Theory
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Temporal Type Theory
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Author : Patrick Schultz
language : en
Publisher: Springer
Release Date : 2019-01-29
Temporal Type Theory written by Patrick Schultz and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-01-29 with Mathematics categories.
This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models. The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action. Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource. It can also serve as a supplemental text for a specialized graduate topics course.
Modal Homotopy Type Theory
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Author : David Corfield
language : en
Publisher: Oxford University Press
Release Date : 2020-02-06
Modal Homotopy Type Theory written by David Corfield and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02-06 with Philosophy categories.
"The old logic put thought in fetters, while the new logic gives it wings." For the past century, philosophers working in the tradition of Bertrand Russell - who promised to revolutionise philosophy by introducing the 'new logic' of Frege and Peano - have employed predicate logic as their formal language of choice. In this book, Dr David Corfield presents a comparable revolution with a newly emerging logic - modal homotopy type theory. Homotopy type theory has recently been developed as a new foundational language for mathematics, with a strong philosophical pedigree. Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy offers an introduction to this new language and its modal extension, illustrated through innovative applications of the calculus to language, metaphysics, and mathematics. The chapters build up to the full language in stages, right up to the application of modal homotopy type theory to current geometry. From a discussion of the distinction between objects and events, the intrinsic treatment of structure, the conception of modality as a form of general variation to the representation of constructions in modern geometry, we see how varied the applications of this powerful new language can be.
An Invitation To Applied Category Theory
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Author : Brendan Fong
language : en
Publisher: Cambridge University Press
Release Date : 2019-07-18
An Invitation To Applied Category Theory written by Brendan Fong 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 2019-07-18 with Computers categories.
Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
Modern Perspectives In Type Theoretical Semantics
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Author : Stergios Chatzikyriakidis
language : en
Publisher: Springer
Release Date : 2017-02-07
Modern Perspectives In Type Theoretical Semantics written by Stergios Chatzikyriakidis and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-02-07 with Language Arts & Disciplines categories.
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.
Categorical Logic And Type Theory
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Author : B. Jacobs
language : en
Publisher: Gulf Professional Publishing
Release Date : 2001-05-10
Categorical Logic And Type Theory written by B. Jacobs and has been published by Gulf Professional Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-05-10 with Computers categories.
This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.
Twenty Five Years Of Constructive Type Theory
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Author : Giovanni Sambin
language : en
Publisher: Clarendon Press
Release Date : 1998-10-15
Twenty Five Years Of Constructive Type Theory written by Giovanni Sambin and has been published by Clarendon Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-10-15 with Mathematics categories.
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.
Sheaf Theory Through Examples
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Author : Daniel Rosiak
language : en
Publisher: MIT Press
Release Date : 2022-10-25
Sheaf Theory Through Examples written by Daniel Rosiak and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-10-25 with Mathematics categories.
An approachable introduction to elementary sheaf theory and its applications beyond pure math. Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more. With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms. Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas.
Formal Semantics In Modern Type Theories
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Author : Stergios Chatzikyriakidis
language : en
Publisher: John Wiley & Sons
Release Date : 2021-02-17
Formal Semantics In Modern Type Theories written by Stergios Chatzikyriakidis and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-17 with Language Arts & Disciplines categories.
This book studies formal semantics in modern type theories (MTTsemantics). Compared with simple type theory, MTTs have much richer type structures and provide powerful means for adequate semantic constructions. This offers a serious alternative to the traditional settheoretical foundation for linguistic semantics and opens up a new avenue for developing formal semantics that is both model-theoretic and proof-theoretic, which was not available before the development of MTTsemantics. This book provides a reader-friendly and precise description of MTTs and offers a comprehensive introduction to MTT-semantics. It develops several case studies, such as adjectival modification and copredication, to exemplify the attractiveness of using MTTs for the study of linguistic meaning. It also examines existing proof assistant technology based on MTT-semantics for the verification of semantic constructions and reasoning in natural language. Several advanced topics are also briefly studied, including dependent event types, an application of dependent typing to event semantics.
Mathematics For Future Computing And Communications
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Author : Liao Heng
language : en
Publisher: Cambridge University Press
Release Date : 2021-12-16
Mathematics For Future Computing And Communications written by Liao Heng 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 2021-12-16 with Computers categories.
A panorama of new ideas in mathematics that are driving innovation in computing and communications.
Text Time And Context
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Author : Richard P. Meier
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-10-20
Text Time And Context written by Richard P. Meier 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 2009-10-20 with Language Arts & Disciplines categories.
Carlota S. Smith was a key figure in linguistic research and a pioneering woman in generative linguistics. This selection of papers focuses on the research into tense, aspect, and discourse that Smith completed while Professor of Linguistics at the University of Texas at Austin. Smith’s early work in English syntax is still cited today, and her early career also yielded key research on language acquisition by young children. Starting in the mid-1970s, after her move to UT, she embarked on her most important line of research. In numerous papers - the first of which was published in 1975 - and in a very important 1991 book (The Parameter of Aspect), Smith analyzed how languages encode time and how they encode the ways events and situations occur over time. Smith’s work on the expression of time in language is notable because of its careful analyses of a number of quite different languages, including not only English and French, but also Russian, Mandarin, and Navajo. Inspired by a year in France in the early 1970s, Smith began to analyze the differing ways in which languages encode time and how they encode the ways events and situations occur over time. In doing so, she developed her signature ‘two-component’ theory of aspect. This model of temporal aspect provided an excellent framework for graduate students seeking to analyze the temporal systems of an array of languages, including under-described languages that are so much the focus of research in UT’s Linguistics Department. Selected by Carlota Smith herself and by her longtime friends and colleagues, this book contains her 1980 piece on temporal structures in discourse, her 1986 comparison of the English and French aspectual systems, a 1996 paper on the aspect system in Navajo (an increasingly-endangered language which Smith worked to preserve), and her 1980 and 1993 papers on the child’s acquisition of tense and aspect. Smith, who died in 2007, was a trailblazer in her field whose broad interests fed into her scholarly research. She was an avid reader who sought to bring the analytic tools of linguistics to the humanistic study of literature, by examining the syntactic and pragmatic principles which underlie literary effects. Her research on rhetorical and temporal effects in context was integrated into her last book, Modes of Discourse (2003). The current volume of articles covers much of her most fruitful work on the way in which language is used to express time, and will be essential reading for many working and studying in linguistics generally and in semantics particularly.