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Author: Bernhard Banaschewski
Publisher:
ISBN:
Category : Frames (Combinatorial analysis)
Languages : en
Pages : 114
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Book Description
Author: Bernhard Banaschewski
Publisher:
ISBN:
Category : Frames (Combinatorial analysis)
Languages : en
Pages : 114
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Book Description
Author: Jorge Picado
Publisher: Springer Nature
ISBN: 3030534790
Category : Mathematics
Languages : en
Pages : 296
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Book Description
This book is the first systematic treatment of this area so far scattered in a vast number of articles. As in classical topology, concrete problems require restricting the (generalized point-free) spaces by various conditions playing the roles of classical separation axioms. These are typically formulated in the language of points; but in the point-free context one has either suitable translations, parallels, or satisfactory replacements. The interrelations of separation type conditions, their merits, advantages and disadvantages, and consequences are discussed. Highlights of the book include a treatment of the merits and consequences of subfitness, various approaches to the Hausdorff's axiom, and normality type axioms. Global treatment of the separation conditions put them in a new perspective, and, a.o., gave some of them unexpected importance. The text contains a lot of quite recent results; the reader will see the directions the area is taking, and may find inspiration for her/his further work. The book will be of use for researchers already active in the area, but also for those interested in this growing field (sometimes even penetrating into some parts of theoretical computer science), for graduate and PhD students, and others. For the reader's convenience, the text is supplemented with an Appendix containing necessary background on posets, frames and locales.
Author: John Harrison
Publisher: Springer Science & Business Media
ISBN: 1447115910
Category : Computers
Languages : en
Pages : 193
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Book Description
This book discusses the use of the real numbers in theorem proving. Typ ically, theorem provers only support a few 'discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification of float ing point hardware and hybrid systems. It also allows the formalization of many more branches of classical mathematics, which is particularly relevant for attempts to inject more rigour into computer algebra systems. Our work is conducted in a version of the HOL theorem prover. We de scribe the rigorous definitional construction of the real numbers, using a new version of Cantor's method, and the formalization of a significant portion of real analysis. We also describe an advanced derived decision procedure for the 'Tarski subset' of real algebra as well as some more modest but practically useful tools for automating explicit calculations and routine linear arithmetic reasoning. Finally, we consider in more detail two interesting application areas. We discuss the desirability of combining the rigour of theorem provers with the power and convenience of computer algebra systems, and explain a method we have used in practice to achieve this. We then move on to the verification of floating point hardware. After a careful discussion of possible correctness specifications, we report on two case studies, one involving a transcendental function.
Author: Klaus Mainzer
Publisher: World Scientific
ISBN: 9811236496
Category : Mathematics
Languages : en
Pages : 425
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Book Description
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.
Author: Jorge Picado
Publisher: Springer Science & Business Media
ISBN: 3034801548
Category : Mathematics
Languages : en
Pages : 412
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Book Description
Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the twenties offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent. Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the fifties.It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory. No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories.
Author: Laura Crosilla
Publisher: Oxford University Press
ISBN: 0198566514
Category : Mathematics
Languages : en
Pages : 371
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Book Description
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.
Author: Maria Manuel Clementino
Publisher: Springer Nature
ISBN: 303084319X
Category : Mathematics
Languages : en
Pages : 266
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Book Description
This book provides an introduction to some key subjects in algebra and topology. It consists of comprehensive texts of some hours courses on the preliminaries for several advanced theories in (categorical) algebra and topology. Often, this kind of presentations is not so easy to find in the literature, where one begins articles by assuming a lot of knowledge in the field. This volume can both help young researchers to quickly get into the subject by offering a kind of « roadmap » and also help master students to be aware of the basics of other research directions in these fields before deciding to specialize in one of them. Furthermore, it can be used by established researchers who need a particular result for their own research and do not want to go through several research papers in order to understand a single proof. Although the chapters can be read as « self-contained » chapters, the authors have tried to coordinate the texts in order to make them complementary. The seven chapters of this volume correspond to the seven courses taught in two Summer Schools that took place in Louvain-la-Neuve in the frame of the project Fonds d’Appui à l’Internationalisation of the Université catholique de Louvain to strengthen the collaborations with the universities of Coimbra, Padova and Poitiers, within the Coimbra Group.
Author: Macario Vives, Sergio
Publisher: Publicacions de la Universitat Jaume I
ISBN: 8480218800
Category : Mathematics
Languages : en
Pages : 184
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Book Description
En las últimas décadas, la Topología se ha revelado como una poderosa herramienta para acometer diferentes problemas relacionados con un amplio espectro de ciencias aplicadas más allá de las matemáticas, como Economía, Inteligencia Artificial, Ciencias de la Computación o Sistemas Dinámicos. El presente volumen recoge las ponencias del Workshop in Applied Topology WiAT¿12, celebrado en junio de 2012 en la Universitat Jaume I, en el que participaron diferentes grupos de investigación del área de la Topología General y sus Aplicaciones.
Author: John Robert Harrison
Publisher:
ISBN:
Category : Automatic theorem proving
Languages : en
Pages : 164
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Book Description
Abstract: "This thesis discusses the use of the real numbers in theorem proving. Typically, theorem provers only support a few 'discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification of floating point hardware and hybrid systems. It also allows the formalization of many more branches of classical mathematics, which is particularly relevant for attempts to inject more rigour into computer algebra systems. Our work is conducted in a version of the HOL theorem prover. We describe the rigorous definitional construction of the real numbers, using a new version of Cantor's method, and the formalization of a significant portion of real analysis. We also describe an advanced derived decision procedure for the 'Tarski subset' of real algebra as well as some more modest but practically useful tools for automating explicit calculations and routine linear arithmetic reasoning. Finally, we consider in more detail two interesting application areas. We discuss the desirability of combining the rigour of theorem provers with the power and convenience of computer algebra systems, and explain a method we have used in practice to achieve this. We then move on to the verification of floating point hardware. After a careful discussion of possible correctness specifications, we report on two case studies, one involving a transcendental function. We aim to show that a theory of real numbers is useful in practice and interesting in theory, and that the 'LCF style' of theorem proving is well suited to the kind of work we describe. We hope also to convince the reader that the kind of mathematics needed for applications is well within the abilities of current theorem proving technology."
Author: Stefano Berardi
Publisher: Springer Science & Business Media
ISBN: 9783540617808
Category : Computers
Languages : en
Pages : 310
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Book Description
This volume contains a refereed selection of revised full papers chosen from the contributions presented during the Third Annual Workshop held under the auspices of the ESPRIT Basic Research Action 6453 Types for Proofs and Programs. The workshop took place in Torino, Italy, in June 1995. Type theory is a formalism in which theorems and proofs, specifications and programs can be represented in a uniform way. The 19 papers included in the book deal with foundations of type theory, logical frameworks, and implementations and applications; all in all they constitute a state-of-the-art survey for the area of type theory.
