Special Topics in Mathematics for Computer Scientists

Sets, Categories, Topologies and Measures

Ernst-Erich Doberkat (Autor)

Buch | Hardcover

XX, 719 Seiten

2015 | 1st ed. 2015
Springer International Publishing (Verlag)
978-3-319-22749-8 (ISBN)

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This textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems. Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Gödel's Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics. Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.

Prof. Dr. Ernst-Erich Doberkat, Inhaber des Lehrstuhls für Software-Technologie an der Technischen Universität Dortmund. Arbeitsgebiete: Praktische Informatik, Logik, Angewandte Mathematik. Seit mehreren Jahren hält er Vorlesungen zu Haskell und ist ein erfahrener Lehrbuchautor.

Preface.- 1 The Axiom of Choice and Some of Its Equivalents.- 2 Categories.- 3 Topological Spaces.- 4 Measures for Probabilistic Systems.- List of Examples.- References.- Index.

"The book is absolutely packed with ideas and theorems, classical results, well-knowns as well as obscure pieces of mathematics, familiar and less familiar axioms, and generally many many facts, ranging from trivial to deep and pivotal. ... book is probably the most original I've seen in terms of its contents. ... ideas and presentation is clearly derived and dictated by the author's renowned expertise in the area, producing a very unique point of view on an active area of research." (Ittay Weiss, MAA Reviews, maa.org, April, 2017)

"This volume is targeted at giving sound mathematical foundations for advanced topics in theoretical computer science ... . The author always gives accurate definitions, complete proofs and lots of different examples clarifying and motivating the concepts. Each chapter has lots of exercises sometimes pointing to further developments. ... there is a comprehensive list of references to examples and counterexamples for various propositions and assertions given in the text. The style is always motivating and sometimes humorous." (Dieter Riebesehl, zbMATH 1334.68002, 2016)

Erscheint lt. Verlag	23.11.2015
Zusatzinfo	XX, 719 p.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	1271 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
Schlagworte	Category Theory, Homological Algebra • coalgebras over general and probabilistic spaces • Computer Science • examples for the use of the Axiom of Choice • Mathematical logic and formal languages • Mathematical Logic and Foundations • measurable spaces and Markov transition systems • proof techniques in games and measure theory • proof techniques in topology and categories • topologies for use in computer science
ISBN-10	3-319-22749-1 / 3319227491
ISBN-13	978-3-319-22749-8 / 9783319227498
Zustand	Neuware