Special Topics in Mathematics for Computer Scientists

1. Front Matter

   Pages i-xx

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2. The Axiom of Choice and Some of Its Equivalents

   • Ernst-Erich Doberkat

   Pages 1-108

3. Categories

   • Ernst-Erich Doberkat

   Pages 109-279

4. Topological Spaces

   • Ernst-Erich Doberkat

   Pages 281-425

5. Measures for Probabilistic Systems

   • Ernst-Erich Doberkat

   Pages 427-683

6. Back Matter

   Pages 685-719

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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.

• coalgebras over general and probabilistic spaces
• examples for the use of the Axiom of Choice
• 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

“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)

• Lewackerstr. 6 b, Math ++ Software, Bochum, Germany

  Ernst-Erich Doberkat

Ernst-Erich Doberkat:
- doctorate degree in mathematics from the University of Paderborn
- habilitation in computer science from the University of Hagen
- associate professor of mathematics and computer science at Clarkson University in Potsdam, NY
- full professor of software technology and adjunct professor of mathematics at various universities in Germany between 1985 and 2014
- lectures given at universities in Germany, Italy, China and the US

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