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Special Topics in Mathematics for Computer Scientists - Sets, Categories, Topologies and Measures (Paperback, Softcover reprint of the original 1st ed. 2015)
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Special Topics in Mathematics for Computer Scientists - Sets, Categories, Topologies and Measures (Paperback, Softcover reprint of the original 1st ed. 2015)
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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, Goedel'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.
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