Ernst-Erich Doberkat Knihy

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

Focusing on the intersection of coalgebraic logic and stochastic systems, this work explores how these concepts enhance modeling in concurrency theory and semantics. By leveraging results from coalgebras, universal algebra, and category theory, it reveals new insights into system behavior that traditional probabilistic methods may overlook. The integration of categorical reasoning offers innovative perspectives on transition systems and modal logics, making significant contributions to the field of systems modeling.

Focusing on stochastic relations, this book lays the groundwork for Markov transition systems through essential mathematical concepts from topology, measure theory, and categories. It explores congruences and morphisms, applying them to monoidal structures while defining bisimilarity and behavioral equivalence. Additionally, it delves into coalgebra theory related to subprobability functors and includes case studies on software architecture, the converse of stochastic relations, and average case analysis of algorithms.

Erzeugende Funktionen werden als zentrales Werkzeug in der Kombinatorik und Theoretischen Informatik vorgestellt. Anhand zahlreicher Beispiele wird demonstriert, wie eine Folge reeller Zahlen durch eine einzige Funktion repräsentiert werden kann. Das Buch bietet eine umfassende Einführung in die Techniken zur Gewinnung und Manipulation dieser Funktionen und behandelt wichtige Folgen sowie ihre entsprechenden erzeugenden Funktionen.