Collecting information previously scattered throughout the vast literature, including the authorвЂ™s own research, Stochastic Relations: Foundations for Markov Transition Systems  develops the theory of stochastic relations as a basis for Markov transition systems.    After an introduction to the basic mathematical tools from topology, measure theory, and categories, the book examines the central topics of congruences and morphisms, applies these to the monoidal structure, and defines bisimilarity and behavioral equivalence within this framework. The author views developments from the general theory of coalgebras in the context of the subprobability functor. These tools show that bisimilarity and behavioral and logical equivalence are the same for general modal logics and for continuous time stochastic logic with and without a fixed point operator.    With numerous problems and several case studies, this book is an invaluable study of an important aspect of computer science theory.

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