Stochastic Relations: Foundations for Markov Transition Systems
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Summary
Collecting information previously scattered throughout the vast literature, including the author’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.
Table of Contents
Preface
A Gentle Tutorial to All Things Considered
Introduction
Measurable Spaces
Polish and Analytic Spaces
Measurable Selectors
Probability Measures
Categories
Stochastic Relations as Monads
Introduction
The Manes Monad
The Giry Monad
Case Study: Architectural Modeling through Monads
Eilenberg–Moore Algebras for Stochastic Relations
Introduction
Characterization through Equivalence Relations
Positive Convex Structures
Algebras through Positive Convex Structures
Examples
The Left Adjoint
The Existence of Semi-Pullbacks
Introduction
A Road Map
Extending Semi-Pullbacks of Measures
The Existence of Semi-Pullbacks
Congruences and Bisimulations
Introduction
Smooth Equivalence Relations
Factoring
Bisimulations
Behavioral Equivalence and a Portmanteau
2-Bisimulations
Simple Relations
Case Study: The Converse of a Stochastic Relation
Case Study: Simple Relations for Counting
Interpreting Modal and Temporal Logics
Introduction
Modal Logics
Projective Limits for Interpreting Temporal Logics
F-Bisimulations for CSL
Logical Equivalence for μCSL
Appendix: Notations
Categories
Spaces
Other
Bibliography
Index
Bibliographic Notes appear at the end of each chapter.
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