Stochastic Relations: Foundations for Markov Transition Systems, 1st Edition (Hardback) book cover

Stochastic Relations

Foundations for Markov Transition Systems, 1st Edition

By Ernst-Erich Doberkat

Chapman and Hall/CRC

376 pages | 16 B/W Illus.

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pub: 2007-05-17
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Description

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.

About the Series

Chapman & Hall/CRC Studies in Informatics Series

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Subject Categories

BISAC Subject Codes/Headings:
COM000000
COMPUTERS / General
MAT000000
MATHEMATICS / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis