Stochastic Relations: Foundations for Markov Transition Systems Book

Preface     xi
Acknowledgments     xxiii
A Gentle Tutorial to All Things Considered     1
Introduction     1
Measurable Spaces     2
Polish and Analytic Spaces     7
Metric Spaces     8
Polish Spaces: Elementary Properties     11
Manipulating Polish Topologies     17
Analytic Spaces     21
Measurable Selectors     32
Probability Measures     36
Regularity and Tightness     38
Weak Topology     41
Disintegration     51
Applications of the [pi]-[lambda]-Theorem     57
Projective Systems     59
Categories     62
The Subprobability Functor     65
Natural Transformations     70
Adjunctions, Monads, Algebras and the Kleisli Construction     72
Bibliographic Notes     81
Stochastic Relations as Monads     83
Introduction     83
The Manes Monad     85
The Giry Monad     87
Adding a Monoid     87
Stochastic Relations     91
Case Study: Architectural Modeling through Monads     95
A First Example     99
First Steps     101
The Basic Construction     105
Stratifying Graphs     109
The General Case     112
System Evolution     120
Related Approaches     125
Bibliographic Notes     128
Eilenberg-Moore Algebras for Stochastic Relations     131
Introduction     131
Characterization through Equivalence Relations     133
Preparations     133
Positive Convex Partitions     135
Smooth Relations     139
Positive Convex Structures     143
Algebras through Positive Convex Structures     144
Examples     147
Monad Multiplication     148
The Finite Case     148
The Unit Interval     150
Barycenter     150
The Left Adjoint     152
Bibliographic Notes     155
The Existence of Semi-Pullbacks     157
Introduction     157
A Road Map     158
Extending Semi-Pullbacks of Measures     162
The Compact Case     164
The General Polish Case     167
The Existence of Semi-Pullbacks     173
The Polish Case     173
The Analytic Case     175
Bibliographic Notes     177
Congruences and Bisimulations     179
Introduction     179
Smooth Equivalence Relations     181
Invariant Borel Sets     182
Operations on Smooth Relations     184
A Confluence Property     190
Simulation Equivalence     194
Factoring     198
Congruences     199
Isomorphism Theorems     202
Bisimulations     205
Behavioral Equivalence and a Portmanteau     215
2-Bisimulations     217
Simple Relations     223
The Polish Case     224
The Analytic Case     225
Case Study: The Converse of a Stochastic Relation     229
Converse Relations     233
Preserving Bisimilarity     238
Case Study: Simple Relations for Counting     242
Left to Right Maxima     242
Williams' Algorithm to Construct Heaps     244
Bibliographic Notes     248
Interpreting Modal and Temporal Logics     251
Introduction     251
Modal Logics     253
Examples      256
Refinements     259
Bisimulations for Kripke Models     263
Projective Limits for Interpreting Temporal Logics     271
Setting the Stage: Infinite Paths     272
Independence and Zeno Paths     279
F-Bisimulations for CSL     283
Interpreting the Logic     284
Definition and Properties of [rho superscript F]     286
Closure Operations     292
F-Bisimulations     294
Logical Equivalence for [micro]CSL     298
The Logic [micro]CSL     299
Models for [micro]CSL and Their Morphisms     300
Interpreting [micro]CSL     303
Congruences     310
Logical Equivalence and Bisimilarity     316
Bibliographic Notes     325
Notations     327
Categories     327
Spaces     328
Other     328
Bibliography     331
Index     339