Abstraction, Refinement and Proof for Probabilistic Systems

By: Annabelle McIver, Charles Carroll Morgan

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Hardcover | 19 November 2004

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408 Pages

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23.5 x 15.88 x 2.54

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The book is a focused survey on probabilistic program semantics, conceived to tell a coherent story with a uniform notation. It is grouped into three themes: Part I is for 'users' of the techniques who will be developing actual programs; Part II gives mathematical foundations intended for those studying exactly how it was done and how to build semantic structures/models in their own work; and Part III describes a very 'hot' research direction, temporal logic and model checking.

Topics and features:

- introduces readers to very up-to-date research in the mathematics of rigorous development of randomized (probabilistic) algorithms

- illustrates by example the typical steps necessary in computer science to build a mathematical model of any programming paradigm

- presents results of a large and integrated body of research in the area of 'quantitative' program logics

An advanced research survey monograph, integrating three major topic areas: random/probabilistic algorithms, assertion-based program reasoning, and refinement programming models. Essential foundation topic for modern sequential programming methodology.

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Published: 19th November 2010

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Preface	p. v
List of definitions etc.	p. xvii
Probabilistic guarded commands	p. 1
Introduction to pGCL	p. 3
Sequential program logic	p. 4
The programming language pGCL	p. 7
An informal computational model for pGCL	p. 11
Behind the scenes: elementary probability theory	p. 16
Basic syntax and semantics of pGCL	p. 18
Healthiness and algebra for pGCL	p. 28
Healthiness example: modular reasoning	p. 32
Interaction of probabilistic- and demonic choice	p. 34
Summary	p. 35
Chapter notes	p. 36
Probabilistic loops: invariants and variants	p. 37
Introduction: loops via recursion	p. 38
Probabilistic invariants	p. 39
Probabilistic termination	p. 40
Invariance and termination together: the loop rule	p. 42
Three examples of probabilistic loops	p. 44
The Zero-One Law for termination	p. 53
Probabilistic variant arguments for termination	p. 54
Termination example: self-stabilisation	p. 56
Uncertain termination	p. 61
Proper post-expectations	p. 63
Bounded vs. unbounded expectations	p. 68
Informal proof of the loop rule	p. 74
Chapter notes	p. 77
Case studies in termination	p. 79
Rabin's choice coordination	p. 79
The dining philosophers	p. 88
The general random ""jump""	p. 99
Chapter notes	p. 105
Probabilistic data refinement: the steam boiler	p. 107
Introduction: refinement of datatypes	p. 107
Data refinement and simulations	p. 108
Probabilistic datatypes: a worked example	p. 110
A safety-critical application: the steam boiler	p. 117
Summary	p. 123
Chapter notes	p. 124
Semantic structures	p. 127
Theory for the demonic model	p. 129
Deterministic probabilistic programs	p. 130
The sample space, random variables and expectations	p. 133
Probabilistic deterministic transformers	p. 135
Relational demonic semantics	p. 137
Regular transformers	p. 141
Healthiness conditions for probabilistic programs	p. 145
Characterising regular programs	p. 149
Complementary and consistent semantics	p. 154
Review: semantic structures	p. 157
Chapter notes	p. 164
The geometry of probabilistic programs	p. 165
Embedding distributions in Euclidean space	p. 166
Standard deterministic programs	p. 166
Probabilistic deterministic programs	p. 167
Demonic programs	p. 168
Refinement	p. 169
Nontermination and sub-distributions	p. 171
Expectations and touching planes	p. 172
Refinement seen geometrically	p. 174
The geometry of the healthiness conditions	p. 175
Sublinearity corresponds to convexity	p. 176
Truncated subtraction	p. 177
A geometrical proof for recursion	p. 177
Chapter notes	p. 180
Proved rules for probabilistic loops	p. 181
Introduction	p. 182
Partial loop correctness	p. 184
Total loop correctness	p. 186
Full proof of the loop rule	p. 189
Probabilistic variant arguments	p. 191
Finitary completeness of variants	p. 193
Do-it-yourself semantics	p. 195
Summary	p. 214
Chapter notes	p. 216
The transformer hierarchy	p. 217
Introduction	p. 217
Infinite state spaces	p. 219
Deterministic programs	p. 221
Demonic programs	p. 224
Angelic programs	p. 227
Standard programs	p. 231
Summary	p. 238
Chapter notes	p. 242
Advanced topics	p. 243
Quantitative temporal logic: an introduction	p. 245
Modal and temporal logics	p. 245
Standard temporal logic: a review	p. 247
Quantitative temporal logic	p. 252
Temporal operators as games	p. 254
Summary	p. 262
Chapter notes	p. 263
The quantitative algebra of qTL	p. 265
The role of algebra	p. 265
Quantitative temporal expectations	p. 268
Quantitative temporal algebra	p. 277
Examples: demonic random walkers and stumblers	p. 283
Summary	p. 289
Chapter notes	p. 291
The quantitative modal -calculus qM, and games	p. 293
Introduction to the -calculus	p. 293
Quantitative -calculus for probability	p. 295
Logical formulae and transition systems	p. 295
Two interpretations of qM	p. 298
Example	p. 301
Proof of equivalence of interpretations	p. 304
Summary	p. 308
Chapter notes	p. 310
Appendices, bibliography and indexes	p. 311
Alternative approaches	p. 313
Probabilistic Hoare-triples	p. 313
A programming logic of distributions	p. 316
Supplementary material	p. 321
Some algebraic laws of probabilistic programs	p. 321
Loop rule for demonic iterations	p. 328
Further facts about probabilistic wp and wlp	p. 331
Infinite state spaces	p. 332
Linear-programming lemmas	p. 341
Further lemmas for eventually	p. 342
Bibliography	p. 345
Index of citations	p. 357
General index	p. 361
Table of Contents provided by Publisher. All Rights Reserved.

Abstraction, Refinement and Proof for Probabilistic Systems

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