By Annabelle McIver
Probabilistic thoughts are more and more being hired in computing device courses and structures simply because they could raise potency in sequential algorithms, permit another way nonfunctional distribution purposes, and make allowance quantification of hazard and defense commonly. This makes operational versions of ways they paintings, and logics for reasoning approximately them, super important.
Abstraction, Refinement and facts for Probabilistic Systems provides a rigorous method of modeling and reasoning approximately desktops that include likelihood. Its foundations lie in conventional Boolean sequential-program logic—but its extension to numeric instead of basically true-or-false judgments takes it a lot extra, into parts equivalent to randomized algorithms, fault tolerance, and, in disbursed platforms, almost-certain symmetry breaking. The presentation starts off with the typical "assertional" sort of application improvement and maintains with expanding specialization: half I treats probabilistic application good judgment, together with many examples and case stories; half II units out the precise semantics; and half III applies the method of complicated fabric on temporal calculi and two-player games.
Topics and features:
* provides a basic semantics for either likelihood and demonic nondeterminism, together with abstraction and knowledge refinement
* Introduces readers to the newest mathematical study in rigorous formalization of randomized (probabilistic) algorithms * Illustrates by means of instance the stairs worthwhile for development a conceptual version of probabilistic programming "paradigm"
* Considers result of a wide and built-in examine workout (10 years and carrying on with) within the modern region of "quantitative" software logics
* contains worthy chapter-ending summaries, a entire index, and an appendix that explores replacement approaches
This available, targeted monograph, written by way of overseas specialists on probabilistic programming, develops a necessary starting place subject for contemporary programming and structures improvement. Researchers, computing device scientists, and complex undergraduates and graduates learning programming or probabilistic platforms will locate the paintings an authoritative and crucial source text.
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Additional resources for Abstraction, Refinement and Proof for Probabilistic Systems
G. “cheating,” because with it Dijkstra successfully conﬁned the second-order reasoning to just the place where it was required. 2. Probabilistic invariants 39 For probabilistic programs, by analogy, we look not necessarily for the greatest pre-expectation but merely for one that is suﬃciently small. Whether standard or probabilistic, the general techniques for loops involve “invariants” and “variants,” and we now consider their probabilistic versions. In the rest of Part I we use single capital letters A, B, · · · , P, Q, · · · for expressions over the state, that is Boolean-valued predicates, real-valued expectations etc.
We see several examples of such invariants in Sec. 5. 1 . 5) that expectations are non-negative.
The sets of states are predicates, and the program is being regarded as a predicate transformer. 15 Probabilistic programs become functions from probabilistic postconditions to probabilistic weakest preconditions — we call them postexpectations and greatest pre-expectations. The corresponding generalisation in the game is as follows. Rather than placing winning markers on the board, we place money — rather than strictly winning or losing, the player simply keeps whatever money he ﬁnds in his ﬁnal square.