By Annabelle McIver

ISBN-10: 0387401156

ISBN-13: 9780387401157

Probabilistic strategies are more and more being hired in machine courses and structures simply because they could elevate potency in sequential algorithms, permit in a different way nonfunctional distribution functions, and make allowance quantification of hazard and protection commonly. This makes operational types of ways they paintings, and logics for reasoning approximately them, tremendous important.

* Abstraction, Refinement and evidence for Probabilistic Systems* provides a rigorous method of modeling and reasoning approximately desktops that include chance. Its foundations lie in conventional Boolean sequential-program logic—but its extension to numeric instead of simply true-or-false judgments takes it a lot extra, into components reminiscent of randomized algorithms, fault tolerance, and, in disbursed platforms, almost-certain symmetry breaking. The presentation starts with the typical "assertional" form of application improvement and maintains with expanding specialization: half I treats probabilistic application common sense, together with many examples and case experiences; half II units out the certain semantics; and half III applies the method of complex fabric on temporal calculi and two-player games.

Topics and features:

* offers a normal semantics for either likelihood and demonic nondeterminism, together with abstraction and information refinement

* Introduces readers to the newest mathematical learn in rigorous formalization of randomized (probabilistic) algorithms * Illustrates through instance the stairs important for development a conceptual version of probabilistic programming "paradigm"

* Considers result of a wide and built-in study workout (10 years and carrying on with) within the modern region of "quantitative" software logics

* contains necessary chapter-ending summaries, a entire index, and an appendix that explores substitute approaches

This available, concentrated monograph, written via overseas professionals on probabilistic programming, develops a vital origin subject for contemporary programming and platforms improvement. Researchers, machine scientists, and complicated undergraduates and graduates learning programming or probabilistic structures will locate the paintings an authoritative and crucial source text.

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**Extra info for Abstraction, Refinement and Proof for Probabilistic Systems**

**Example text**

117]. postE is no longer an expectation even in the weak sense: it not linear. postE. This deﬁnition is one of the main features of this approach. postE2 . And clearly non-negativity continues to hold. The characterisations of expectations given above for the simpler cases might suggest that non-negative and sublinear functionals uniquely determine a set of probability distributions — and, in Chap. 1 Syntax Let prog range over programs and p over real number expressions taking values between zero and one inclusive; assume that x stands for a list of distinct variables, and expr for a list of expressions (of the same length as x 24 She was working in a much more general context.

The three “curtain” variables ac, cc, pc are of type {A, B, C}. Written in full, the ﬁrst three statements would be pc: = A pc: = B pc: = C; cc: = A 1 ⊕ (cc: = B 1 ⊕ cc: = C); 3 2 ac: ∈ {A, B, C} − {pc, cc} . The fourth statement is written using ∈ just for convenience — in fact it executes deterministically, since cc and ac are guaranteed to be diﬀerent at that point. 1. The “Monty Hall” program point to curtain cc. The host then tries to get the contestant to change his choice, showing that the prize is not behind some other curtain ac — which means that either the contestant has chosen it already or it is behind the other closed curtain.

22) for consistency with embedded Booleans. Obvious possibilities for & are multiplication ∗ and minimum min, and each of those has its uses; but neither satisﬁes anything like a generalisation of conjunctivity. Return for example to the program of Fig. 1, and consider its second statement cc: = (A @ 1 3 | B@ 1 3 | C @ 13 ) . [cc = A] . Thus probabilistic programs do not distribute min in general, and we must ﬁnd something else. 6. Healthiness and algebra for pGCL 31 whose right-hand side is inspired by sublinearity when c0 , c1 , c2 : = 1, 1, 1.

### Abstraction, Refinement and Proof for Probabilistic Systems by Annabelle McIver

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