Abstraction, Refinement and Proof for Probabilistic Systems by Annabelle McIver

By Annabelle McIver

Probabilistic innovations are more and more being hired in desktop courses and structures simply because they could bring up potency in sequential algorithms, let differently nonfunctional distribution purposes, and make allowance quantification of possibility and protection mostly. This makes operational versions of ways they paintings, and logics for reasoning approximately them, tremendous important.

Abstraction, Refinement and facts for Probabilistic Systems offers a rigorous method of modeling and reasoning approximately computers that include likelihood. Its foundations lie in conventional Boolean sequential-program logic—but its extension to numeric instead of in basic terms true-or-false judgments takes it a lot additional, into components reminiscent of randomized algorithms, fault tolerance, and, in dispensed structures, almost-certain symmetry breaking. The presentation starts with the normal "assertional" form of application improvement and keeps with expanding specialization: half I treats probabilistic application good judgment, together with many examples and case reports; half II units out the distinctive semantics; and half III applies the method of complicated fabric on temporal calculi and two-player games.

Topics and features:

* offers a basic semantics for either chance and demonic nondeterminism, together with abstraction and information refinement

* Introduces readers to the most recent mathematical examine in rigorous formalization of randomized (probabilistic) algorithms * Illustrates through instance the stairs useful for construction 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 zone of "quantitative" software logics

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

This available, targeted monograph, written by means of overseas experts on probabilistic programming, develops a necessary beginning subject for contemporary programming and platforms 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 info for Abstraction, Refinement and Proof for Probabilistic Systems

Sample text

PostE . It means “execute prog1 with probability at least p1 , and prog2 with probability at least p2 . . ”31 If the probabilities sum to 1 exactly, then it is a simple N -way probabilistic branch; if there is a deficit 1−Σi pi , it gives the probability of aborting. When all the programs progi are assignments with the same left-hand side, say x: = expri , we write even more briefly x: = (expr1 @ p1 | · · · | exprN @ pN ) . Variations on p ⊕ — By prog ⊕p prog we mean prog general we write prog p ⊕p prog for prog prog prog prog @ @ @ p⊕ prog, and in p p 1 − (p+p ) , the program that executes prog with probability at least p and prog 29 One predicate entails another, written |=, just when it implies the other in all states.

In this deterministic game,14 play becomes a succession of die rolls, taking the player from square to square; termination (no card) and winning (marker) are defined as before. e. demonically); the card is turned over and a die roll (probability) determines which of its listed alternatives to take. In the probabilistic game one can ask for the greatest guaranteed probability of winning; as in the standard case, the prediction will vary depending on the initial square. ) 13 A glance at Fig. 1 (p.

Variations on p ⊕ — By prog ⊕p prog we mean prog general we write prog p ⊕p prog for prog prog prog prog @ @ @ p⊕ prog, and in p p 1 − (p+p ) , the program that executes prog with probability at least p and prog 29 One predicate entails another, written |=, just when it implies the other in all states. 30 See Sec. 3 for an example of the vertical notation. 31 It is “at least p ” because if the probabilities sum to less than one there will be an i “aborting” component, which might behave like progi .

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