Read e-book online Abstraction, Refinement and Proof for Probabilistic Systems PDF

By Annabelle McIver

Probabilistic concepts are more and more being hired in desktop courses and structures simply because they could elevate potency in sequential algorithms, let another way nonfunctional distribution purposes, and make allowance quantification of probability and safeguard ordinarily. This makes operational types of ways they paintings, and logics for reasoning approximately them, tremendous important.

Abstraction, Refinement and facts for Probabilistic Systems provides a rigorous method of modeling and reasoning approximately computers that comprise 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 additional, into parts akin to randomized algorithms, fault tolerance, and, in dispensed structures, almost-certain symmetry breaking. The presentation starts with the well-known "assertional" sort of software improvement and maintains with expanding specialization: half I treats probabilistic application common sense, together with many examples and case reports; half II units out the unique 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 most recent mathematical learn in rigorous formalization of randomized (probabilistic) algorithms * Illustrates by means of instance the stairs worthy 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 sector of "quantitative" software logics

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

This obtainable, centred monograph, written through overseas gurus on probabilistic programming, develops an important beginning subject for contemporary programming and platforms improvement. Researchers, desktop scientists, and complex 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

Sample text

Above assumption about postE, postE as above, but for postE ✷ This kind of reasoning is nothing new for standard programs, and indeed is usually taken for granted (although its formal justification appeals to conjunctivity). 8 Interaction of probabilisticand demonic choice We conclude with some illustrations of the interaction of demonic and probabilistic choice. Consider two variables x, y, one chosen demonically and the other probabilistically. Suppose first that x is chosen demonically and y probabilistically, and take post-expectation [x = y].

1 Example: an inductive termination argument Proper post-expectations . . . . . . . . 1 The martingale revisited . . . . . . Bounded vs. unbounded expectations . . . . . 1 Unbounded invariants: a counter-example . Informal proof of the loop rule . . . . . . . . . . . . . . . . 38 2. 1 Introduction: loops via recursion We saw in Chap. 1 that iteration is a special case of recursion. postE cannot be given a purely syntactic definition for general recursive prog — the definition given earlier (Fig.

38 2. 1 Introduction: loops via recursion We saw in Chap. 1 that iteration is a special case of recursion. postE cannot be given a purely syntactic definition for general recursive prog — the definition given earlier (Fig. 3) is semantic, a least fixed-point over expectation transformers. It does give us an algebraic property of recursive programs, viz. 6) below however for an example of its use in spite of that. 1) we have immediately the property that do pred → prog od = ( prog ; do pred → prog od ) if pred else skip .

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