Download Abstraction, Refinement and Proof for Probabilistic Systems by Annabelle McIver PDF

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By Annabelle McIver

Probabilistic options are more and more being hired in laptop courses and structures simply because they could elevate potency in sequential algorithms, allow differently nonfunctional distribution purposes, and make allowance quantification of threat and safeguard quite often. This makes operational types of the way 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 in basic terms true-or-false judgments takes it a lot extra, into components akin to randomized algorithms, fault tolerance, and, in dispensed structures, almost-certain symmetry breaking. The presentation starts with the conventional "assertional" variety of application improvement and keeps with expanding specialization: half I treats probabilistic application common sense, together with many examples and case stories; half II units out the special semantics; and half III applies the method of complex fabric on temporal calculi and two-player games.

Topics and features:

* provides a common 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 by way of instance the stairs valuable for construction 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 quarter of "quantitative" software logics

* comprises invaluable chapter-ending summaries, a entire index, and an appendix that explores replacement approaches

This available, targeted monograph, written via foreign professionals on probabilistic programming, develops a vital beginning subject for contemporary programming and structures improvement. Researchers, desktop 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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Example text

We see later that the finiteness requirement is so that our programs will be continuous (Footnote 60 on p. 71); and in some cases — for example, the third and fourth statements of the program shown in Fig. 1 — we rely on type information for that finiteness. 19) which matches Dijkstra’s original formulation more closely [Dij76]. 12). Q if pred else R , and is called a fixed point because placing Q in the expression does not alter its value — this is the mathematical equivalent of “and the same again” when the loop returns to its starting point for potentially more iterations.

In general any amount of money can be placed in a square, and that is the key to allowing a smooth sequential composition of programs at the logical level — for if the program game of Fig. prog to the greatest pre-expectation table for game. {4, 5} ) , and that table contains non-integer values (for example 50p). Another reason for allowing arbitrary values in R≥ is that using only standard postconditions ({0, 1}-valued) — equivalently, using explicit probabilities (recall the important fact above) — is not discriminating enough when nondeterminism is present: certain programs are identified that should be distinguished, and the semantics becomes non-compositional.

Probabilistic variant arguments for termination . . Termination example: self-stabilisation . . . . 1 Variations on the ring . . . . . . . Uncertain termination . . . . . . . . . 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 . . . . . . . . . .

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