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

By Annabelle McIver

Probabilistic options are more and more being hired in machine courses and structures simply because they could elevate potency in sequential algorithms, allow in a different way nonfunctional distribution purposes, and make allowance quantification of hazard and security generally. This makes operational types of ways they paintings, and logics for reasoning approximately them, super important.

Abstraction, Refinement and evidence for Probabilistic Systems provides a rigorous method of modeling and reasoning approximately desktops that comprise 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 additional, into parts similar to randomized algorithms, fault tolerance, and, in disbursed platforms, almost-certain symmetry breaking. The presentation starts off with the common "assertional" type of application improvement and keeps with expanding specialization: half I treats probabilistic software good judgment, together with many examples and case experiences; half II units out the distinct 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 worthy 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 sector of "quantitative" application logics

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

This obtainable, concentrated monograph, written by way of overseas specialists on probabilistic programming, develops an important starting place subject for contemporary programming and structures improvement. Researchers, laptop 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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Additional info for Abstraction, Refinement and Proof for Probabilistic Systems

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Prog in each case transforms an expression in the program variables: that is, we give a procedure for calculating the greatest pre-expectation by purely syntactic manipulation. prog are then of type TS. The expression-based view is more convenient in an introduction, and for the treatment of specific programs; the function-based view is more convenient (and, for recursion, necessary) for general properties of expectation transformers. In this chapter and the rest of Part I we retain the 38 A partial order differs from the familiar “total” orders like “≤” in that two elements can be “incomparable”; the most common example is subset ⊆ between sets, which satisfies reflexivity (a set is a subset of itself), anti-symmetry (two sets cannot be subsets of each other without being the same set) and transitivity (one set within a second within a third is a subset of the third directly as well).

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.

Invariance and termination together: the loop rule . Three examples of probabilistic loops . . . . . 1 The martingale . . . . . . . . . 2 Probabilistic amplification . . . . . . 3 Faulty factorial . . . . . . . . . The Zero-One Law for termination . . . . . 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 .

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