Read e-book online Abstraction, Refinement and Proof for Probabilistic Systems PDF
By Annabelle McIver
Probabilistic strategies are more and more being hired in desktop courses and structures simply because they could elevate potency in sequential algorithms, let in a different way nonfunctional distribution functions, and make allowance quantification of possibility and security often. This makes operational types of the way they paintings, and logics for reasoning approximately them, tremendous important.
Abstraction, Refinement and evidence for Probabilistic Systems offers a rigorous method of modeling and reasoning approximately desktops that comprise chance. 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 equivalent to randomized algorithms, fault tolerance, and, in dispensed structures, almost-certain symmetry breaking. The presentation starts with the common "assertional" kind of application improvement and maintains with expanding specialization: half I treats probabilistic software good judgment, together with many examples and case stories; 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:
* provides a normal semantics for either chance and demonic nondeterminism, together with abstraction and knowledge refinement
* Introduces readers to the most recent mathematical examine in rigorous formalization of randomized (probabilistic) algorithms * Illustrates through instance the stairs helpful for construction a conceptual version of probabilistic programming "paradigm"
* Considers result of a wide and built-in learn workout (10 years and carrying on with) within the modern zone of "quantitative" software logics
* contains worthwhile chapter-ending summaries, a finished index, and an appendix that explores substitute approaches
This obtainable, centred monograph, written by way of foreign specialists on probabilistic programming, develops an important starting place subject for contemporary programming and platforms improvement. Researchers, computing device 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
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.
38 39 40 42 44 44 46 51 53 54 56 59 61 62 63 64 68 71 74 Chapter notes . . . . . . . . . . . . . . 12 Introduction: loops via recursion . . . . . . Probabilistic invariants . . . . . . . . . Probabilistic termination . . . . . . . . Invariance and termination together: the loop rule . Three examples of probabilistic loops . . . . . 1 The martingale . . . . . . . . . 2 Probabilistic ampliﬁcation . . . . . . 3 Faulty factorial . . . . . .
3 for this deﬁnition. 24 1. Introduction to pGCL ≡ ≡ (1/2)(1/2 + 0/2) + (1/2)(0/2 + 1/2) 1/2 . 20) and what does that mean on its own? It must be given the ﬁrst interpretation, that is as an expected worth, since “will establish [x = H] /2 + [x = T ] /2” makes no sense. Thus it means the expected value of the expression [x = H] /2 + [x = T ] /2 after executing the program x: = H 12 ⊕ x: = T , which the calculation goes on to show is in fact 1/2. 5 Semantics The probabilistic semantics is derived from generalising the standard semantics in the way suggested in Sec.
Abstraction, Refinement and Proof for Probabilistic Systems by Annabelle McIver