By Annabelle McIver

Probabilistic strategies are more and more being hired in laptop courses and structures simply because they could elevate potency in sequential algorithms, let differently nonfunctional distribution functions, and make allowance quantification of danger and safeguard more often than not. This makes operational versions of ways 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 in simple terms true-or-false judgments takes it a lot additional, into parts comparable to randomized algorithms, fault tolerance, and, in disbursed platforms, almost-certain symmetry breaking. The presentation starts off with the conventional "assertional" sort 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 certain semantics; and half III applies the method of complicated fabric on temporal calculi and two-player games.

Topics and features:

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

* Introduces readers to the newest mathematical examine in rigorous formalization of randomized (probabilistic) algorithms * Illustrates via instance the stairs invaluable for development 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

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

This obtainable, concentrated monograph, written by means of foreign specialists on probabilistic programming, develops a vital origin subject for contemporary programming and structures improvement. Researchers, machine scientists, and complex undergraduates and graduates learning programming or probabilistic structures will locate the paintings an authoritative and crucial source text.

**Read Online or Download Abstraction, Refinement and Proof for Probabilistic Systems PDF**

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**Extra resources for Abstraction, Refinement and Proof for Probabilistic Systems**

**Sample text**

Then 52 Lem. prog is deﬁned for them; the proof of that can be given by direct reference to the deﬁnition of wp over the model, as set out in Chap. 5. g. 3). 9. ([x = 1] /2 + [x = 2] /2) ([1 = 1] /2 + [1 = 2] /2) min ([2 = 1] /2 + [2 = 2] /2) (1/2 + 0/2) min (0/2 + 1/2) 1/2 , from which we see that program establishes x = y with probability at least 1/2: no matter which value is assigned to x, with probability 1/2 the second command will assign the same to y. Now suppose instead that it is the second choice that is demonic.

An informal computational model for pGCL 15 The post-expectation: Final state Payoﬀ awarded if this state reached 0 0 1 0 2 0 3 0 4 £1 5 £1 6 0 4 1 5 1 The probability of winning (ending on a £1) (from Fig. 2. 16 We are not limited to £1 coins for indicating postconditions — that is only an artefact of embedding standard postconditions into the probabilistic world. 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.

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.