By Annabelle McIver

Illustrates by way of instance the common steps valuable in machine technological know-how to construct a mathematical version of any programming paradigm .

Presents result of a wide and built-in physique of study within the sector of 'quantitative' application logics.

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**Additional info for Abstraction, Refinement and Proof for Probabilistic Systems (Monographs in Computer Science)**

**Example text**

16 For deterministic (yet probabilistic) programs, the card-game model and the associated transformers are essentially Kozen’s original construction [Koz81, Koz85]. We have added demonic (and later angelic) nondeterminism. 16 1. 4 Behind the scenes: elementary probability theory In probability theory, an event is a subset of some given sample space S, so that the event is said to have occurred if the sampled value is in that set; a probability distribution Pr over the sample space is a function from its events into the closed interval [0, 1], giving for each event the probability of its occurrence.

Let the state space be S. 2 Expectation space S is deﬁned ES := The space of expectations over (S → R≥ , ), where the entailment relation , as we have seen, is inherited pointwise from the normal ≤ ordering in R≥ . The expectation-transformer model for programs is TS := (ES ← ES, ) , where we write the functional arrow backward just to emphasise that such transformers map ﬁnal post-expectations to initial pre-expectations, and where the reﬁnement order is derived pointwise from entailment on ES.

That “healthiness condition” [Dij76] is used to prove many general properties of programs. postE2 c0 , which property of prog is called sublinearity. Truncated subtraction deﬁned x 43 Footnote y := is (x − y) max 0 , 50 on p. 33 explains how typing might be propagated this way. satisfy monotonicity too, which is implied by conjunctivity. 45 Having discovered a probabilistic analogue of conjunctivity, we naturally ask for an analogue of disjunctivity. That turns out to be “super-linearity” — which when combined with sublinearity gives (just) linearity, and is characteristic of deterministic probabilistic programs, just as disjunctivity (with conjunctivity) characterises deterministic standard programs.