Computer programming and formal systems by P. Braffort

By P. Braffort

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Bringing simple common sense out of the educational darkness into the sunshine of day, Paul Tomassi makes good judgment totally obtainable for somebody trying to come to grips with the complexities of this hard topic. together with student-friendly workouts, illustrations, summaries and a thesaurus of phrases, good judgment introduces and explains:

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Example text

On the purely logical side, we should expect that any statement from a reliable source is indeed true. This allows us to write reli → ϕi to connect the auxiliary variable RELi with ϕi . With Φ + = {rel1 → ϕ1 , . . , reln → ϕn } we denote the set of all such material implications, from which we obtain a probabilistic argumentation system A + = (V ∪ W, LV ∪W , Φ + ,W, P) with W = {REL1 , . . , RELn } and P as defined above. This allows us then to compute the degrees of support and possibility for the conclusion ψ and to use them as lower and upper bounds for the target interval Y .

If we assume the least restrictive interval Xi = [0, 1] to represent a totally incompetent source, and similarly the most restrictive interval Xi = [xi , xi ] to represent a totally competent source, then ui − i surely represents the source’s degree of incompetence, from which we obtain P(compi ) = 1 − (ui − i ) = 1 − ui + i for the marginal probability of compi . Following a similar line of reasoning, we first obtain P(compi ∧ honi ) = i for the combined event compi ∧ honi of a reliable source, which then leads to P(honi ) = i P(compi ) = i 1 − u i + li for the marginal probability of honi .

Note that the independence assumption, on which Dempster’s rule is based, has raised quite some criticism with regard to the appropriateness of the rule and the theory as a whole (Zadeh, 1979). In probabilistic argumentation, these criticisms are circumvented by not explicitly formulating Dempster’s rule and thus by not giving it such a fundamental role. Another major difference is the fact that the notions of belief and plausibility in the Dempster-Shafer theory are often entirely detached from a probabilistic interpretation (for example in Smets’ Transferable Belief Model (Smets and Kennes, 1994)), whereas degrees of support and possibility are probabilities by definition.

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