We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the proposi-tional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatiza-tion and show that the problem of deciding satistiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatiza-tions, we give a complete axiomatization for reasoning about Boolean combina-tions of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.

The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition under which the ideal principle should be applied is encapsulated as the Fundamental Inequality, which in broad terms states that the principle is valid when the data are random, relative to every contemplated hypothesis and also these hypotheses are random relative to the (universal) prior. Basically, the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability, and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized. If we restrict the model class to the finite sets then application of the ideal principle turns into Kolmogorov's mi...

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This paper introduces an initial account of a formal methodology for specification-based black-box verification testing of software artefacts against their specifications, as well as for validation testing of specifications against the so-called application concept [14]

We study the modal logic ML r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of ML r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the nite frame property and its satisfiability problem is in EXPSPACE. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic.

A limited but basic problem in the foundations of statistics is the following: Given a parametric model, given perhaps some observations from the model, but given no prior information about the parameters ("total ignorance"), what can we say about the occurrence of a specified event A under this model in the future (prediction problem)? Or, as probabilities are often described in terms of bets, how can we bet on A? Bayesian solutions are internally consistent and fully conditional on the observed data, but their ties to the observed reality and their frequentist properties can be arbitrarily bad (unless, of course, the assumed prior distribution happens to be the true prior). Frequentist solutions are generally not possible with ordinary probabilities; but it is possible to define "successful bets" (using upper and lower probabilities), which even lead out of the state of total ignorance in an objective learning process converging to the true probability model. A special variant (su...

The heritage of logical empirism is manifold. One of its most important parts is the attempt of one of its late members, CARL GUSTAV HEMPEL, to explicate the notion of scientific explanation. The debate which Hempel's explanation models has cau-sed was probably the greatest in the philosophy of science since the 1940ies. It

1.2 Kripke frames and structures................................... 4 1.3 The standard translations into first- and second-order logic.................. 5 1.4 Theories, equivalence and definability.............................. 6 1.5 Polyadic modalities........................................ 8 2 Bisimulation and basic model constructions.............................. 8

The label semantics linguistic representation framework is introduced as an alternative approach to computing and modelling with words, based on the concept of appropriateness rather than graded truth. A possible worlds interpretation of label semantics is then proposed. In this model it is shown that certain assumptions about the underlying possible worlds can result in appropriateness measures that correspond to well known t-norms and t-conorms for conjunctions and disjunctions of linguistic labels.