6 research outputs found

    Belief as Willingness to Bet

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    We investigate modal logics of high probability having two unary modal operators: an operator KK expressing probabilistic certainty and an operator BB expressing probability exceeding a fixed rational threshold c≥12c\geq\frac 12. Identifying knowledge with the former and belief with the latter, we may think of cc as the agent's betting threshold, which leads to the motto "belief is willingness to bet." The logic KB.5\mathsf{KB.5} for c=12c=\frac 12 has an S5\mathsf{S5} KK modality along with a sub-normal BB modality that extends the minimal modal logic EMND45\mathsf{EMND45} by way of four schemes relating KK and BB, one of which is a complex scheme arising out of a theorem due to Scott. Lenzen was the first to use Scott's theorem to show that a version of this logic is sound and complete for the probability interpretation. We reformulate Lenzen's results and present them here in a modern and accessible form. In addition, we introduce a new epistemic neighborhood semantics that will be more familiar to modern modal logicians. Using Scott's theorem, we provide the Lenzen-derivative properties that must be imposed on finite epistemic neighborhood models so as to guarantee the existence of a probability measure respecting the neighborhood function in the appropriate way for threshold c=12c=\frac 12. This yields a link between probabilistic and modal neighborhood semantics that we hope will be of use in future work on modal logics of qualitative probability. We leave open the question of which properties must be imposed on finite epistemic neighborhood models so as to guarantee existence of an appropriate probability measure for thresholds c≠12c\neq\frac 12.Comment: Removed date from v1 to avoid confusion on citation/reference, otherwise identical to v

    The complexity of probabilistic EL

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    We analyze the complexity of subsumption in probabilistic variants of the description logic EL. In the case where probabilities apply only to concepts, we map out the borderline between tractability and EXPTIME-completeness. One outcome is that any probability value except zero and one leads to intractability in the presence of general TBoxes, while this is not the case for classical TBoxes. In the case where probabilities can also be applied to roles, we show PSPACEcompleteness. This result is (positively) surprising as the best previously known upper bound was 2-EXPTIME and there were reasons to believe in completeness for this class

    Probabilistic description logics for subjective uncertainty

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    We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable

    Reasoning in Many Dimensions : Uncertainty and Products of Modal Logics

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    Probabilistic Description Logics (ProbDLs) are an extension of Description Logics that are designed to capture uncertainty. We study problems related to these logics. First, we investigate the monodic fragment of Probabilistic first-order logic, show that it has many nice properties, and are able to explain the complexity results obtained for ProbDLs. Second, in order to identify well-behaved, in best-case tractable ProbDLs, we study the complexity landscape for different fragments of ProbEL; amongst others, we are able to identify a tractable fragment. We then study the reasoning problem of ontological query answering, but apply it to probabilistic data. Therefore, we define the framework of ontology-based access to probabilistic data and study the computational complexity therein. In the final part of the thesis, we study the complexity of the satisfiability problem in the two-dimensional modal logic KxK. We are able to close a gap that has been open for more than ten years

    Modal probability, belief, and actions

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    International audienceWe investigate a modal logic of probability with a unary modal operator expressing that a proposition is more probable than its negation. Such an operator is not closed under conjunction, and its modal logic is therefore non-normal. Within this framework we study the relation of probability with other modal concepts: belief and action. We focus on the evolution of belief, and propose an integration of revision. For that framework we give a regression algorithm
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