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NP-containment for the coherence test of assessments of conditional probability: a fuzzy logical approach

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Abstract

In this paper we investigate the problem of testing the coherence of an assessment of conditional probability following a purely logical setting. In particular we will prove that the coherence of an assessment of conditional probability χ can be characterized by means of the logical consistency of a suitable theory T χ defined on the modal-fuzzy logic FP k (RŁΔ) built up over the many-valued logic RŁΔ. Such modal-fuzzy logic was previously introduced in Flaminio (Lecture Notes in Computer Science, vol. 3571, 2005) in order to treat conditional probability by means of a list of simple probabilities following the well known (smart) ideas exposed by Halpern (Proceedings of the eighth conference on theoretical aspects of rationality and knowledge, pp 17–30, 2001) and by Coletti and Scozzafava (Trends Logic 15, 2002). Roughly speaking, such logic is obtained by adding to the language of RŁΔ a list of k modalities for “probably” and axioms reflecting the properties of simple probability measures. Moreover we prove that the satisfiability problem for modal formulas of FP k (RŁΔ) is NP-complete. Finally, as main result of this paper, we prove FP k (RŁΔ) in order to prove that the problem of establishing the coherence of rational assessments of conditional probability is NP-complete.

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  1. Department of Mathematics and Computer Science, Pian dei Mantellini 44, 53100, Siena, Italy

    Tommaso Flaminio

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  1. Tommaso Flaminio
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Correspondence to Tommaso Flaminio.

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Flaminio, T. NP-containment for the coherence test of assessments of conditional probability: a fuzzy logical approach. Arch. Math. Logic 46, 301–319 (2007). https://doi.org/10.1007/s00153-007-0045-3

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  • DOI: https://doi.org/10.1007/s00153-007-0045-3

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