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Canonical Extension of Possibility Measures to Boolean Algebras of Conditionals

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12897))

Abstract

In this paper we study conditional possibility measures within the algebraic setting of Boolean algebras of conditional events. More precisely, we focus on the possibilistic version of the Strong Conditional Event Problem, introduced for probabilities by Goodman and Nguyen, and solved in finitary terms in a recent paper by introducing the so-called Boolean algebras of conditionals. Our main result shows that every possibility measure on a finite Boolean algebra can be canonically extended to an unconditional possibility measure on the resulting Boolean algebra of conditionals, in such a way that the canonical extension and the conditional possibility, determined in usual terms by any continuous t-norm, coincide on every basic conditional expression.

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Notes

  1. 1.

    A t-norm \(*\) has no zero-divisors when \(x * y = 0\) implies either \(x = 0\) or \(y = 0\), see e.g. [16].

  2. 2.

    Called T-conditional possibility in [5, 6].

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Acknowledgments

The authors are grateful to the anonymous reviewers for their comments and suggestions. Flaminio acknowledges partial support by the Spanish Ramón y Cajal research program RYC-2016- 19799. Godo is indebted to Karim Tabia for helpful discussions on conditional possibility and acknowledges support by the Spanish project ISINC (PID2019-111544GB-C21). Ugolini receives funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 890616 (H2020-MSCA-IF-2019).

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Authors and Affiliations

  1. IIIA - CSIC, 08193, Bellaterra, Spain

    Tommaso Flaminio, Lluis Godo & Sara Ugolini

Authors
  1. Tommaso Flaminio
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  2. Lluis Godo
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  3. Sara Ugolini
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Correspondence to Lluis Godo .

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  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Jiřina Vejnarová

  2. Insight Centre for Data Analytics, School of Computer Science and IT University College Cork, Cork, Ireland

    Nic Wilson

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Flaminio, T., Godo, L., Ugolini, S. (2021). Canonical Extension of Possibility Measures to Boolean Algebras of Conditionals. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_39

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  • DOI: https://doi.org/10.1007/978-3-030-86772-0_39

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