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Propositional Variable Forgetting and Marginalization: Semantically, Two Sides of the Same Coin

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Foundations of Information and Knowledge Systems (FoIKS 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14589))

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Abstract

This paper investigates variable forgetting and marginalization in propositional logic. We show that for finite signatures and infinite signatures, variable forgetting and marginalization are corresponding operations, i.e., they yield semantically equivalent outputs for respective complementary inputs. This observation holds for formulas and also for sets of formulas. For formulas, both operations, variable forgetting and marginalization, are shown to be compatible with disjunctions, but not with conjunction, implication and negation. For general sets of formulas, a consequence is that the element-wise application of these operations to a set of formulas and the application to a formula equivalent to this set are not equivalent in general. However, for every deductively closed set \( X \), we show that the element-wise application of variable forgetting or marginalization, respectively, and the application to any formula equivalent to \( X \) are equivalent. This latter observation is important because deductively closed sets play an important role in many areas, e.g., in logic-based approaches to knowledge representation and databases.

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Notes

  1. 1.

    The proofs are available at: https://kai-sauerwald.de/pub/FoIKS2024.pdf.

  2. 2.

    We thank the anonymous reviewer for phrasing this interrelation so nicely.

  3. 3.

    Closure operators satisfy (Monotonicity), (Idempotency) and (Extensitivity), i.e., \( X\subseteq Cl(X) \).

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Acknowledgements

We thank the reviewers for their constructive and helpful comments. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Priority Research Program “Intentional Forgetting in Organizations” (SPP 1921; grant BE 1700/10-1 awarded to Christoph Beierle and grant KE 1413/10-1 awarded to Gabriele Kern-Isberner).

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Authors and Affiliations

  1. FernUniversität in Hagen, 58084, Hagen, Germany

    Kai Sauerwald & Christoph Beierle

  2. Technische Universität Dortmund, 44227, Dortmund, Germany

    Gabriele Kern-Isberner

Authors
  1. Kai Sauerwald
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  2. Christoph Beierle
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  3. Gabriele Kern-Isberner
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Correspondence to Kai Sauerwald .

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Editors and Affiliations

  1. Leibniz University Hannover, Hanover, Germany

    Arne Meier

  2. TU Wien, Wien, Austria

    Magdalena Ortiz

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Sauerwald, K., Beierle, C., Kern-Isberner, G. (2024). Propositional Variable Forgetting and Marginalization: Semantically, Two Sides of the Same Coin. In: Meier, A., Ortiz, M. (eds) Foundations of Information and Knowledge Systems. FoIKS 2024. Lecture Notes in Computer Science, vol 14589. Springer, Cham. https://doi.org/10.1007/978-3-031-56940-1_8

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  • DOI: https://doi.org/10.1007/978-3-031-56940-1_8

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