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A Logic to Reason About f-Indices of Inclusion over Ł\(_n\)

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Fuzzy Logic and Technology, and Aggregation Operators (EUSFLAT 2023, AGOP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14069))

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Abstract

In this paper we provide a sound and complete logic to formalise and reason about f-indices of inclusion. The logic is based on finite-valued Łukasiewicz logic Ł\(_n\) and its S5-like modal extension S5(Ł\(_n\)) with additional unary operators.

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Notes

  1. 1.

    For the sake of simplicity, along this paper we will use the same symbol to denote both a logical language \(\mathcal{L}\) and its corresponding set of formulas \(Fm_\mathcal{L}\) built in the usual way. This will be done with no danger of confusion.

  2. 2.

    Actually, we will henceforth identify both notations \(\sigma \) and (w, S) to indicate this map, and we can even write \(\sigma = (w, S)\).

  3. 3.

    Indeed, if \(w(\varphi ) = r_0\), then \(w(\psi ) = \max _r \min ( w(\varDelta (\varphi \equiv \overline{r}), \tau (r)) = \)

    \( = \max (\max _{r \ne r_0} w(\varDelta (\varphi \equiv \overline{r}) \wedge \overline{\tau (r)}), w(\varDelta (\varphi \equiv \overline{r_0}) \wedge \overline{\tau (r_0)}) = \max (0, \min (1, \tau (r_0)) =\)

    \( = 0 \vee \tau (r_0) = \tau (w(\varphi ))\).

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Acknowledgments

Flaminio and Godo acknowledge partial support by the MOSAIC project (EU H2020-MSCA-RISE-2020 Project 101007627) and by the Spanish project PID2019-111544GB-C21/AEI/10.13039/501100011033. Madrid and Ojeda-Aciego acknowledge partial support by the project VALID (PID2022-140630NB-I00 funded by MCIN/AEI/10.13039/501100011033), and by Plan Propio de la Universidad de Málaga.

Author information

Authors and Affiliations

  1. IIIA-CSIC, 08193, Bellaterra, Spain

    Tommaso Flaminio & Lluis Godo

  2. Departamento de Matemática Aplicada, Universidad de Málaga, 29071, Málaga, Spain

    Nicolás Madrid & Manuel Ojeda-Aciego

Authors
  1. Tommaso Flaminio
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  2. Lluis Godo
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  3. Nicolás Madrid
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  4. Manuel Ojeda-Aciego
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Corresponding author

Correspondence to Tommaso Flaminio .

Editor information

Editors and Affiliations

  1. University of the Balearic Islands, Palma, Spain

    Sebastia Massanet

  2. University of Oviedo, Gijón, Spain

    Susana Montes

  3. University of the Balearic Islands, Palma, Spain

    Daniel Ruiz-Aguilera

  4. University of the Balearic Islands, Palma, Spain

    Manuel González-Hidalgo

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Flaminio, T., Godo, L., Madrid, N., Ojeda-Aciego, M. (2023). A Logic to Reason About f-Indices of Inclusion over Ł\(_n\). In: Massanet, S., Montes, S., Ruiz-Aguilera, D., González-Hidalgo, M. (eds) Fuzzy Logic and Technology, and Aggregation Operators. EUSFLAT AGOP 2023 2023. Lecture Notes in Computer Science, vol 14069. Springer, Cham. https://doi.org/10.1007/978-3-031-39965-7_44

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  • DOI: https://doi.org/10.1007/978-3-031-39965-7_44

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  • Online ISBN: 978-3-031-39965-7

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