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Many-Valued Dynamic Object-Oriented Inheritance and Approximations

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Book cover Rough Sets (IJCRS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12872))

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Abstract

The majority of contemporary software systems are developed using object-oriented tools and methodologies, where constructs like classes, inheritance and objects are first-class citizens. In the current paper we provide a novel formal framework for many-valued object-oriented inheritance in rule-based query languages. We also relate the framework to rough set-like approximate reasoning. Rough sets and their generalizations have intensively been studied and applied. However, the mainstream of the area mainly focuses on the context of information and decision tables. Therefore, approximations defined in the much richer object-oriented contexts generalize known approaches.

Supported by the Polish National Science Centre grant 2017/27/B/ST6/02018.

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Notes

  1. 1.

    This technique is actually used, e.g., in SAT solver-based implementations of Asp.

  2. 2.

    Observe that removing the value from these orderings result in orderings shown in Fig. 1.

  3. 3.

    Each rule is treated as a separate information source. In 4ql ignorance and inconsistencies may be resolved using heuristic/non-monotonic rules or additional information sources. For simplicity we skip these aspects here. For details see [14, 15].

  4. 4.

    \(\nu \) is a function specific to a logic in question. The requirements (4) and (5) make the considered many-valued logics compatible with \(K_3\) on connectives \(\lnot _{},\wedge ,\vee \).

  5. 5.

    According to a convention used in rule languages, comma in rules’ premises (bodies) is interpreted as a conjunction.

  6. 6.

    We assume here type compatibility in the sense of o\(^{\mathrm {n}}\) QL [25].

  7. 7.

    Recall that we assume that objects are created using classes as patterns. Therefore we require that actual parameters determine a unique object represented by a nested structure.

  8. 8.

    For clarity, the unknown facts in ‘smartphone’ are listed explicitly.

  9. 9.

    In fact, it is compatible with belief fusion in 4ql and belief bases of [8].

  10. 10.

    Recall that the domains we deal with are finite.

  11. 11.

    Observe that approximations are classical two-valued sets.

  12. 12.

    The assumption that is a designated truth value is not artificial – see, e.g., the Priest logic [20].

  13. 13.

    Since approximations are two-valued sets, we list elements belonging to a set.

  14. 14.

    The complexity of computing truth values returned by connectives is typically \(\mathcal {O} (1)\).

  15. 15.

    In fact, tableaux became a dominant verification technique in Semantic Web.

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Authors and Affiliations

  1. Institute of Informatics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Szałas

  2. Department of Computer and Information Science, Linköping University, 581 83, Linköping, Sweden

    Andrzej Szałas

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  1. Andrzej Szałas
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Editors and Affiliations

  1. University of Winnipeg, Winnipeg, MB, Canada

    Sheela Ramanna

  2. Ghent University, Gent, Belgium

    Chris Cornelis

  3. University of Milano-Bicocca, Milan, Italy

    Davide Ciucci

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Szałas, A. (2021). Many-Valued Dynamic Object-Oriented Inheritance and Approximations. In: Ramanna, S., Cornelis, C., Ciucci, D. (eds) Rough Sets. IJCRS 2021. Lecture Notes in Computer Science(), vol 12872. Springer, Cham. https://doi.org/10.1007/978-3-030-87334-9_10

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  • DOI: https://doi.org/10.1007/978-3-030-87334-9_10

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