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What’s in a Relation? Logical Structures of Modes of Granulation

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Rough Sets (IJCRS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11103))

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Abstract

Granulation can be though of as a conceptual grid based on given knowledge, while approximation is the process of forming new knowledge through an available conceptual grid. In a wider sense, approximating is an operation required when a “scale” is used to determine something which does not fit exactly with the “precision” enabled by that scale. One can find instances of this dialectic between granulation and approximation in different fields spanning from data mining to story understanding, from pattern recognition to machine learning. We use the term “scale” in a general sense. Granulation is a sort of “conceptual scale”. Granules are groups of items (or points) of a given universe of discourse formed by means of knowledge which has been acquired or hypothesized and stored, that is, an established knowledge. From now on, we use the terms “granule” and “neighbourhood”, as well as “granulation” and “neighbourhood system”, interchangeably.

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Notes

  1. 1.

    In formal topology, it is called a basic pair or a Chu space.

  2. 2.

    In Rough Set Theory the following operartors have been introduced by [4] and independently in [11].

  3. 3.

    Sometimes this term denotes what we call a relational system.

  4. 4.

    The combination of quantifiers suggests an investigation of the relationships between the formal properties of the above operators and those in the hexagon of opposition which are obtained by similar combinations (see [2]).

  5. 5.

    If \(R(U)=U'\) then R is said to be right-total, or surjective, or that \(R^\smile \) is serial. \(R^\smile (U')=U\) means that R is left-total or serial.

  6. 6.

    From now on the interested reader is addressed to [12] and its bibliography.

  7. 7.

    Proposition 3 amends point (iv) of Corollary 1 of [9] and point (ii) of Facts 3 of [10], which state also the converse implication, erroneously.

  8. 8.

    Notice that if \(U'\ne U\) and one substitutes \(\wp (U')\) for \(\wp (U)\) then a more general picture is obtained. However, the result of the more specific case can be translated into the more general case by means of a map from U to \(U'\).

  9. 9.

    Details may be found in [12]. Pay attention that in that book R(x) is denoted as \({\mathcal N}_x\) and property systems are called “basic neighbourhood pairs”, in the context of pre-topological formal spaces. A simplified proof can be found in [9].

  10. 10.

    This approach was pioneered in [5]. In [8] neighbourhood systems result from families of relational systems and two approximation operators “according to n relations” were introduced. Neighbourhood systems not fulfilling N1 were investigated in [6].

  11. 11.

    Cf. [12], where a more complete notion of a pre-topological formal system is defined, together with a classification of such systems.

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Authors and Affiliations

  1. Research Group on Knowledge and Communication Models, Rome, Italy

    Piero Pagliani

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  1. Piero Pagliani
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Correspondence to Piero Pagliani .

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Editors and Affiliations

  1. University of Warsaw, Warsaw, Poland

    Hung Son Nguyen

  2. Faculty of Information Technology, Vietnam National University, Hanoi, Vietnam

    Quang-Thuy Ha

  3. School of Information Science, Southwest Jiaotong University, Chengdu, China

    Tianrui Li

  4. Institute of Computer Science, University of Silesia, Sosnowiec, Poland

    Małgorzata Przybyła-Kasperek

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Pagliani, P. (2018). What’s in a Relation? Logical Structures of Modes of Granulation. In: Nguyen, H., Ha, QT., Li, T., Przybyła-Kasperek, M. (eds) Rough Sets. IJCRS 2018. Lecture Notes in Computer Science(), vol 11103. Springer, Cham. https://doi.org/10.1007/978-3-319-99368-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-99368-3_4

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