Generalized Rough Logics with Rough Algebraic Semantics

2012 ◽  
pp. 183-193
Author(s):  
Jianhua Dai
Keyword(s):  
Rough Set ◽  
Stone Algebra ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Approximation Space ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Double Stone Algebra ◽  
Unary Operator ◽  
Soundness And Completeness

The collection of the rough set pairs <lower approximation, upper approximation> of an approximation (U, R) can be made into a Stone algebra by defining two binary operators and one unary operator on the pairs. By introducing a more unary operator, one can get a regular double Stone algebra to describe the rough set pairs of an approximation space. Sequent calculi corresponding to the rough algebras, including rough Stone algebras, Stone algebras, rough double Stone algebras, and regular double Stone algebras are proposed in this paper. The sequent calculi are called rough Stone logic (RSL), Stone logic (SL), rough double Stone logic (RDSL), and double Stone Logic (DSL). The languages, axioms and rules are presented. The soundness and completeness of the logics are proved.

Generalized Rough Logics with Rough Algebraic Semantics

2010 ◽  
Vol 4 (2) ◽  
pp. 35-49 ◽  
Author(s):  
Jianhua Dai
Keyword(s):  
Rough Set ◽  
Stone Algebra ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Approximation Space ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Double Stone Algebra ◽  
Unary Operator ◽  
Soundness And Completeness

The collection of the rough set pairs <lower approximation, upper approximation> of an approximation (U, R) can be made into a Stone algebra by defining two binary operators and one unary operator on the pairs. By introducing a more unary operator, one can get a regular double Stone algebra to describe the rough set pairs of an approximation space. Sequent calculi corresponding to the rough algebras, including rough Stone algebras, Stone algebras, rough double Stone algebras, and regular double Stone algebras are proposed in this paper. The sequent calculi are called rough Stone logic (RSL), Stone logic (SL), rough double Stone logic (RDSL), and double Stone Logic (DSL). The languages, axioms and rules are presented. The soundness and completeness of the logics are proved.

scholarly journals On Some Types of Covering-Based ℐ , T -Fuzzy Rough Sets and Their Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Mohammed Atef ◽  
José Carlos R. Alcantud ◽  
Hussain AlSalman ◽  
Abdu Gumaei
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Practical Application ◽  
Fuzzy Rough Sets ◽  
Fuzzy Rough Set ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Numerical Example ◽  
Variable Precision

The notions of the fuzzy β -minimal and maximal descriptions were established by Yang et al. (Yang and Hu, 2016 and 2019). Recently, Zhang et al. (Zhang et al. 2019) presented the fuzzy covering via ℐ , T -fuzzy rough set model ( FC ℐ T FRS ), and Jiang et al. (Jiang et al., in 2019) introduced the covering through variable precision ℐ , T -fuzzy rough sets ( CVP ℐ T FRS ). To generalize these models in (Jiang et al., 2019 and Zhang et al. 2019), that is, to improve the lower approximation and reduce the upper approximation, the present paper constructs eight novel models of an FC ℐ T FRS based on fuzzy β -minimal (maximal) descriptions. Characterizations of these models are discussed. Further, eight types of CVP ℐ T FRS are introduced, and we investigate the related properties. Relationships among these models are also proposed. Finally, we illustrate the above study with a numerical example that also describes its practical application.

Generalized Rough Sets

2016 ◽  
pp. 223-246
Author(s):  
Mona Hosny ◽  
Ali Kandil ◽  
Osama A. El-Tantawy ◽  
Sobhy A. El-Sheikh
Keyword(s):  
Rough Set ◽  
Rough Set Theory ◽  
Order Relation ◽  
Boundary Region ◽  
Current Approach ◽  
Partially Order ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Relation Ideal ◽  
Lower And Upper Approximations

This chapter concerns construction of a new rough set structure for an ideal ordered topological spaces and ordered topological filters. The approximation space approached depend on general binary relation, partially order relation, ideal and filter concepts. Properties of lower and upper approximation are extended to an ideal order topological approximation spaces. The main aim of the rough set theory is reducing the bouwndary region by increasing the lower approximation and decreasing the upper approximation. So, in this chapter different methods are proposed to reduce the boundary region. Comparisons between the current approximations and the previous approximations (El-Shafei et al.,2013) are introduced. It's therefore shown that the current approximations are more generally and reduce the boundary region by increasing the lower approximation and decreasing the upper approximation. The lower and upper approximations satisfy some properties in analogue of Pawlak's spaces (Pawlak, 1982). Moreover, we give several examples for comparison between the current approach and (El-Shafei et al., 2013).

scholarly journals De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

2021 ◽  
Author(s):  
Paul Égré ◽  
Lorenzo Rossi ◽  
Jan Sprenger
Keyword(s):  
Proof Theory ◽  
Future Research ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Truth Conditions ◽  
Indicative Conditionals ◽  
Tarski Algebra ◽  
Soundness And Completeness ◽  
Tableau Calculi ◽  
The One

AbstractIn Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1935, 1944) on the one hand, and by Cooper (Inquiry, 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic, 49, 245–260, 2008) on the other. Here we provide the proof theory for the resulting logics and , using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: allows for algebraic completeness, but not for the construction of a canonical model, while fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.

scholarly journals Idealization of j-approximation spaces

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 287-301
Author(s):  
Mona Hosny
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Current Work ◽  
Core Concept ◽  
Approximation Spaces ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Generalized Rough Set ◽  
The Ideal

The current work concentrates on generating different topologies by using the concept of the ideal. These topologies are used to make more thorough studies on generalized rough set theory. The rough set theory was first proposed by Pawlak in 1982. Its core concept is upper and lower approximations. The principal goal of the rough set theory is reducing the vagueness of a concept to uncertainty areas at their borders by increasing the lower approximation and decreasing the upper approximation. For the mentioned goal, different methods based on ideals are proposed to achieve this aim. These methods are more accurate than the previous methods. Hence it is very interesting in rough set context for removing the vagueness (uncertainty).

scholarly journals Seleksi Fitur Pada Klasifikasi Multi-label Menggunakan Proportional Feature Rough Selector

2021 ◽  
Vol 8 (4) ◽  
pp. 2084-2094
Author(s):  
Vilat Sasax Mandala Putra Paryoko
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Boundary Region ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Tv Shows

Proportional Feature Rough Selector (PFRS) merupakan sebuah metode seleksi fitur yang dikembangkan berdasarkan Rough Set Theory (RST). Pengembangan ini dilakukan dengan merinci pembagian wilayah dalam set data menjadi beberapa bagian penting yaitu lower approximation, upper approximation dan boundary region. PFRS memanfaatkan boundary region untuk menemukan wilayah yang lebih kecil yaitu Member Section (MS) dan Non-Member Section (NMS). Namun PFRS masih hanya digunakan dalam seleksi fitur pada klasifikasi biner dengan tipe data teks. PFRS ini juga dikembangkan tanpa memperhatikan hubungan antar fitur, sehingga PFRS memiliki potensi untuk ditingkatkan dengan mempertimbangkan korelasi antar fitur dalam set data. Untuk itu, penelitian ini bertujuan untuk melakukan penyesuaian PFRS untuk bisa diterapkan pada klasifikasi multi-label dengan data campuran yakni data teks dan data bukan teks serta mempertimbangkan korelasi antar fitur untuk meningkatkan performa klasifikasi multi-label. Pengujian dilakukan pada set data publik yaitu 515k Hotel Reviews dan Netflix TV Shows. Set data ini diuji dengan menggunakan empat metode klasifikasi yaitu DT, KNN, NB dan SVM. Penelitian ini membandingkan penerapan seleksi fitur PFRS pada data multi-label dengan pengembangan PFRS yaitu dengan mempertimbangkan korelasi. Hasil penelitian menunjukkan bahwa penggunaan PFRS berhasil meningkatkan performa klasifikasi. Dengan mempertimbangkan korelasi, PFRS menghasilkan peningkatan akurasi hingga 23,76%. Pengembangan PFRS juga menunjukkan peningkatan kecepatan yang signifikan pada semua metode klasifikasi sehingga pengembangan PFRS dengan mempertimbangkan korelasi mampu memberikan kontribusi dalam meningkatkan performa klasifikasi.

scholarly journals Further Study of MultigranulationT-Fuzzy Rough Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Wentao Li ◽  
Xiaoyan Zhang ◽  
Wenxin Sun
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Approximation Space ◽  
Fuzzy Rough Sets ◽  
Fuzzy Rough Set ◽  
Upper Approximation ◽  
Fuzzy Approximation ◽  
Approximation Operators ◽  
Special Instance

The optimistic multigranulationT-fuzzy rough set model was established based on multiple granulations underT-fuzzy approximation space by Xu et al., 2012. From the reference, a natural idea is to consider pessimistic multigranulation model inT-fuzzy approximation space. So, in this paper, the main objective is to make further studies according to Xu et al., 2012. The optimistic multigranulationT-fuzzy rough set model is improved deeply by investigating some further properties. And a complete multigranulationT-fuzzy rough set model is constituted by addressing the pessimistic multigranulationT-fuzzy rough set. The full important properties of multigranulationT-fuzzy lower and upper approximation operators are also presented. Moreover, relationships between multigranulation and classicalT-fuzzy rough sets have been studied carefully. From the relationships, we can find that theT-fuzzy rough set model is a special instance of the two new types of models. In order to interpret and illustrate optimistic and pessimistic multigranulationT-fuzzy rough set models, a case is considered, which is helpful for applying these theories to practical issues.

scholarly journals Rough Hyperfilters in Po-LA-Semihypergroups

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Ferdaous Bouaziz ◽  
Naveed Yaqoob
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Lower Approximation ◽  
Upper Approximation

This paper concerns the study of hyperfilters of ordered LA-semihypergroups, and presents some examples in this respect. Furthermore, we study the combination of rough set theory and hyperfilters of an ordered LA-semihypergroup. We define the concept of rough hyperfilters and provide useful examples on it. A rough hyperfilter is a novel extension of hyperfilter of an ordered LA-semihypergroup. We prove that the lower approximation of a left (resp., right, bi) hyperfilter of an ordered LA-semihypergroup becomes left (resp., right, bi) hyperfilter of an ordered LA-semihypergroup. Similarly we prove it for upper approximation.

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

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