Abstract
Zdzisław Pawlak influenced our thinking about uncertainty by borrowing the idea of approximation from geometry and topology and carrying those ideas into the realm of knowledge engineering. In this way, simple and already much worn out mathematical notions, gained a new life given to them by new notions of decision rules and algorithms, complexity problems, and problems of optimization of relations and rules. In his work, the author would like to present his personal remembrances of how his work was influenced by Zdzisław Pawlak interlaced with discussions of highlights of research done in enliving classical concepts in new frameworks, and next, he will go to more recent results that stem from those foundations, mostly on applications of rough mereology in behavioral robotics and classifier synthesis via granular computing.
L.T. Polkowski—An invited Fellow IRSS talk.
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- 1.
Results on topology of rough sets can be best found in author’s [4].
- 2.
The pair IS = (U,A) will be called an information system; each \(a_n\in A\) maps U into a set V of possible values.
- 3.
\(Cl_{\tau }\) is the closure operator and \(Int_{\tau }\) is the interior operator with respect to a topology \(\tau \).
- 4.
\([u]_n\) is the \(Ind_n\)-class of u.
- 5.
\(dist(x,A)=min_{y\in A}d(x,y)\).
- 6.
This theorem comes from the chapter by the author in [3].
- 7.
The upper approximation of a set \(X\subseteq U\) with respect to a partition P on U is \(\bigcup \{q\in P: q\cap X\ne \emptyset \}\).
- 8.
To acquaint oneself with this theory it is best to read Lesniewski [2]. This is a rendering by E. Luschei of the original work Foundations of Set Theory. Polish Scientific Circle. Moscow 1916.
- 9.
Please see relevant chapters in Polkowski [5].
- 10.
Please see Polkowski L., Osmialowski P. [8].
- 11.
- 12.
An information system \({\varvec{IS}}\,{\varvec{=}}\,{\varvec{(U,A)}}\) augmented by a new attribute \({\varvec{d}}\,{\varvec{:}}\,{\varvec{U}}\,\rightarrow \,{\varvec{V}}\), the decision, is called the decision system \({\varvec{DS}}\,{\varvec{=}}\,{\varvec{(U,A,d)}}\).
- 13.
The philosophical term ‘thing’ is reserved for beings of virtual character possibly not present in the given information/decision system.
- 14.
In a decision system (U, A, d), for \(u\in U\), the information set of u is \(Inf(u)=\{(a, a(u)): a\in A\cup \{d\}\)}.
- 15.
MI is the Michalski index. \(MI=\frac{1}{2} \cdot aex + \frac{1}{4}\cdot aex^2+ \frac{1}{2}\cdot cex - \frac{1}{4}\cdot aex\cdot cex\).
- 16.
A detailed account please find in Polkowski, Nowak [7].
- 17.
In order to split the data set into parts of which one is GB-self-contained and the other GB-vacuous, we propose the DIM matrix.
- 18.
A relaxed idea of convex combinations of objects lies in approximating only parts of data objects with training objects, see Artiemjew, Nowak, Polkowski [1].
References
Artiemjew, P., Nowak, B., Polkowski, L.: A classifier based on the dual indiscernibility matrix. In: Dregvaite, G., Damasevicius, R. (eds.) Forthcoming in Communications in Computer and Information Science, Proceedings ICIST 2016, CCIS639, pp. 1–12. Springer (2016). doi:10.1007/978-3-319-46254-7_30
Lesniewski, S.: On the foundations of mathematics. Topoi 2, 7–52 (1982)
Polkowski, L.: Approximate mathematical morphology. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization, pp. 151–162. Springer, Singapore (1999)
Polkowski, L.T.: Rough Sets. Mathematical Foundations. Springer, Physica, Heidelberg (2002)
Polkowski, L.T.: Approximate Reasoning by Parts. An Introduction to Rough Mereology. Springer, Switzerland (2011)
Polkowski, L., Artiemjew, P.: Granular Computing in Decision Approximation. An Application of Rough Mereology. Springer, Switzerland (2015)
Polkowski, L., Nowak, B.: Betweenness, Łukasiewicz rough inclusion, Euclidean representations in information systems, hyper-granules, conflict resolution. Fundamenta Informaticae 147(2-3) (2016)
Polkowski, L., Osmialowski, P.: Navigation for mobile autonomous robots and their formations. An application of spatial reasoning induced from rough mereological geometry. In: Barrera, A. (ed.) Mobile Robots Navigation, pp. 329–354. Intech, Zagreb (2010)
Acknowledgements
This is in remembrance of Prof. Zdzisław Pawlak on the 10th anniversary of His demise. To organizers of IJCRS 2016 Prof. Richard Weber and Dr. Dominik Ślȩzak DSc my thanks go for the invitation to talk henceforth the occasion to remember. To referees my thanks go for their useful remarks.
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Polkowski, L.T. (2016). Rough Sets of Zdzisław Pawlak Give New Life to Old Concepts. A Personal View .... . In: Flores, V., et al. Rough Sets. IJCRS 2016. Lecture Notes in Computer Science(), vol 9920. Springer, Cham. https://doi.org/10.1007/978-3-319-47160-0_4
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