938 research outputs found
Approximations from Anywhere and General Rough Sets
Not all approximations arise from information systems. The problem of fitting
approximations, subjected to some rules (and related data), to information
systems in a rough scheme of things is known as the \emph{inverse problem}. The
inverse problem is more general than the duality (or abstract representation)
problems and was introduced by the present author in her earlier papers. From
the practical perspective, a few (as opposed to one) theoretical frameworks may
be suitable for formulating the problem itself. \emph{Granular operator spaces}
have been recently introduced and investigated by the present author in her
recent work in the context of antichain based and dialectical semantics for
general rough sets. The nature of the inverse problem is examined from
number-theoretic and combinatorial perspectives in a higher order variant of
granular operator spaces and some necessary conditions are proved. The results
and the novel approach would be useful in a number of unsupervised and semi
supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings,
Springe
Formalizing Two Generalized Approximation Operators
Rough sets, developed by Pawlak [15], are important tool to describe situation of incomplete or partially unknown information. In this article we give the formal characterization of two closely related rough approximations, along the lines proposed in a paper by GomoliÅska [2]. We continue the formalization of rough sets in Mizar [1] started in [6].Adam Grabowski - Institute of Informatics, University of BiaÅystok, PolandMichaÅ Sielwiesiuk - Institute of Informatics, University of BiaÅystok, PolandGrzegorz Bancerek, CzesÅaw ByliÅski, Adam Grabowski, Artur KorniÅowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261ā279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Anna GomoliÅska. A comparative study of some generalized rough approximations. Fundamenta Informaticae, 51:103ā119, 2002.Adam Grabowski. Automated discovery of properties of rough sets. Fundamenta Informaticae, 128:65ā79, 2013. doi:10.3233/FI-2013-933.Adam Grabowski. Lattice theory for rough sets ā a case study with Mizar. Fundamenta Informaticae, 147(2ā3):223ā240, 2016. doi:10.3233/FI-2016-1406.Adam Grabowski. Formalization of generalized almost distributive lattices. Formalized Mathematics, 22(3):257ā267, 2014. doi:10.2478/forma-2014-0026.Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21ā28, 2004.Adam Grabowski. Relational formal characterization of rough sets. Formalized Mathematics, 21(1):55ā64, 2013. doi:10.2478/forma-2013-0006.Adam Grabowski. Binary relations-based rough sets ā an automated approach. Formalized Mathematics, 24(2):143ā155, 2016. doi:10.1515/forma-2016-0011.Adam Grabowski and Magdalena JastrzÄbska. A note on a formal approach to rough operators. In Marcin S. Szczuka and Marzena Kryszkiewicz et al., editors, Rough Sets and Current Trends in Computing ā 7th International Conference, RSCTC 2010, Warsaw, Poland, June 28-30, 2010. Proceedings, volume 6086 of Lecture Notes in Computer Science, pages 307ā316. Springer, 2010. doi:10.1007/978-3-642-13529-3_33.Adam Grabowski and Magdalena JastrzÄbska. Rough set theory from a math-assistant perspective. In Rough Sets and Intelligent Systems Paradigms, International Conference, RSEISP 2007, Warsaw, Poland, June 28ā30, 2007, Proceedings, pages 152ā161, 2007. doi:10.1007/978-3-540-73451-2_17.Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5ā10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300ā314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.Adam Grabowski and Christoph Schwarzweller. Rough Concept Analysis - theory development in the Mizar system. In Asperti, Andrea and Bancerek, Grzegorz and Trybulec, Andrzej, editor, Mathematical Knowledge Management, Third International Conference, MKM 2004, Bialowieza, Poland, September 19ā21, 2004, Proceedings, volume 3119 of Lecture Notes in Computer Science, pages 130ā144, 2004. doi:10.1007/978-3-540-27818-4_10. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19-21, 2004.Jouni JƤrvinen. Lattice theory for rough sets. Transactions of Rough Sets, VI, Lecture Notes in Computer Science, 4374:400ā498, 2007.Eliza Niewiadomska and Adam Grabowski. Introduction to formal preference spaces. Formalized Mathematics, 21(3):223ā233, 2013. doi:10.2478/forma-2013-0024.ZdzisÅaw Pawlak. Rough sets. International Journal of Parallel Programming, 11:341ā356, 1982. doi:10.1007/BF01001956.Y.Y. Yao. Two views of the theory of rough sets in finite universes. International Journal of Approximate Reasoning, 15(4):291ā317, 1996. doi:10.1016/S0888-613X(96)00071-0.William Zhu. Generalized rough sets based on relations. Information Sciences, 177: 4997ā5011, 2007.26218319
Automatic Ordinals
We prove that the injectively omega-tree-automatic ordinals are the ordinals
smaller than . Then we show that the injectively
-automatic ordinals, where is an integer, are the ordinals
smaller than . This strengthens a recent result of Schlicht
and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite
word -automatic ordinals. As a by-product we obtain that the
hierarchy of injectively -automatic structures, n>0, which was
considered in [Finkel-Todorcevic12], is strict.Comment: To appear in a Special Issue on New Worlds of Computation 2011 of the
International Journal of Unconventional Computing. arXiv admin note: text
overlap with arXiv:1111.150
Metric Semantics and Full Abstractness for Action Refinement and Probabilistic Choice
This paper provides a case-study in the field of metric semantics for probabilistic programming. Both an operational and a denotational semantics are presented for an abstract process language L_pr, which features action refinement and probabilistic choice. The two models are constructed in the setting of complete ultrametric spaces, here based on probability measures of compact support over sequences of actions. It is shown that the standard toolkit for metric semantics works well in the probabilistic context of L_pr, e.g. in establishing the correctness of the denotational semantics with respect to the operational one. In addition, it is shown how the method of proving full abstraction --as proposed recently by the authors for a nondeterministic language with action refinement-- can be adapted to deal with the probabilistic language L_pr as well
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