On the Compactness Property of Mereological Spaces

• Autorzy: Polkowski Lech

Identyfikatory

DOI

10.3233/FI-2020-1893

• Języki publikacji: EN

Abstrakty

EN

Continuing our work on mass-based rough mereologies, we make use of the Stone representation theorem for complete Boolean algebras and we exhibit the existence of a finite base in each mereological space. Those bases in turn allow for the introduction of distributed mereologies; regarding each element of the base as a mereological space, we propose a mechanism for fusing those mereological spaces into a global distributed mereological space. We define distributed mass-assignments and rough inclusions pointing to possible applications.

Słowa kluczowe

EN

distributed mereology; mass assignment; mereology; rough inclusion; rough mereology; Stone representation; Stone space; compactness theorem

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2020
• Tom: Vol. 172, nr 1
• Strony: 73--95
• Opis fizyczny: Bibliogr. 35 poz.

Twórcy

autor

Polkowski Lech

• Department of Mathematics and Computer Science, Chair of Mathematical Methods in Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna str. 54, 10-710 Olsztyn, Poland

Bibliografia

• [1] Polkowski L. Introducing mass-based rough mereology in a mereological universe with relations to fuzzy logics and a generalization of the Łukasiewicz logical foundations of probability. Fundamenta Informaticae, 2019. 165: 1-23. ISSN:0169-2968 (P), ISSN:1875-8681 (E). doi:10.3233/FI-2019-1752.
• [2] Łukasiewicz, J. Die Logischen Grundlagen der Wahrscheinlichkeitsrechnung. Drukarnia Polska in Kraków, 1913.
• [3] Łukasiewicz J. Logical Foundations of Probability Theory. In: Jan Łukasiewicz. Selected Works. Studies in Logic and Foundations of Mathematics series. Borkowski, L. (ed.). North Holland - Polish Scientific Publishers (PWN), 1970 pp.1-63. ISBN:978-0720422528.
• [4] Stone M. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 1936. 40: 37-111.
• [5] Gleason A M. Projective topological spaces. Ill. J. Math., 1958. 2: 482-489. Print ISSN:0019-2082.
• [6] Izadi A, Stock K M, Guesgen H W. Multidimensional Region Connection Calculus. URL http://qrg.northwestern.edu/qr2017/papers/QR2017_paper_8.pdf. Accessed 2019-02-10.
• [7] Polkowski L. Approximate Reasoning with Parts. An Introduction to Rough Mereology. ISRL vol. 20. Springer-Verlag, 2011. ISBN:978-3-642-22278-8. doi:10.1007/978-3-642-22279-5.
• [8] Cohn A. G., Bennett B., Gooday J, Gotts N. RCC: A calculus for region-based qualitative spatial reasoning. Geoinformatica, 1997. 1:275-316.
• [9] Polkowski L, Ośmiałowski P. Spatial reasoning with applications to mobile robotics. In: Mobile Robots Motion Planning. New Challenges. Xi-Jing. (ed.). I-Tech, 2008 pp. 433-453. ISBN 10.5772/8987.
• [10] Polkowski L, Ośmiałowski P. Navigation for mobile autonomous robots and their formations: An application of spatial reasoning induced from rough mereological geometry. In: Mobile Robots Navigation. Barrera A. (ed.). In-Tech, 2010 pp. 39-354. Identifier-ark ark://13960/t53f6226g.
• [11] Łukasiewicz J. Aristotle’s Syllogistics from the Standpoint of Modern Formal Logic. 2nd. edition. Lejewski, C. (ed.). Oxford University Press, 1957. ISBN:0824069242. doi:978-0198241447.
• [12] Polkowski L. Formal granular calculi based on rough inclusions (A feature talk). In: IEEE/GrC Proceedings. Tsinghua University, Beijing, 2005 pp. 57-62. ISBN 0-7803-9017-2.
• [13] Leśniewski S. Foundations of the General Theory of Sets (in Polish). The Popławski Printing Shop in Moscow, 1916.
• [14] Leśniewski S. Collected Works I. Nijhoff Int. Philosophy Series 44/1. Surma S, Srzednicki J T, Barnett D I, Rickey V F. (eds.). Springer Netherlands, 1992. ISBN:978-0792315124.
• [15] Casati R, Varzi A C. Parts and Places. The Structures of Spatial Representation. The MIT Press, 1999. ISBN:978-0262517072.
• [16] Simons P. Parts: A Study in Ontology. Clarendon Press, 2000. ISBN:978-0199241460.
• [17] Pietruszczak A. Metamereology. The Nicolaus Copernicus University Scientific Publishing House 2018. ISBN:978-83-231-3975-1. doi:10.12775/3961-4.
• [18] Polkowski L. Mereology in Engineering and Computer Science. In: Mereology and the Sciences. Parts and Wholes in the Contemporary Scientific Context. Calosi C, Graziani P. (eds.). Springer Synthese Library vol. 371. Springer International Publishers, 2015 pp. 217-292. ISBN:978-3-319-05356-1. doi:10.1007/978-3-319-05356-1.
• [19] Iwanuś B. On Leśniewski’s Elementary Ontology. Studia Logica, 1973. 31(1): 7-72. Print ISSN:0039-3215. doi:10.1007/BF02120531.
• [20] Tarski A. On the foundations of Boolean Algebra. In: Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923 to 1938. Woodger J K. (ed.). Clarendon Press, 1956. doi:10.2307/2216869.
• [21] Clay R E. Relation of Lesniewski’s Mereology to Boolean algebra. J. Symbolic Logic, 1974. 39(4): 638-648. doi:10.2307/2272847.
• [22] Tarski A. Zur Grundlegen der Booleschen Algebra I. Fund. Math, 1935. 24: 177-198. ISSN:0016-2736.
• [23] Varzi A. Mereology. In: Stanford Encyclopedia of Philosophy. URL https://plato.stanford.edu/entries/mereology/. Accessed 2018-12-28.
• [24] Polkowski L, Skowron A. Rough mereology. In: ISMIS’94 Proceedings. LNCS vol. 869, 1994 pp. 85-94. ISBN 3-540-58495-1.
• [25] Polkowski L, Skowron A. Rough mereology: A new paradigm for approximate reasoning. Int. J. Approx. Reasoning, 1997. 15(4): 333-365. doi:10.1016/60888-615X(96)00072-2.
• [26] Hájek P. Metamathematics of Fuzzy Logic. Springer Netherlands, 1998. ISBN:978-94-011-5300-3. doi:0.1007/978-94-011-5300-3.
• [27] Polkowski L, Artiemjew P. Granular Computing in Decision Approximation. Springer International Publishing Switzerland, 2015. ISBN:978-3-319-12880-1. doi:10.1007/978-3-319-12880-1.
• [28] Nicolas D. The logic of mass expressions. In: Stanford Encyclopedia of Philosophy. URL https://plato.stanford.edu/entries/logic-massexpress/. Accessed 2018-12-28.
• [29] Łukasiewicz J, Tarski A. Untersuchungen uber den Aussagenkalk ul. C. R. Soc. Sci. Lettr. Varsovie, 1930. 23, pp 1-21.
• [30] Engelking R. General Topology. Helderman Verlag, 1989. ISBN: 3-88538-006-4.
• [31] Kuratowski K, Mostowski A. Set Theory. PWN (Polish Scientific Publishers), 1968.
• [32] Givant S, Halmos P. Introduction to Boolean Algebras. Springer Science+Business Media, 2009. ISBN:978-0-387-40293-2. doi:10.1007/978-0-387-68436-9.
• [33] Clarke B L. A calculus of individuals based on connection. Notre Dame Journal of Formal Logic, 1981. 22(3), pp 204-218. doi:10.1305/ndjfl/1093883455.
• [34] Dempster A. P. Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statistics, 1976. 38(2), pp 325-331. doi:10.1214/aoms/1177698950.
• [35] Shafer G. A Mathematical Theory of Evidence. Princeton University Press, 1976. ISBN:9780691100425.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).