Introducing Mass-based Rough Mereology in a Mereological Universe with Relations to Fuzzy Logics and a Generalization of the Łukasiewicz Logical Foundations of Probability*
Affiliations: Department of Mathematics and Computer Science, Chair of Mathematical Methods in Computer Science, University of Warmia and Mazury in Olsztyn, Olsztyn, Poland. [email protected]; ORCID no.0000-0002-6990-3959
Correspondence:
[†]
Address for correspondence: 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.
Note: [*] Part of this work was presented at the CS&P workshop at Humboldt Universität zu Berlin on September 23-26, 2018 [24].
Abstract: We investigate a model for rough mereology based reasoning in which things in the universe of mereology are endowed with positive masses. We define the mass based rough inclusion and establish its properties. This model does encompass inter alia set theoretical universes of finite sets with masses as cardinalities, probability universes with masses as probabilities of possible events, sets of satisfiable formulas with values of satisfiability, measurable bounded sets in Euclidean n-spaces with n-dimensional volume as mass, in particular complete Boolean algebras of regular open or closed sets – the playground for spatial reasoning and geographic information systems1. We define a mass-based rough mereological theory (in short mRM-theory). We demonstrate affinities of the mass-based rough mereological mRM-theory with classical many-valued (‘fuzzy’) logics of Łukasiewicz, Gödel and Goguen and we generalize the theses of logical foundations of probability as given by Łukasiewicz. We give an abstract version of the Bayes theorem which does extend the classical Bayes theorem as well as the proposed by Łukasiewicz logical version of the Bayes formula. We also establish an abstract form of the betweenness relation which has proved itself important in problems of data analysis and behavioral robotics. We address as well the problem of granulation of knowledge in decision systems by pointing to the most general set of conditions a thing has to satisfy in order to be included into a formally defined granule of knowledge, the notion instrumental in our approach to data analysis. We address the problem of applications by pointing to our work on intelligent robotics in which the mass interpreted as the relative area of a planar region is basic for definition of a rough inclusion on regular open/closed regions as well as in definition of the notion of betweenness crucial for a strategy for navigating teams of robots.
Keywords: mereology, mass assignment, rough inclusion, the Bayes theorem, fuzzy logics, logical foundations of probability, betweenness, granularity
