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Introducing Mass-based Rough Mereology in a Mereological Universe with Relations to Fuzzy Logics and a Generalization of the Łukasiewicz Logical Foundations of Probability

Polkowski Lech

Fundamenta Informaticae

2019

Vol. 166, nr 3

227--249

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 systems. 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.

Betweenness, Łukasiewicz Rough Inclusions, Euclidean Representations in Information Systems, Hyper-granules and Conflict Resolution

Polkowski L. T., Nowak B.

Fundamenta Informaticae

2016

Vol. 147, nr 2/3

337--352

In this work, we approach the problem of data analysis from a new angle: we investigate a relational method of separation of data into disjoint sub–data employing a modified betweenness relation, successfully applied by us in the area of behavioral robotics, and, we set a scheme for applications to be studied. The effect of the action by that relation on data is selection of a sub–data, say, ‘kernel’ with the property that each thing in it is a convex combination, in a sense explained below, of some other things in the kernel. One can say that kernel thus exhibited is ‘self–closed’. Algorithmically, this is achieved by means of a new construct, called by us a ‘dual indiscernibility matrix’. On the other hand, the complement to kernel consists of things in the data, which have some attribute values not met in any other thing. It is proper to call this complement to kernel the residuum. We examine both the kernel and the residuum from the point of view of quality of classification into decision classes for a few standard data sets from the UC Irvine Repository finding the results very satisfactory. Conceptually, our work is set in the framework of rough set theory and rough mereology and the main tool in inducing of the betweenness relation is the Łukasiewicz rough inclusion. Apart from the classification problem, we propose some strategies for conflict resolution based on concepts introduced in this work, and in this way we continue conflict analysis in rough set framework initiated by Zdzisław Pawlak.