Fundamenta Informaticae - Volume 166, issue 3

Authors: Bisht, Raj Kishor | Nishida, Taishin Yasunobu | Yamamoto, Kouhei

Article Type: Research Article

Abstract: We introduce word matrices and word matrix rewriting systems. A word matrix over an alphabet ∑ of k letters is a k × n rectangular matrix with entries of nonnegative integers. A word matrix represents a set of words. The i -th row corresponds to the i -th letter in ∑ (thus the letters in ∑ are ordered as a 1 , . . . , a k ). Each column vector represents a set of words which have the vector as the Parikh vector. The word matrix represents the concatenation of all sets represented by column …vectors. A word matrix rewriting system consists of a set of rewriting rules which rewrite word matrices and an initial word matrix (the axiom). A word matrix rewriting system generates a set of word matrices by iterated applications of rules to the axiom and generates a language which is the union of all sets represented by the matrices. A word matrix rewriting system is a kind of (restricted) parallel rewriting system with scattered context dependency and hence can generate non-context-free languages, e.g., {an bn cn | 1 ≤ n }. We define variants of word matrix rewriting systems and show an infinite hierarchy among them. Some language theoretical problems, including relations among Chomsky hierarchy, closure properties, and decision properties, are considered. Show more

Keywords: restricted parallel rewriting, Parikh vector, regular languages, context-free languages

DOI: 10.3233/FI-2019-1800

Citation: Fundamenta Informaticae, vol. 166, no. 3, pp. 199-226, 2019

Authors: Polkowski, Lech

Article Type: Research Article

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. Show more

Keywords: mereology, mass assignment, rough inclusion, the Bayes theorem, fuzzy logics, logical foundations of probability, betweenness, granularity

DOI: 10.3233/FI-2019-1801

Citation: Fundamenta Informaticae, vol. 166, no. 3, pp. 227-249, 2019

Authors: Tiwari, Surabhi | Singh, Pankaj Kumar

Article Type: Research Article

Abstract: In this paper, we have constructed topological structures on rough sets by choosing the path of proximity relations on approximation spaces. So, by this virtue of purpose, we have used rough metric to define nearness concept between rough sets. Some basic results have been proved on this new nearness structure named as rough proximity. The study is well supported by examples. Finally, the theory is developed to construct the compactification of a rough proximity space.

Keywords: Rough sets, proximity spaces, grill, ultra filter, cluster, clan

DOI: 10.3233/FI-2019-1802

Citation: Fundamenta Informaticae, vol. 166, no. 3, pp. 251-271, 2019

Authors: Wang, Zhaohao | Yue, Huifang | Deng, Jianping

Article Type: Research Article

Abstract: Uncertainty measures are an important tool for analyzing data. There is the uncertainty of a rough set caused by its boundary region in rough set models. Thus the uncertainty measurement issue is also an important topic for rough set theory. Shannon entropy has been introduced into rough set theory. However, there are relatively few studies on the uncertainty measure in generalized rough set models. We know that the boundary region of a rough set is closely related to the upper and lower approximations in rough set models. In this paper, from the viewpoint of the upper and lower approximations, we …propose new uncertainty measures, the upper rough entropy and the lower rough entropy, in generalized rough set models. Then we focus on the investigations of the upper rough entropy, and give the concepts of the upper joint entropy, the upper conditional entropy and the mutual information with respect to a general binary relation. Some important properties of these measures are obtained. The connections among these measures are given. Furthermore, comparing with the existing uncertainty measures, the upper rough entropy has high distinguishing degree. Theoretical analysis and experimental results show that the proposed entropy is better effective than some existing measures. Show more

Keywords: Generalized rough set, Uncertainty measure, Rough entropy, Granularity

DOI: 10.3233/FI-2019-1803

Citation: Fundamenta Informaticae, vol. 166, no. 3, pp. 273-296, 2019