Transactions on Rough Sets XXI

Jan Łukasiewicz was one of leading logicians of the XX-th century, universally regarded as the father of many-valued logics which proved to be the language for many paradigms of Computer Science and Artificial Intelligence, inventor of the Polish notation whose dual, the Reverse Polish notation has become implemented in computers and calculators, renowned historian of logic especially of logics of Stoics school and of Aristotle, twice the Rector of Warsaw University in academic years 1922/23 and 1931/32, Minister of Religious Beliefs and Public Enlightenment in the Paderewski cabinet in 1919, earlier in Ministry of Education in provisional Jan Kanty Steczkowski cabinet in 1918, born in Lwów at the time of autonomization of Galicia, student at the Lwów University at the time of its start toward flourishing, in Warsaw between 1915 and 1944, then an exile in Germany, Belgium and finally in Ireland, far from dear Lwów and Poland. He was one of pillars of the world famous Warsaw School of Logic alongside of Warsaw School od Mathematics, Lwów School of Mathematics, Warsaw - Lwów School of Philosophy together with Kazimierz Twardowski, Alfred Tarski, Stanisław Leśniewski, Stefan Banach, Hugo Steinhaus, Juliusz Schauder, Stanisław Mazur, Stanisław Ulam, Wacław Sierpiński, Kazimierz Kuratowski, Stefan Mazurkiewicz, Adolf Lindenbaum, Mordechaj Wajsberg, Bolesław Sobociński and many others. They worked in often difficult conditions, living through two world wars, regional conflicts, many of them lost all their possessions and archives, forced to rebuild their lives anew, often overseas, but always devoted to Poland and its causes

This article presents an approach to performing the task of visual search in the context of descriptive topological spaces. The presented algorithm forms the basis of a descriptive visual search system (DVSS) that is based on the guided search model (GSM) that is motivated by human visual search. This model, in turn, consists of the bottom-up and top-down attention models and is implemented within the DVSS in three distinct stages. First, the bottom-up activation process is used to generate saliency maps and to identify salient objects. Second, perceptual objects, defined in the context of descriptive topological spaces, are identified and associated with feature vectors obtained from a VGG deep learning convolutional neural network. Lastly, the top-down activation process makes decisions on whether the object of interest is present in a given image through the use of descriptive patterns within the context of a descriptive topological space. The presented approach is tested with images from the ImageNet ILSVRC2012 and SIMPLIcity datasets. The contribution of this article is a descriptive pattern-based visual search algorithm.

We examine double successive approximations on a set, which we denote by \(L_2L_1, \ U_2U_1, U_2L_1,\) \(L_2U_1\) where \(L_1, U_1\) and \(L_2, U_2\) are based on generally non-equivalent equivalence relations \(E_1\) and \(E_2\) respectively, on a finite non-empty set V. We consider the case of these operators being given fully defined on its powerset . Then, we investigate if we can reconstruct the equivalence relations which they may be based on. Directly related to this, is the question of whether there are unique solutions for a given defined operator and the existence of conditions which may characterise this. We find and prove these characterising conditions that equivalence relation pairs should satisfy in order to generate unique such operators.

In one perspective, the main theme of this research revolves around the inverse problem in the context of general rough sets that concerns the existence of rough basis for given approximations in a context. Granular operator spaces and variants were recently introduced by the present author as an optimal framework for anti-chain based algebraic semantics of general rough sets and the inverse problem. In the framework, various sub-types of crisp and non-crisp objects are identifiable that may be missed in more restrictive formalism. This is also because in the latter cases concepts of complementation and negation are taken for granted - while in reality they have a complicated dialectical basis. This motivates a general approach to dialectical rough sets building on previous work of the present author and figures of opposition. In this paper dialectical rough logics are invented from a semantic perspective, a concept of dialectical predicates is formalized, connection with dialetheias and glutty negation are established, parthood analyzed and studied from the viewpoint of classical and dialectical figures of opposition by the present author. The proposed method become more geometrical and encompass parthood as a primary relation (as opposed to roughly equivalent objects) for algebraic semantics.

Spatial reasoning concerns a language in which spatial objects are described and argued about. Within the plethora of approaches, we single out the one set in the framework of mereology - the theory of concepts employing the notion of a part as the primitive one. Within mereology, we can choose between the approach based on part as the basic notion or the approach based on the notion of a connection from which the notion of a part is defined. In this work, we choose the former approach modified to the rough mereology version in which the notion of a part becomes ‘fuzzified’ to the notion of a part to a degree. The prevalence of this approach lies in the fact that it does allow for quantitative assessment of relations among spatial objects in distinction to only qualitative evaluation of those relations in case of other mereology based approaches.

In this work, we introduce sections on mereology based reasoning, covering part and connection based variants as well as rough mereology in order to provide the Reader with the conceptual environment we work in. We recapitulate shortly those approaches along with based on them methods for spatial reasoning. We then introduce the mereological approach in the topological context used in spatial reasoning, i.e., in collections of regular open or regular closed sets known to form complete Boolean algebras. In this environment, we create a logic for reasoning about parts and degrees of inclusion based on an abstract notion of a mass which generalizes geometric measure of area or volume and extends in the abstract manner the Lukasiewicz logical rendering of probability calculus. We give some applications, notably, we extend the relation of betweenness applied by us earlier in robot navigation and we give it the abstract characterization.

This article presents similarity based reasoning approach for recognition of compound objects. It contains mathematical foundations for comparators theory as well as comparators network theory. It shows also three different practical applications in field of image recognition, text recognition and risk recognition.

The paper presents a new generation of Rseslib library - a collection of rough set and machine learning algorithms and data structures in Java. It provides algorithms for discretization, discernibility matrix, reducts, decision rules and for other concepts of rough set theory and other data mining methods. The third version was implemented from scratch and in contrast to its predecessor it is available as a separate open-source library with API and with modular architecture aimed at high reusability and substitutability of its components. The new version can be used within Weka and with a dedicated graphical interface. Computations in Rseslib 3 can be also distributed over a network of computers.