Rough Sets and Intelligent Systems - Professor Zdzisław Pawlak in Memoriam

This chapter is dedicated to the memory of Professor Zdzisław Pawlak, founder of the Polish school of Artificial Intelligence and one of the pioneers in Computer Engineering and Computer Science with worldwide influence.

To capture the spirit of Professor Pawlak’s creative genius, this chapter contains testimonies of many collaborators, colleagues and friends pointing to Professor’s scientific achievements and his personal qualities. In short, we present Professor Pawlak as a truly great scientist, teacher and human being.

In the development of rough set theory and applications, one can distinguish three main stages. At the beginning, the researchers concentrated on descriptive properties such as reducts of information systems preserving indiscernibility relations or description of concepts or classifications. Next, they moved to applications of rough sets in machine learning, pattern recognition and data mining. After gaining some experiences, they developed foundations for inductive reasoning leading to, for example, inducing classifiers. While the first period was based on the assumption that objects are perceived by means of partial information represented by attributes, the second period was based on the assumption that information about the approximated concepts is partial too. Approximation spaces and searching strategies for relevant approximation spaces were recognized as the basic tools for rough sets. Important achievements both in theory and applications were obtained using Boolean reasoning and approximate Boolean reasoning applied, for example, in searching for relevant features, discretization, symbolic value grouping, or, in more general sense, in searching for relevant approximation spaces. Nowadays, we observe that a new period is emerging in which two new important topics are investigated: (i) strategies for discovering relevant (complex) contexts of analysed objects or granules, what is strongly related to information granulation process and granular computing, and (ii) interactive computations on granules. Both directions are aiming at developing tools for approximation of complex vague concepts, such as behavioural patterns or adaptive strategies, making it possible to achieve the satisfactory qualities of realized interactive computations. This chapter presents this development from rudiments of rough sets to challenges, for example, related to ontology approximation, process mining, context inducing or Perception-Based Computing (PBC). The approach is based on Interactive Rough-Granular Computing (IRGC).

We discuss work of Zdzisław Pawlak in the area of databases and the extension of that work to the theory of rough sets. In particular, we look at his motivations for introducing information storage and retrieval systems and how this, eventually, led to rough sets theory.

We present a rough set data analysis software jMAF. It employs java Rough Set (jRS) library in which are implemented data analysis methods provided by the (variable consistency) Dominance-based Rough Set Approach (DRSA). The chapter also provides some basics of the DRSA and of its variable consistency extension.

This chapter is devoted to the study of an extension of dynamic programming approach that allows sequential optimization of exact decision rules relative to the length and coverage. It contains also results of experiments with decision tables from UCI Machine Learning Repository.

The aim of this work is to describe the generic properties of a visual system without hard coding the environment.

We explore an extension of rough set theory based on probability theory. Lower and upper approximations, the basic ideas of rough set theory, are generalized by adding two parameters, denoted by alpha and beta. In our experiments, for different pairs of alpha and beta, we induced three types of rules: positive, boundary, and possible. The quality of these rules was evaluated using ten-fold cross-validation on five data sets. The main results of our experiments are that there is no significant difference in quality between positive and possible rules and that boundary rules are the worst.

We introduce neighborhoods of samples to granulate the universe and use the neighborhood granules to approximate classification, thus they derived a model of neighborhood rough sets. Some machine learning algorithms, including boundary sample selection, feature selection and rule extraction, were developed based on the model.

Ill-known sets are subsets whose members are not known exactly. They can be represented by a family of subsets that can be true. When each subset is assigned a possible degree, the ill-known set is called a graded ill-known set. In this chapter, we focus on manipulations of graded ill-known sets, a possibility distribution on the power set. Two fuzzy sets on the universe called lower and upper approximations are uniquely defined from a graded ill-known set. On the contrary, a graded ill-known set is not uniquely determined by given lower and upper approximations but the maximal one is. Under a certain condition, we explicitly represent the maximal graded ill-known set having given lower and upper approximations. To utilize graded ill-known sets in decision and information sciences, possibility and necessity measures of graded ill-known sets are described. Simple computation formulae of possibility and necessity measures of graded ill-known sets are shown when lower and upper approximations are given.

The problem of approximating an arbitrary relation by a relation with desired properties is formally defined and analysed. Two special cases, approximation by partial orders and approximation by equivalence relations are discussed in detail.

An interesting and important problem of how similar (and/or dissimilar) are the voting procedures (social choice functions) is dealt with. First, we extend our previous qualitative-type analysis based on rough sets theory which makes it possible to partition the set of voting procedures considered into some subsets within which the voting procedures are indistinguishable, that is, (very) similar. Then, we propose an extension of those analyses towards a quantitative evaluation of a degree of similarity. We use some known measures of similarity and dissimilarity for binary patterns (strings) exemplified by the one based on the Hamming distance and that due to Jaccard-Needham. The use of both the measures makes it possible to obtain a quantitative evaluation of how similar the particular voting procedures are. This quantitative evaluation, when combined with a more qualitative evaluation obtained by using rough sets, provide a comprehensive view of similarity.

We present algebraic formalisms for different kinds of information systems, viz. deterministic, incomplete, and non-deterministic. Algebraic structures generated from these information systems are considered and corresponding abstract algebras are proposed. Representation theorems for these classes of abstract algebras are proved, which lead us to equational logics for deterministic, incomplete, and non-deterministic information systems.

Rough set methods are often employed for reducting decision rules. The specific techniques involving rough sets can be carried out in a computational manner. However, they are quite demanding when it comes computing overhead. In particular, it becomes problematic to compute all minimal length decision rules while dealing with a large number of decision rules. This results in an NP-hard problem. To address this computational challenge, in this study, we propose a method of DNA rough-set computing composed of computational DNA molecular techniques used for decision rule reducts. This method can be effectively employed to alleviate the computational complexity of the problem.

In the chapter we present a tool for reasoning about covering-based rough sets in the form of three-valued logic in which the value t corresponds to the positive region of a set, the value f — to the negative region and the undefined value u — to the boundary of a given set. Atomic formulas of the logic represent either membership of objects of the universe in rough sets or the subordination relation between objects generated by the covering underlying the approximation space, and complex formulas are built out of the atomic ones using three-valued Kleene connectives. We give a strongly sound sequent calculus for the logic defined in this way and prove its strong completeness for a subset of its language.

Support vector techniques were proposed by Vapnik as an alternative to neural networks for solving non-linear problems. The concepts of margins in support vector techniques provides a natural relationship with the rough set theory. This chapter describes rough set theoretic extensions of support vector technologies for classification, prediction, and clustering. The theoretical formulations of rough support vector machines, rough support vector regression, and rough support vector clustering are supported with a summary of experimental results.

The current chapter is devoted to roughification. In the most general setting, we intend the term roughification to refer to methods/techniques of constructing equivalence/similarity relations adequate for Pawlak-like approximations. Such techniques are fundamental in rough set theory. We propose and investigate novel roughification techniques. We show that using the proposed techniques one can often discern objects indiscernible by original similarity relations, what results in improving approximations. We also discuss applications of the proposed techniques in granulating relational databases and concept learning. The last application is particularly interesting, as it shows an approach to concept learning which is more general than approaches based solely on information and decision systems.

This chapter commemorates the work of Zdzisław Pawlak as a painter with the focus on the subtleties that come to light in considering the symmetries in his paintings. Specifically, this chapter considers how merotopic distance functions can be used as an aid to visual perception in determining the nearness of Zdzisław Pawlak’s paintings. Eventually, the study of the resemblance of perceptual fragments found in nature (e.g., collections of falling snow flakes) in the poem How Near? by Z. Pawlak and J.F. Peters in 2002 led to the discovery of descriptively near sets by J.F. Peters in 2007 and a merotopological approach to measuring the nearness of collections of subsets recently introduced by J.F. Peters, S.A. Naimpally and S. Tiwari. The main contribution of this chapter is the introduction of an approach to measuring the nearness or apartness of Z. Pawlak’s paintings in terms of the merotopic distances between collections of neighbourhoods in digital picture regions-of-interest. This study includes a consideration of ε-approach nearness spaces as frameworks in the search for patterns in digital pictures. An application of the proposed approach to measuring visual image nearness is reported relative to resemblances between Z. Pawlak’s paintings of waterscapes that span more than a half century, starting in 1954. This study offers a partial answer to the question How near are Zdzisław Pawlak’s paintings?

This chapter presents a granular concept hierarchy (GCH) construction and mapping of the hierarchy for granular knowledge. A GCH is comprised of multilevel granular concepts with their hierarchy relations. A rough set based approach is proposed to induce the approximation of a domain concept hierarchy of an information system. A sequence of attribute subsets is selected to partition a granularity, hierarchically. In each level of granulation, reducts and core are applied to retain the specific concepts of a granule whereas common attributes are applied to exclude the common knowledge and generate a more general concept. A granule description language and granule measurements are proposed to enable mapping for an appropriate granular concept that represents sufficient knowledge so solve problem at hand. Applications of GCH are demonstrated through learning of higher order decision rules.

This chapter discusses a correspondence between the core ideas of rough sets and medical differential diagnosis. Classically, a disease is defined as a set of symptoms, each of which gives the degree of confidence and coverage for the diagnosis. Diagnostic procedure mainly consists of the following three procedures: First, focusing mechanism (characterization) selects the candidates of differential diagnosis by using a set of symptoms. Secondly, additional set of symptoms make a differential diagnosis among the selected candidates. Finally, complications of other disease will be considered by symptoms which cannot be explained by the final candidates. This chapter mainly focuses on the first and second process and shows that these processes correspond to rules extracted by upper and lower approximation of supporting set of a given disease.

In the chapter we present rough set theory against the background of recent philosophical discussions about vagueness and empirical sciences. Weiner, in her article about this topic, discusses the supervaluationist semantics of vague predicates and its criticism offered by Fodor and Lepore. She argues that neither the former nor latter approach is consistent with the scientific methodology of dealing with vague concepts such as “obese”. In actual fact, it is Frege’s philosophical approach that concepts must have sharp boundaries, which is the closest to scientific practice. In this context, rough set theory can be viewed as a modified supervaluationist semantics. To be more precise, rough sets provide a modal version of this semantics, where the super-truth is replaced by a local one. However, there are flies in the ointment: firstly, rough set theory is philosophically weaker than supervaluationism (in consequence, more vulnerable to the criticism of Fodor and Lepore); secondly, Weiner’s arguments concerning scientific methods apply to rough sets as well. Yet there is also good news: this philosophical weakness stays actually in full accordance with scientific practice. Thus, rough set theory may be seen as a supervaluationism shifted toward the scientific methodology. In the chapter we shall make a further step into this direction and also present how rough set theory would be like when made fully consistent with the scientific approach to vague predicates. In other words, we also offer a Fregean rough set methodology.