Zoltán Ernő Csajbók

Partial approximative set theory: A generalization of the rough set theory

There are close links between mathematical morphology and rough set theory. Both theories are successfully applied among others to image processing and pattern recognition. This paper presents a new generalization of the classical rough set theory, called the partial approximative set theory (PAST). According to Pawlaks classic rough set theory, the vagueness of a subset of a finite universe is defined by the difference of its upper and lower approximations with respect to an equivalence relation on the universe. There are two most natural ways of the generalization of this idea. Namely, the equivalence relation is replaced by either any other type of binary relations on the universe or an arbitrary covering of the universe. In this paper, our starting point will be an arbitrary family of subsets of an arbitrary universe, neither that it covers the universe nor that the universe is finite will be assumed. We will give some reasons why this new approach is worth studying, and put our discussions into an overall treatment, called the general approximation framework.

A General Set Theoretic Approximation Framework

To approximate sets a number of theories have been appeared for the last decades. Starting up from some general theoretical pre-conditions we give a set of minimum requirements against as the lower and upper approximations. We provide a characterization of them within the proposed general set theoretic approximation framework finding out their compound nature.

On Definability and Approximations in Partial Approximation Spaces

In this paper, we discuss the relationship occurring among the basic blocks of rough set theory: approximations, definable sets and exact sets. This is done in a very general framework, named Basic Approximation Space that generalizes and encompasses previous known definitions of Approximation Spaces. In this framework, the lower and upper approximation as well as the boundary and exterior region are independent from each other. Further, definable sets do not coincide with exact sets, the former being defined “a priori” and the latter only “a posteriori” on the basis of the approximations. The consequences of this approach in the particular case of partial partitions are developed and a discussion is started in the case of partial coverings.

Membranes with boundaries

Active cell components involved in real biological processes have to be close enough to a membrane in order to be able to pass through it. Rough set theory gives a plausible opportunity to model boundary zones around cell-like formations. However, this theory works within conventional set theory, and so to apply its ideas to membrane computing, first, we have worked out an adequate approximation framework for multisets. Next, we propose a two---component structure consisting of a P system and an approximation space for multisets. Using the approximation technique, we specify the closeness around membranes, even from inside and outside, via boundaries in the sense of multiset approximations. Then, we define communication rules within the P system in such a way that they operate in the boundary zones solely. The two components mutually cooperate.

From Vagueness to Rough Sets in Partial Approximation Spaces

Vagueness has a central role in the motivation basis of rough set theory. Expressing vagueness, after Frege, Pawlak’s information-based proposal was the boundary regions of sets. In rough set theory, Pawlak represented boundaries by the differences of upper and lower approximations and defined exactness and roughness of sets via these differences. However, defining exactness/roughness of sets have some possibilities in general. In this paper, categories of vagueness, i.e., different kinds of rough sets, are identified in partial approximation spaces. Their formal definitions and intuitive meanings are given under sensible restrictions.

Simultaneous Anomaly and Misuse Intrusion Detections Based on Partial Approximative Set Theory

Nowadays, it is already a banality that people run their applications in a complex open computing environment including allsorts of interconnected devices. In order to meet the network security challenge in nonprofessional human environments, Intrusion Detection Systems (IDS) have to be designed. Intrusion detection techniques are categorized into anomaly and misuse detection. To describe the outlined problem, we focus solely on externally observable executions generated by the observed system. Thus, we need some sort of tool being able to discover acceptable and unacceptable patterns in execution traces. Such a tool may be the rough set theory. According to the rough set theory, the vagueness of a subset of a finite universe U is defined by the difference of its upper and lower approximations with respect to a partition of U. In this paper, our starting point will be an arbitrary family of subsets of an arbitrary U, neither that it covers U nor that U is finite will be assumed. This new approach is called the partial approximative set theory. We will apply this theory to build an IDS which is simultaneously able to detect anomaly and misuse intrusions.

A security model for personal information security management based on partial approximative set theory

Nowadays, computer users especially run their applications in a complex open computing environment which permanently changes in the running time. To describe the behavior of such systems, we focus solely on externally observable execution traces generated by the observed computing system. In these extreme circumstances the pattern of sequences of primitive actions (execution traces) which is observed by an external observer cannot be designed and/or forecast in advance. We have also taken into account in our framework that security policies are partial-natured. To manage the outlined problem we need tools which are approximately able to discover secure or insecure patterns in execution traces based on presupposes of computer users. Rough set theory may be such a tool. According to it, the vagueness of a subset of a finite universe U is defined by the difference of its lower and upper approximations with respect to a partition of the universe U. Using partitions, however, is a very strict requirement. In this paper, our starting point will be an arbitrary family of subsets of U. Neither that this family of sets covers the universe nor that the universe is finite will be assumed. This new approach is called the partial approximative set theory. We will apply it to build up a new security model for distributed software systems solely focusing on their externally observable executions and to find out whether the observed system is secure or not.

General tool-based approximation framework based on partial approximation of sets

Let us assume that we observe a class of objects and have some well-defined features with which an observed object possesses or not. In real life, two relevant groups of objects can be established determined by our current and necessarily constrained knowledge. In particular, a group whose elements really possess a feature in question, and another group whose elements substantially do not possess the same feature. In practice, as a rule, we can observe a feature of objects via only tools with which we are able to judge easily whether an object possesses a property or not. Of course, a property ascertained by tools does not coincide with a feature completely. To manage this problem, we propose a general tool-based approximation framework based on partial approximation of sets in which a positive feature and its negative one of any proportion of the observed objects can simultaneously be approximated.

An Adequate Representation of Medical Data Based on Partial Set Approximation

Computer aided medical diagnosis and treatment require an adequate representation of uncertain or imperfect medical data. There are many approaches dealing with such type of data. Pawlak proposed a new method called rough set theory. In this paper, beyond classical and recent methods, the authors propose a basically new approach. It relies on a generalization of rough set theory, namely, the partial covering of the universe of objects. It adequately reflects the partial nature of real–life problems. This new approach called the partial approximation of sets is presented as well as its medical informatics application is demonstrated.

General set approximation and its logical applications

To approximate sets a number of theories have appeared for the last decades. Starting up from some general theoretical pre-conditions the authors give a set of minimum requirements for the lower and upper approximations and define general partial approximation spaces. Then, these spaces are applied in logical investigations. The main question is what happens in the semantics of the first-order logic when the approximations of sets as semantic values of predicate parameters are used instead of sets as their total interpretations. On the basis of defined partial interpretations, logical laws relying on the defined general set-theoretical framework of set approximation are investigated.