Tamás Mihálydeák

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.

Partial first-order logic with approximative functors based on properties

In the present paper a logically exact way is presented in order to define approximative functors on object level in the partial first-order logic relying on approximation spaces. By the help of defined approximative functors one can determine what kind of approximations has to be taken into consideration in the evaluating process of a formula. The representations of concepts (properties) of our available knowledge can be used to approximate not only any concept (property) but any relation. In the last section lower and upper characteristic matrixes are introduced. These can be very useful in different applications.

Rough Clustering Generated by Correlation Clustering

Correlation clustering relies on a relation of similarity and the generated cost function. If the similarity relation is a tolerance relation, then not only one optimal partition may exist: an object can be approximated from lower and upper side with the help of clusters containing the given object and belonging to different partitions. In practical cases there is no way to take into consideration all optimal partitions. The authors give an algorithm which produces near optimal partitions and can be used in practical cases to avoid the combinatorial explosion. From the practical point of view it is very important, that the system of sets appearing as lower or upper approximations of objects can be taken as a system of base sets of general partial approximation spaces.

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.

Rough Classification Based on Correlation Clustering

In this article we propose a two-step classification method. At the first step it constructs a tolerance relation from the data, and at second step it uses correlation clustering to construct the base sets, which are used at the classification of the objects. Besides the exposition of the theoretical background we also show this method in action: we present the details of the classification of the well-known iris data set. Moreover we frame some open question due this kind of classification.

Partial first-order logic relying on optimistic, pessimistic and average partial membership functions

One of the common features of decision‐theoretic rough set models is that they rely on total background (available) knowledge in the sense that the knowledge covers the discourse universe. In the proposed framework the author gives up this requirement and allows that available knowledge about the discourse universe may be partial. It is shown by introducing optimistic, average and pessimistic partial membership functions that a decision‐theoretic rough set model can be based on a very general version of partial approximation spaces. Dierent membership functions may serve as a base of the semantics of a partial first‐order logic. The proposed logical system gives an exact possibility to introduce dierent semantic notions of logical consequence relations which can be used in order to make clear the consequences of our decisions.

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.

Correlation clustering by contraction

We suggest an effective method for solving the problem of correlation clustering. This method is based on an extension of a partial tolerance relation to clusters. We present several implementation of this method using different data structures, and we show a method to speed up the execution by a quasi-parallelism.

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.