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The paper presents a generalization of the classical rough set theory, called the partial approximative set theory (PAST). According to Pawlak's 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 σ-algebra generated by an equivalence relation on U . There are two most natural ways of the generalization of this idea. In particular, the equivalence relation is replaced by either any other type of binary relations on U or an arbitrary covering of U . In this paper, our starting point will be a partial covering of an arbitrary universe. In general, the family of sets neither covers the universe nor forms a σ-algebra. We will put our discussions into an overall treatment called the general set theoretic approximation framework. We will investigate under what conditions our generalized upper and lower approximation pair forms Galois connection.

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Abstract. 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.

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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.