Hierarchical modelling in searching for complex patterns: constrained sums of information systems

One of the main tasks in granular computing is to develop calculi of information granules (Zadeh 1997, Skowron and Stepaniuk 2001). Information systems used in rough set theory are particular kinds of information granules. In the paper, we introduce and study operations on such information granules basic for reasoning in distributed systems of information granules. The operations are called constrained sums. They are developed by interpreting infomorphisms between classifications (Barwise and Seligman 1997). In Skowron et al. (2003), we have shown that classifications (Barwise and Seligman 1997) and information systems (Pawlak 1991) are, in a sense, equivalent. We also extend the results included in Skowron et al. (2003) on applications of approximation spaces to study properties of infomorphisms. Operations, called constrained sums seem to be very important in searching for patterns in data mining (e.g. in spatio-temporal reasoning) (Kloesgen and Zytkow 2002) or in a more general sense in generating relevant granules for approximate reasoning using calculi on information granules (Polkowski and Skowron 1999, Skowron et al. 2003). This paper is organized as follows. In section 2, we present basic concepts. In section 3, we introduce sums of information systems and approximation spaces. In section 4, we discuss constrained sums of information systems and hierarchical