Modeling rough granular computing based on approximation spaces

The results reported in this paper create a step toward the rough set-based foundations of data mining and machine learning. The approach is based on calculi of approximation spaces. In this paper, we present the summarization and extension of our results obtained since 2003 when we started investigations on foundations of approximation of partially defined concepts (see, e.g., [2], [3], [7], [37], [20], [21], [5], [42], [39], [38] and [40]). We discuss some important issues for modeling granular computations aimed at inducing compound granules relevant for solving problems such as approximation of complex concepts or selecting relevant actions (plans) for reaching target goals. The problems discussed in this article are crucial for building computer systems that assist researchers in scientific discoveries in many areas such as biology. In this paper, we present foundations for modeling of granular computations inside of system that is based on granules called approximation spaces. Our approach is based on the rough set approach introduced by Pawlak [24] and [25]. Approximation spaces are fundamental granules used in searching for relevant complex granules called as data models, e.g., approximations of complex concepts, functions or relations. In particular, we discuss some issues that are related to generalizations of the approximation space introduced in [33] and [34]. We present examples of rough set-based strategies for the extension of approximation spaces from samples of objects onto a whole universe of objects. This makes it possible to present foundations for inducing data models such as approximations of concepts or classifications analogous to the approaches for inducing different types of classifiers known in machine learning and data mining. Searching for relevant approximation spaces and data models are formulated as complex optimization problems. The proposed interactive, granular computing systems should be equipped with efficient heuristics that support searching for (semi-)optimal granules.