New incremental clustering framework based on induction as inverted deduction

A new incremental clustering framework is presented, the basis of which is the induction as inverted deduction. Induction is inherently risky because it is not truth-preserving. If the clustering is considered as an induction process, the key to build a valid clustering is to minimize the risk of clustering. From the viewpoint of modal logic, the clustering can be described as Kripke frames and Kripke models which are reflexive and symmetric. Based on the theory of modal logic, its properties can be described by system B in syntax. Thus, the risk of clustering can be calculated by the deduction relation of system B and proximity induction theorem described. Since the new proposed framework imposes no additional restrictive conditions of clustering algorithm, it is therefore a universal framework. An incremental clustering algorithm can be easily constructed by this framework from any given nonincremental clustering algorithm. The experiments show that the lower the a priori risk is, the more effective this framework is. It can be demonstrated that this framework is generally valid.