This chapter shows the subsumption computation techniques for a LOGIDATA+ schema allowing cyclic definitions for classes. The formal framework LOGIDATA_CYC*, which extends LOGIDATA* to perform taxonomic reasoning in the presence of cyclic class definitions is introduced. It includes the notions of possible instances of a schema; legal instance of a schema, defined as the greatest fixed-point of possible instances; subsumption relation. On the basis of this framework, the definitions of coherent type and consistent class are introduced and the necessary algorithms to detect incoherence and compute subsumption in a LOGIDATA+ schema are given. Some examples of subsumption computation show its feasibility for schema design and validation.