The system F , the well-known second-order polymorphic typed -calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type systems with subtyping by weakening F [CP94, KS92], and also by reinforcing or extending it [Vor94a, Vor94b, Vor95]. However, for the moment, these extensions lack the important proof-theoretic minimum type property, which holds for F and guarantees that each typable term has the minimum type, being a subtype of any other type of the term in the same context [CG92, Vor94c]. As a preparation step to introducing the extensions of F with the minimum type property and the decidable term typing relation (which we do in [Vor94e]), we define and study here the hierarchies of decidable extensions of the F subtyping relation. We demonstrate conditions providing that each theory in a hierarchy: 1. ext...