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...

. The theory of finite trees in finite signature \Sigma is axiomatized by the simple set of axioms E : 1. 8x x 6= t(x) for every non-variable term t(x) containing x, 2. 8x 8y ( f(x) = f(y) , x = y ) for every f 2 \Sigma , 3. 8x 8y (f(x) 6= g(y)) for different f , g 2 \Sigma , plus the usual equality axioms, plus the following Domain Closure Axiom: 8x f2\Sigma 9z ( x = f(z) ) (DCA) postulating that every element of a model is in the range of some (perhaps 0-ary) function, i.e., there are no isolated elements. The theory E [ (DCA) has numerous applications in Automated deduction, Constraint solving, Unification theory, Logic programming, Database theory. It was proved complete by Maher [Mah88] using the straightforward quantifier elimination, and also by Lescanne and Comon [CL89] by a direct transformational method. Earlier Kunen [Kun87] proved that E (without (DCA)) is complete in the case of infinite signatures with constants (this case is much more simple), again by...