Structural properties for feasibly computable classes of type two

This paper treates classes in the polynomial hierarchy of type two, , that were first developed by Townsend as a natural extension of the Meyer-Stockmeyer polynomial hierarchy in complexity theory. For these classes, it is discussed whether each of them has the extension property and the three recursion-theoretic properties: separation, reduction, and pre-wellordering. This paper shows that every , lacks the pre-wellordering property by using a probabilistic argument on constant-depth Boolean circuits. From the assumption NP = coNP it follows by a pruning argument that has the separation and extension properties.