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Complexity for Type-2 Relations

If Σ is an alphabet, then a type-2 functional is a (partial) function whose arguments are elements of Σ* and functions from Σ* into Σ*. Type-2 relations are the domains of such functionals. We consider the natural extension of the class of polynomial time functions to include type-2 functionals, and a variant, in which the time bounds depend only on the string arguments. Using these, we define two possible extensions of the (relativized) polynomial hierarchy to include type-2 relations. We consider some topological notions that seem «naturally» associated with time and space bounded computations of oracle Turing machines, and we give topological characterizations of several classes of type-2 relations