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Axiomatizing Resource Bounds for Measure
Resource-bounded measure is a generalization of classical Lebesgue measure
that is useful in computational complexity. The central parameter of
resource-bounded measure is the {\it resource bound} , which is a class
of functions. When is unrestricted, i.e., contains all functions with
the specified domains and codomains, resource-bounded measure coincides with
classical Lebesgue measure. On the other hand, when contains functions
satisfying some complexity constraint, resource-bounded measure imposes
internal measure structure on a corresponding complexity class.
Most applications of resource-bounded measure use only the
"measure-zero/measure-one fragment" of the theory. For this fragment,
can be taken to be a class of type-one functions (e.g., from strings to
rationals). However, in the full theory of resource-bounded measurability and
measure, the resource bound also contains type-two functionals. To
date, both the full theory and its zero-one fragment have been developed in
terms of a list of example resource bounds chosen for their apparent utility.
This paper replaces this list-of-examples approach with a careful
investigation of the conditions that suffice for a class to be a
resource bound. Our main theorem says that every class that has the
closure properties of Mehlhorn's basic feasible functionals is a resource bound
for measure.
We also prove that the type-2 versions of the time and space hierarchies that
have been extensively used in resource-bounded measure have these closure
properties. In the course of doing this, we prove theorems establishing that
these time and space resource bounds are all robust.Comment: Changed one referenc