Axiomatizing Resource Bounds for Measure

TL;DR

This paper develops a rigorous framework for resource bounds in resource-bounded measure, replacing example-based approaches with a set of closure properties that characterize resource bounds, including those used in time and space hierarchies.

Contribution

It provides a characterization of resource bounds for measure via closure properties, unifying various classes under a common theoretical framework.

Findings

Every class with Mehlhorn's closure properties is a resource bound.

Type-2 time and space hierarchies have these closure properties.

The resource bounds used in measure are robust and well-founded.

Abstract

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} $\Delta$, which is a class of functions. When $\Delta$ 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 $\Delta$ 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, $\Delta$ 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 $\Delta$ 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 $\Delta$ to be a resource bound. Our main theorem says that every class $\Delta$ 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.