Axiomatizing Resource Bounds for Measure

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 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.Keywords: basic feasible functionals, computational complexity, resource-bounded measure, type-two functionals IntroductionResource-bounded measure is a generalization of classical Lebesgue measure theory that allows us to quantify the "sizes" (measures) of interesting subsets of various complexity classes. This quantitative capability has been useful in computational complexity because it has intersected informatively with reducibilities, completeness, randomization, circuit-size, and many other central ideas of complexity theory. Resource-bounded ⋆ A preliminary version of the paper was presented in the Workshop on Logic in Computational Complexity, 2009.

Fractals in complexity and geometry

“…We use Δ to represent a function class that serves as a resource bound. (To be precise, a resource bound is a class of type-2 functional in order to have a complete theory of resourcebounded measure and measurability [40]. In here, we take the measurability for granted and only discuss measure, in particular, measure 0 and avoid type-2 computation by doing so.…”