Resource Bounded Measure

Abstract

A general theory of resource-bounded measurability and measure is developed. Starting from any feasible probability measure $\nu$ on the Cantor space $\C$ and any suitable complexity class $C \subseteq \C$, the theory identifies the subsets of $\C$ that are $\nu$-measurable in $C$ and assigns measures to these sets, thereby endowing $C$ with internal measure-theoretic structure. Classes to which the theory applies include various exponential time and space complexity classes, the class of all decidable languages, and the Cantor space itself, on which the resource-bounded theory is shown to agree with the classical theory. The sets that are $\nu$-measurable in $C$ are shown to form an algebra relative to which $\nu$-measure is well-behaved. This algebra is also shown to be complete and closed under sufficiently uniform infinitary unions and intersections, and $\nu$-measure in $C$ is shown to have the appropriate additivity and monotone convergence properties with respect to such infinitary operations. A generalization of the classical Kolmogorov zero-one law is proven, showing that when $\nu$ is any feasible coin-toss probability measure on $\C$, every set that is $\nu$-measurable in $C$ and (like most complexity classes) invariant under finite alterations must have $\nu$-measure 0 or $\nu$-measure 1 in $C$. The theory is presented here is based on resource-bounded martingale splitting operators, which are type-2 functionals, each of which maps $\N \times {\cal D}_\nu$ into ${\cal D}_\nu \times {\cal D}_\nu$, where ${\cal D}_\nu$ is the set of all $\nu$-martingales. This type-2 aspect of the theory appears to be essential for general $\nu$-measure in complexity classes $C$, but the sets of $\nu$-measure 0 or 1 in C are shown to be characterized by the success conditions for martingales (type-1 functions) that have been used in resource-bounded measure to date.