Resource-bounded measure

Abstract

A general theory of resource-bounded measurability and measure is developed. Starting from any feasible probability measure /spl nu/ on the Canter space C (the set of all decision problems) and any suitable complexity class C/spl sube/C, the theory identifies the subsets of C that are /spl nu/-measurable in C and assigns measures to these sets, thereby endowing C with internal measure-theoretic structure. Classes C to which the theory applies include various exponential time and space complexity classes, the class of all decidable languages, and the Canter space C itself, on which the resource-bounded theory is shown to agree with the classical theory. The sets that are /spl nu/-measurable in C are shown to form an algebra relative to which /spl nu/-measure is well-behaved (monotone, additive, etc.). This algebra is also shown to be complete (subsets of measure 0 sets are measurable) and closed under sufficiently uniform infinitary unions and intersections, and /spl 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 /spl nu/ is any feasible coin-toss (i.e., product) probability measure on C, every set that is /spl nu/-measurable in C and (like most complexity classes) invariant under finite alterations must have /spl nu/-measure 0 or /spl nu/-measure 1 in C. The theory presented here is based on resource-bounded martingale splitting operators, which are type-2 functionals, each of which maps N/spl times/D/sub /spl nu// into D/sub /spl nu///spl times/D/sub /spl nu//, where D/sub /spl nu// is the set of all /spl nu/-martingales. This type-2 aspect of the theory appears to be essential for general /spl nu/-measure in complexity classes C, but the sets of /spl 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.