Bias Invariance of Small Upper Spans

Abstract

The resource-bounded measures of certain classes of languages are shown to be invariant under certain changes in the underlying probability measure. Specifically, for any real number δ > 0, any polynomial-time computable sequence β = (β0, β1, ...) of biases βi Ɛ [δ, 1 - δ], and any class C of languages that is closed upwards or downwards under positive, polynomial-time truth-table reductions with linear bounds on number and length of queries, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 relative to the probability measure given by β. (2) C has p-measure 0 relative to the uniform probability measure. The analogous equivalences are established for measure in E and measure in E2. (Breutzmann and Lutz [5] established this invariance for classes C that are closed downwards under slightly more powerful reductions, but nothing was known about invariance for classes that are closed upwards.) The proof introduces two new techniques, namely, the contraction of a martingale for one probability measure to a martingale for an induced probability measure, and a new, improved positive bias reduction of one bias sequence to another. Consequences for the BPP versus E problem and small span theorems are derived.