# Bias Invariance of Small Upper Spans

## Abstract

*δ*> 0, any polynomial-time computable sequence

*= (*

**β***β*

_{0},

*β*

_{1}, ...) of biases

*β*

_{i}∈ [

*δ*, 1 −

*δ*], and any class \( \mathcal{C} \) 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)
\( \mathcal{C} \) has p-measure 0 relative to the probability measure given by

.**β** - (2)
\( \mathcal{C} \) has p-measure 0 relative to the uniform probability measure.

The analogous equivalences are established for measure in E and measure in E_{2}. ([5] established this invariance for classes \(
\mathcal{C}
\) 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.

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