Suppose we are given a computably enumerable object arise from algorithmic randomness or computable analysis. We are interested in the strength of oracles which can compute an object that approximates this c.e. object. It turns out that, depending on the type of object, the resulting highness property is either close to, or equivalent to being PA-complete. We examine, for example, dominating a c.e. martingale by an oracle-computable martingale, computing compressions functions for two variants of Kolmogorov complexity, and computing subtrees of positive measure of a given Π 0 1 tree of positive measure without dead ends. We prove a separation result from PAcompleteness for the latter property, called the continuous covering property. We also separate the corresponding principles in reverse mathematics. Contents 1. Introduction 1 2. Properties that imply PA degree 8 3. A K-compression function without PA degree 17 4. The discrete covering property 20 5. The (strong) continuous covering property 22 6. (Weak) strong weak weak Kőnig's lemma 28 References 29

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The authors discuss the role of expert witnesses in the context of conventional understandings of trials at common law. Controversies surrounding the expert's role turn primarily on whether the expert should educate fact finders, as lay witnesses are required to do, or instead should provide conclusions to which the fact finder simply defers. The authors observe that the likelihood of irrational verdicts increases the more fact finders defer to experts and that experts become advocates often enough to make deference fertile ground for abuse. They conclude that the expert who educates better serves the justice system.

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's Fall 2009 IP Workshop Series offered insightful, challenging comments that helped me improve the piece. Lydia Loren and Jan Neuman highlighted weak spots in an early version that helped me focus my efforts. Bill Funk, Tom Merrill, and Jim Speta helped with administrative law, and Ed Brunet andJuliet Stumpf helped with civil procedure. Connie Trela gave a welcome critical read.