Depth, Highness and DNR Degrees

A sequence is Bennett deep [5] if every recursive approximation of the Kolmogorov complexity of its initial segments from above satisfies that the difference between the approximation and the actual value of the Kolmogorov complexity of the initial segments dominates every constant function. We study for different lower bounds r of this difference between approximation and actual value of the initial segment complexity, which properties the corresponding r(n)-deep sets have. We prove that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller choices of r, i.e., r is any recursive order function, we show that depth implies either highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order depth already implies highness. As a corollary, we obtain that weakly-useful sets are either high or DNR. We prove that not all deep sets are high by constructing a low order-deep set. Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition, one obtains a notion which no longer satisfies the slow growth law (which stipulates that no shallow set truth-table computes a deep set); however, under this notion, random sets are not deep (at the unbounded recursive order magnitude). We improve Bennett's result that recursive sets are shallow by proving all K-trivial sets are shallow; our result is close to optimal. For Bennett's depth, the magnitude of compression improvement has to be achieved almost everywhere on the set. Bennett observed that relaxing to infinitely often is meaningless because every recursive set is infinitely often deep. We propose an alternative infinitely often depth notion that doesn't suffer this limitation (called i.o. depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude εn, and construct a π01- class where every member is an i.o. deep set of magnitude εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep of constant magnitude, and that every nonrecursive many-one degree contains such a set

Macrophages in Alzheimer’s disease: the blood-borne identity

Alzheimer’s disease (AD) is a progressive and incurable neurodegenerative disorder clinically characterized by cognitive decline involving loss of memory, reasoning and linguistic ability. The amyloid cascade hypothesis holds that mismetabolism and aggregation of neurotoxic amyloid-β (Aβ) peptides, which are deposited as amyloid plaques, are the central etiological events in AD. Recent evidence from AD mouse models suggests that blood-borne mononuclear phagocytes are capable of infiltrating the brain and restricting β-amyloid plaques, thereby limiting disease progression. These observations raise at least three key questions: (1) what is the cell of origin for macrophages in the AD brain, (2) do blood-borne macrophages impact the pathophysiology of AD and (3) could these enigmatic cells be therapeutically targeted to curb cerebral amyloidosis and thereby slow disease progression? This review begins with a historical perspective of peripheral mononuclear phagocytes in AD, and moves on to critically consider the controversy surrounding their identity as distinct from brain-resident microglia and their potential impact on AD pathology