  We study the relations between the notions of highness, lowness and logical
depth in the setting of complexity theory. We introduce a new notion of polylog
depth based on time bounded Kolmogorov complexity. We show polylog depth
satisfies all basic logical depth properties, namely sets in P are not polylog
deep, sets with (time bounded)-Kolmogorov complexity greater than polylog are
not polylog deep, and only polylog deep sets can polynomially Turing compute a
polylog deep set. We prove that if NP does not have p-measure zero, then NP
contains polylog deep sets. We show that every high set for E contains a
polylog deep set in its polynomial Turing degree, and that there exist
low(E,EXP) polylog deep sets. Keywords: algorithmic information theory;
Kolmogorov complexity; Bennett logical depth.
