  We construct a theory of the derived PD-Hirsch extension of the log
crystalline complex of a log smooth scheme and we construct a fundamental
filtered dga $(H_{{\rm zar},{\rm TW}},P)$ and a fundamental filtered complex
$(H_{\rm zar},P)$ for a simple normal crossing log scheme $X$ over a family of
log points by using the log crystalline method in order to overcome obstacles
arising from the incompatibility of the p-adic Steenbrink complexes in [M] and
[Nak4] with the cup product of the log crystalline complex of $X$. When the
base log scheme is the log point of a perfect field of characteristic $p>0$, we
prove that $(H_{{\rm zar},{\rm TW}},P)$ and $(H_{\rm zar},P)$ is canonically
isomorphic to Kim and Hain's filtered dga and their filtered complex in [KH],
respectively.

Algebraic Geometry