In this paper we present a new notion of what it
means for a problem in $P$ to be inherently sequential. Informally, a
problem $L$ is {\em strictly sequential P-complete\/} if when the best
known {\em sequential\/} algorithm for $L$ has polynomial speedup by
parallelization, this implies that {\em all\/} problems in $P$ have a
polynomial speedup in the parallel setting. The motivation for
defining this class of problems is to try and capture the problems in
$P$ that are truly inherently sequential. Our work extends the
results of Condon who exhibited problems such that if a polynomial
speedup of their best known {\em parallel\/} algorithms could be
achieved, then all problems in $P$ would have polynomial speedup. We
demonstrate one such natural problem, namely the {\em Multiplex-select
Circuit Problem\/} (MCP). MCP has one of the highest degrees of
sequentiality of any problem yet defined. On the way to proving MCP
is strictly sequential $P$-complete, we define an interesting model,
the {\em register stack machine}, that appears to be of independent
interest for exploring pure sequentiality.