  In this paper (part of the author's PhD thesis), we introduce the notions of
semistability and potential semistability of overconvergent F-crystals over an
equal characteristic local field. We establish their equivalence with the
notions of unipotency and quasi-unipotency given by Crew, and to recast the
conjecture that every overconvergent crystal is quasi-unipotent in terms of
potential semistability. Consequences, including an extension of de Jong's
extension theorem for F-crystals to the logarithmic case, will appear in a
subsequent paper.
