  We investigate the complexity of the model checking problem for
intuitionistic and modal propositional logics over transitive Kripke models.
More specific, we consider intuitionistic logic IPC, basic propositional logic
BPL, formal propositional logic FPL, and Jankov's logic KC. We show that the
model checking problem is P-complete for the implicational fragments of all
these intuitionistic logics. For BPL and FPL we reach P-hardness even on the
implicational fragment with only one variable. The same hardness results are
obtained for the strictly implicational fragments of their modal companions.
Moreover, we investigate whether formulas with less variables and additional
connectives make model checking easier. Whereas for variable free formulas
outside of the implicational fragment, FPL model checking is shown to be in
LOGCFL, the problem remains P-complete for BPL.
