  Motivated by description logics, we investigate what happens to the
complexity of modal satisfiability problems if we only allow formulas built
from literals, $\wedge$, $\Diamond$, and $\Box$. Previously, the only known
result was that the complexity of the satisfiability problem for K dropped from
PSPACE-complete to coNP-complete (Schmidt-Schauss and Smolka, 1991 and Donini
et al., 1992). In this paper we show that not all modal logics behave like K.
In particular, we show that the complexity of the satisfiability problem with
respect to frames in which each world has at least one successor drops from
PSPACE-complete to P, but that in contrast the satisfiability problem with
respect to the class of frames in which each world has at most two successors
remains PSPACE-complete. As a corollary of the latter result, we also solve the
open problem from Donini et al.'s complexity classification of description
logics (Donini et al., 1997). In the last section, we classify the complexity
of the satisfiability problem for K for all other restrictions on the set of
operators.
