The Effect of Bounding the Number of Primitive Propositions and the Depth of Nesting on the Complexity of Modal Logic

A well-known result of Ladner says that the satissability problem for K45, KD45, and S5 is NP-complete. This result implicitly assumes that there are innnitely many primitive propositions in the language; it is easy to see that the satissability problem for these logics becomes linear time if there are only nitely many primitive propositions in the language. By way of contrast, we show that the PSPACE-completness results of Ladner and Halpern and Moses hold for the modal logics if there is only one primitive proposition in the language. We go on to examine the eeect on complexity of bounding the depth of nesting of modal operators. If we restrict to nite nesting, then the satissability problem is NP-complete for all the modal logics considered, but S4. If we then further restrict the language to having only nitely many primitive propositions, the complexity goes down to linear time in all cases.