Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic

There has been a great of work on characterizing the complexity of the satisfiability and validity problem for modal logics. In particular, Ladner showed that the validity problem for all logics between K, T, and S4 is {\sl PSPACE}-complete, while for S5 it is {\sl NP}-complete. We show that, in a precise sense, it is \emph{negative introspection}, the axiom $\neg K p \rimp K \neg K p$, that causes the gap. In a precise sense, if we require this axiom, then the satisfiability problem is {\sl NP}-complete; without it, it is {\sl PSPACE}-complete.