A note on the complexity of S4.2

is the modal logic of directed partial pre-orders and/or the modal logic of reflexive and transitive relational frames with a final cluster. It holds a distinguished position in philosophical logic, where it has been advocated as the ‘correct’ logic of knowledge; it has also found interesting applications in the temporal logic of relativistic spacetime and the metamathematics of forcing in set theory. The satisfiability problem for is -complete: this is a result established in an AiML 2004 paper of Shapirovsky [(2004). On PSPACE-decidability in transitive modal logic. In R. A. Schmidt, I. Pratt-Hartmann, M. Reynolds, & H. Wansing (Eds.), Advances in modal logic 5 (pp. 269–287). King's College Publications] where the complexity classification of emerges as a consequence of a very general method for constructing decision procedures for transitive modal logics. We provide here a ‘classical’ proof in the standard Halpern-Moses style of Halpern and Moses [(1992). A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54(2), 319–379]. With little extra work, it is shown that the -completeness result extends to , the multimodal version of . We prove that the -completeness characterisation of monomodal persists even if we restrict ourselves to fragments with bounded modal depth, but the problem is -complete when it is restricted to formulae with modal depth at most one. The complexity of satisfiability for the fragment of the language with a finite number of propositional variables (but unbounded modal depth) remains -hard. For a finite language and bounded modal depth, -satisfiability can be checked in linear time.