Logics with Transitive Accessibility Relations

This chapter is about the model construction problem in classes of models satisfying the constraint of transitivity. We present the modal logics of the class of models where the accessibility relation is transitive (K4), transitive and serial (KD4), and transitive and reflexive (KT4, alias S4). For these logics, the model construction procedure may loop, which contrasts with the simple logics of Chap. 4. Termination can be ensured by means of blocking techniques: basically, the construction is stopped when the labels of a node are identical to those of some ancestor node. In the end of the chapter we present another general termination theorem guaranteeing that the tableau construction does not loop and which applies to all these logics.