KD extends basic normal modal logic K with the axiom

D = []p -> <>p which forces the binary Kripke relation to be serial or total.

That is, for every world w, there is a world v such that w R v holds.

KD is characterised by finite rooted trees where only the leaves have to be reflexive

since all internal nodes have an R-successor.

When you enter a formula A, then "closed" means that A is a

theorem because the tableau for ~ A is closed, and "open" means

that it is not a theorem because the tableau for ~ A is open.