This version of PDL first transforms formulae into a *-normal form defined in the associated paper. The conversion to *-normal form can result in a blow up of the given formula e.g. < * * a > p gets converted to < (a ; *a) ; * (a ; *a) > p. Since this is suboptimal, there is another prover that does not require *-normal form. The rules of that prover are currently being written up.
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.
For an introduction to PDL see http://plato.stanford.edu/entries/logic-dynamic/