Completeness of Certain Bimodal Logics for Subset Spaces

Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states.In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes intersection spaces.We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set.