On the Priorean temporal logic with ‘around now’ over the real line

We consider the temporal language with the Priorean operators G and H expressing that a formula is true at all future times and all past times, plus an operator □ expressing that a formula is true throughout some open interval containing the evaluation time (i.e. it is true ‘around now’). We show that the logic of time based on the real numbers in this language is finitely axiomatizable, answering an implicit question of Shehtman (1993). We also show that the logic has PSPACE-complete complexity, but is not Kripke complete and has no strongly complete axiomatization.