In our contribution, we study the effects of adding past operators to interval temporal logics. We focus our attention on the representative case of Propositional Neighborhood Logic (AAbar for short), taking into consideration different temporal domains. AAbar is the proper fragment of Halpern and Shoham's modal logic of intervals with modalities for Allen's relations meets (future modality) and met by (past modality). We first prove that, unlike what happens with point-based linear temporal logic, AAbar is strictly more expressive than its future fragment A. Then, we show that there is a log-space reduction from the satisfiability problem for AAbar over Z to its satisfiability problem over N. Compared to the corresponding reduction for point-based linear temporal logic, the one for AAbar turns out to be much more involved. Finally, we prove that AAbar is able to separate Q and R.
