Hybrid Metric Propositional Neighborhood Logics with Interval Length Binders

We investigate the question of how much hybrid machinery can be added to the interval neighbourhood logic PNL and its metric extension MPNL without losing the decidability of their satisfiability problem in N. In particular, we consider the natural hybrid extension of MPNL obtained by adding binders on integer variables ranging over lengths of intervals, thus enabling storage of the length of the current interval and further references to it. We show that even a very weak natural fragment of such extensions becomes undecidable, which is somewhat surprising, being in contrast with the decidability of MPNL, which can be seen as a hybrid language with length constraints only involving constants over interval lengths. These results show that MPNL itself is, in this sense, a maximal decidable (weakly) hybrid extension of PNL.