Temporal reasoning plays an important role in artificial intelligence. Temporal logics provide a natural framework for its formalization and implementation. A standard way of enhancing the expressive power of temporal logics is to replace their unidimensional domain by a multidimensional one. In particular, such a dimensional increase can be exploited to obtain spatial counterparts of temporal logics. Unfortunately, it often involves a blow up in complexity, possibly losing decidability. In this paper, we propose a spatial generalization of the decidable metric interval temporal logic RPNL+INT, called Directional Area Calculus (DAC). DAC features two modalities, that respectively capture (possibly empty) rectangles to the north and to the east of the current one, and metric operators, to constrain the size of the current rectangle. We prove the decidability of the satisfiability problem for DAC, when interpreted over frames built on natural numbers, and we analyze its complexity. In addition, we consider a weakened version of DAC, called WDAC, which is expressive enough to capture meaningful qualitative and quantitative spatial properties and computationally better.
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