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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