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In this paper, we focus our attention on the interval temporal logic of the
Allen's relations "meets","begins", and "begun by" (ABBbar for short), interpreted over
natural numbers. We first introduce the logic and we show that it is expressive enough to
model distinctive interval properties, such as accomplishment conditions, to capture basic
modalities of point-based temporal logic, such as the until operator, and to encode relevant
metric constraints. Then, we prove that the satisfiability problem for ABBbar over natural
numbers is decidable by providing a small model theorem based on an original contraction
method. Finally, we prove the EXPSPACE-completeness of the problem
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