  In the plethora of fragments of Halpern and Shoham's modal logic of time
intervals (HS), the logic AB of Allen's relations Meets and Started-by is at a
central position. Statements that may be true at certain intervals, but at no
sub-interval of them, such as accomplishments, as well as metric constraints
about the length of intervals, that force, for instance, an interval to be at
least (resp., at most, exactly) k points long, can be expressed in AB.
Moreover, over the linear order of the natural numbers N, it subsumes the
(point-based) logic LTL, as it can easily encode the next and until modalities.
Finally, it is expressive enough to capture the {\omega}-regular languages,
that is, for each {\omega}-regular expression R there exists an AB formula
{\phi} such that the language defined by R coincides with the set of models of
{\phi} over N. It has been shown that the satisfiability problem for AB over N
is EXPSPACE-complete. Here we prove that, under the homogeneity assumption, its
model checking problem is {\Delta}^p_2 = P^NP-complete (for the sake of
comparison, the model checking problem for full HS is EXPSPACE-hard, and the
only known decision procedure is nonelementary). Moreover, we show that the
modality for the Allen relation Met-by can be added to AB at no extra cost
(AA'B is P^NP-complete as well).
