  Constraint linear-time temporal logic (CLTL) is an extension of LTL that is
interpreted on sequences of valuations of variables over an infinite domain.
The atomic formulas are interpreted as constraints on the valuations. The
atomic formulas can constrain valuations over a range of positions along a
sequence, with the range being bounded by a parameter depending on the formula.
The satisfiability and model checking problems for CLTL have been studied by
Demri and D'Souza. We consider the realizability problem for CLTL. The set of
variables is partitioned into two parts, with each part controlled by a player.
Players take turns to choose valuations for their variables, generating a
sequence of valuations. The winning condition is specified by a CLTL formula --
the first player wins if the sequence of valuations satisfies the specified
formula. We study the decidability of checking whether the first player has a
winning strategy in the realizability game for a given CLTL formula. We prove
that it is decidable in the case where the domain satisfies the completion
property, a property introduced by Balbiani and Condotta in the context of
satisfiability. We prove that it is undecidable over $(\mathbb{Z},<,=)$, the
domain of integers with order and equality. We prove that over
$(\mathbb{Z},<,=)$, it is decidable if the atomic constraints in the formula
can only constrain the current valuations of variables belonging to the second
player, but there are no such restrictions for the variables belonging to the
first player. We call this single-sided games.
