  Constraint LTL, a generalisation of LTL over Presburger constraints, is often
used as a formal language to specify the behavior of operational models with
constraints. The freeze quantifier can be part of the language, as in some
real-time logics, but this variable-binding mechanism is quite general and
ubiquitous in many logical languages (first-order temporal logics, hybrid
logics, logics for sequence diagrams, navigation logics, logics with
lambda-abstraction etc.). We show that Constraint LTL over the simple domain
(N,=) augmented with the freeze quantifier is undecidable which is a surprising
result in view of the poor language for constraints (only equality tests). Many
versions of freeze-free Constraint LTL are decidable over domains with
qualitative predicates and our undecidability result actually establishes
Sigma_1^1-completeness. On the positive side, we provide complexity results
when the domain is finite (EXPSPACE-completeness) or when the formulae are flat
in a sense introduced in the paper. Our undecidability results are sharp (i.e.
with restrictions on the number of variables) and all our complexity
characterisations ensure completeness with respect to some complexity class
(mainly PSPACE and EXPSPACE).
