On the Freeze Quantifier in Constraint LTL: Decidability and Complexity
Résumé
Constraint LTL, a generalization 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, etc.). We show that Constraint LTL over the simple domain (N,=) augmented with the freeze operator is undecidable which is a surprising result regarding 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 quite sharp (\emph{i.e.}~with restrictions on the number of variables) and all our complexity characterizations insure completeness with respect to some complexity class (mainly PSPACE and EXPSPACE)
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