Various logical formalisms with the freeze quantifier have been recently considered to model computer systems even though this is a powerful mechanism that often leads to undecidability. In this article, we study a linear-time temporal logic with pasttime operators such that the freeze operator is only used to express that some value from an infinite set is repeated in the future or in the past. Such a restriction has been inspired by a recent work on spatio-temporal logics that suggests such a restricted use of the freeze operator. We show decidability of finitary and infinitary satisfiability by reduction into the verification of temporal properties in Petri nets by proposing a symbolic representation of models. This is a quite surprising result in view of the expressive power of the logic since the logic is closed under negation, contains future-time and past-time temporal operators and can express the nonce property and its negation. These ingredients are known to lead to undecidability with a more liberal use of the freeze quantifier. The article also contains developments about the relationships between temporal logics with the freeze operator and counter automata as well as reductions into first-order logics over data words.
We introduce an LTL-like logic with atomic formulae built over a constraint language interpreting variables in Z. The constraint language includes periodicity constraints, comparison constraints of the form x = y and x < y, it is closed under Boolean operations and it admits a restricted form of existential quantification. This is the largest set of qualitative constraints over Z known so far, shown to admit a decidable LTL extension. Such constraints are those used for instance in calendar formalisms or in abstractions of counter automata by using congruences modulo some power of two. Indeed, various programming languages perform arithmetic operators modulo some integer. We show that the satisfiability and model-checking problems (with respect to an appropriate class of constraint automata) for this logic are decidable in polynomial space improving significantly known results about its strict fragments. As a by-product, LTL model-checking over integral relational automata is proved complete for polynomial space which contrasts with the known undecidability of its CTL counterpart.
We study decidability and complexity issues for fragments of LTL with Presburger constraints by restricting the syntactic resources of the formulae (the class of constraints, the number of variables and the distance between two states for which counters can be compared) while preserving the strength of the logical operators. We provide a complete picture refining known results from the literature, in some cases pushing forward the known decidability limits. By way of example, we show that model-checking formulae from LTL with quantifier-free Presburger arithmetic over one-counter automata is only PSPACE-complete. In order to establish the PSPACE upper bound, we show that the nonemptiness problem for Büchi one-counter automata taking values in Z and allowing zero tests and sign tests, is only NLOGSPACE-complete.
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