Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL--which includes invariance and time-bounded response properties--is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature.
Existing decidability results for MTL involve placing restrictions on the semantics or the syntax of the logic to circumvent the problem of punctuality. Alur and Henzinger @cite_12 showed that the satisfiability and model-checking problems for MTL relative to a discrete-time semantics are EXPSPACE-complete. Alur, Feder, and Henzinger @cite_15 @cite_0 introduced as a fragment of MTL in which the temporal operators may only be constrained by intervals. They showed that the satisfiability and model-checking problems for MITL relative to a dense-time semantics are also EXPSPACE-complete. Wilke @cite_9 considered MTL over a dense-time semantics with , i.e., the semantics is parameterised by a bound @math on the number of events per unit time interval. He showed that the satisfiability problem is decidable in this semantics and that MTL with existential quantification over propositions is equally expressive as Alur-Dill timed automata.
@cite_0 @cite_9 @cite_15 @cite_12
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Metric temporal logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL-which includes invariance and time-bounded response properties-is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature. The question of the decidability of MTL over infinite words remains open. In this paper, we consider the robust interpretation of Metric Temporal Logic (MTL) formulas over signals that take values in metric spaces. For such signals, which are generated by systems whose states are equipped with non-trivial metrics, for example continuous or hybrid, robustness is not only natural, but also a critical measure of system performance. Thus, we propose multi-valued semantics for MTL formulas, which capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance, @e, from unsatisfiability. We prove that any other signal that remains @e-close to the initial one also satisfies the same MTL specification under the usual Boolean semantics. Finally, our framework is applied to the problem of testing formulas of two fragments of MTL, namely Metric Interval Temporal Logic (MITL) and closed Metric Temporal Logic (clMTL), over continuous-time signals using only discrete-time analysis. The motivating idea behind our approach is that if the continuous-time signal fulfills certain conditions and the discrete-time signal robustly satisfies the temporal logic specification, then the corresponding continuous-time signal should also satisfy the same temporal logic specification. Over the last years there has been an increasing research effort directed towards the automatic verification of infinite state systems, such as timed automata, hybrid automata, data-independent systems, relational automata, Petri nets, lossy channel systems, context-free and push-down processes. We present a method for deciding reachability properties of networks of timed processes. Such a network consists of an arbitrary set of identical timed automata, each with a single real-valued clock. Using a standard reduction from safety properties to reachability properties, we can use our algorithm to decide general safety properties of timed networks. To our knowledge, this is the first decidability result concerning verification of systems that are infinite-state in "two dimensions": they contain an arbitrary set of (identical) processes, and they use infinite data-structures, viz real-valued clocks. We illustrate our method by showing how it can be used to automatically verify Fischer's protocol, a timer-based protocol for enforcing mutual exclusion among an arbitrary number of processes.Finally, we show undecidability of the recurrent state problem: given a state in a timed network, check whether there is a computation of the network visiting the state infinitely often. This implies undecidability of model checking for any temporal logic which is sufficiently expressive to encode the recurrent state problem, such as PTL, CTL, etc. In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.