James Worrell

On the decidability of metric temporal logic

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.

A behavioural pseudometric for probabilistic transition systems

Discrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper, we present a pseudometric on a class of probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is defined via the terminal coalgebra of a functor based on a metric on the space of Borel probability measures on a metric space. States of a probabilistic transition system have distance 0 if and only if they are probabilistic bisimilar. We also characterize our distance function in terms of a real-valued modal logic.

Some Recent Results in Metric Temporal Logic

Metric Temporal Logic ( MTL ) is a widely-studied real-time extension of Linear Temporal Logic. In this paper we survey results about the complexity of the satisfiability and model checking problems for fragments of MTL with respect to different semantic models. We show that these fragments have widely differing complexities: from polynomial space to non-primitive recursive and even undecidable. However we show that the most commonly occurring real-time properties, such as invariance and bounded response, can be expressed in fragments of MTL for which model checking, if not satisfiability, can be decided in polynomial or exponential space.

On the language inclusion problem for timed automata: closing a decidability gap

We consider the language inclusion problem for timed automata: given two timed automata A and B, are all the timed traces accepted by B also accepted by A? While this problem is known to be undecidable, we show here that it becomes decidable if A is restricted to having at most one clock. This is somewhat surprising, since it is well-known that there exist timed automata with a single clock that cannot be complemented. The crux of our proof consists in reducing the language inclusion problem to a reachability question on an infinite graph; we then construct a suitable well-quasi-order on the nodes of this graph, which ensures the termination of our search algorithm. We also show that the language inclusion problem is decidable if the only constant appearing among the clock constraints of A is zero. Moreover, these two cases are essentially the only decidable instances of language inclusion, in terms of restricting the various resources of timed automata.

Towards Quantitative Verification of Probabilistic Transition Systems

It has been argued that Boolean-valued logics and associated discrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper we present a pseudometric on a class of reactive probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is defined via the terminal coalgebra of a functor based on the Hutchinson metric on the space of Borel probability measures on a metric space. We also characterize the distance between systems in terms of a real-valued modal logic.

An Algorithm for Quantitative Verification of Probabilistic Transition Systems

In an earlier paper we presented a pseudometric on the class of reactive probabilistic transition systems, yielding a quantitative notion of behavioural equivalence. The pseudometric is defined via the terminal coalgebra of a functor based on the Hutchinson metric on probability measures. In the present paper we give an algorithm, based on linear programming, to calculate the distance between two states up to prescribed degree of accuracy.

On the decidability and complexity of Metric Temporal Logic over finite words

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.

On the final sequence of a finitary set functor

A standard construction of the final coalgebra of an endofunctor involves defining a chain of iterates, starting at the final object of the underlying category and successively applying the functor. In this paper we show that, for a finitary set functor, this construction always yields a final coalgebra in ω2 = ω + ω steps.

Approximating and computing behavioural distances in probabilistic transition systems

In an earlier paper we presented a pseudometric on the states of a probabilistic transition system, yielding a quantitative notion of behavioural equivalence. The behavioural pseudometric was defined via the terminal coalgebra of a functor based on a metric on Borel probability measures. In the present paper we give a polynomial-time algorithm, based on linear programming, to calculate the distances between states up to a prescribed degree of accuracy.

Revisiting digitization, robustness, and decidability for timed automata

We consider several questions related to the use of digitization techniques for timed automata. These very successful techniques reduce dense-time language inclusion problems to discrete time, but are applicable only when the implementation is closed under digitization and the specification is closed under inverse digitization. We show that, for timed automata, the former (whether the implementation is closed under digitization) is decidable, but not the latter. We also investigate digitization questions in connection with the robust semantics for timed automata. The robust modeling approach introduces a timing fuzziness through the semantic removal of equality testing. Since its introduction half a decade ago, research into the robust semantics has suggested that it yields roughly the same theory as the standard semantics. This paper shows that, surprisingly, this is not the case: the robust semantics is significantly less tractable, and differs from the standard semantics in many key respects. In particular, the robust semantics yields an undecidable (nonregular) discrete-time theory, in stark contrast with the standard semantics. This makes it virtually impossible to apply digitization techniques together with the robust semantics. On the positive side, we show that the robust languages of timed automata remain recursive.