Automaten und Logiken zur Beschreibung zeitabhängiger Systeme

When speaking of a 'real-time system' we are interested in a system's evolution in time where time is viewed as linear and measured in terms of non-negative real numbers. The thesis deals with automata-theoretic models of real-time systems and their description in monadic second-order and temporal logics. A parametrized automaton model is introduced and for this model a logical description in terms of a family of existential monadic second-order logics is obtained. This characterization is used to give a logical description of the behaviour of well-known models of real-time systems: timed automata (Alur & Dill), timed automata with halting feature, and linear hybrid automata. The corresponding logics incorporate distance, duration, and integration formulas, respectively. For instance, timed automata are captured by the {\em monadic logic of relative distance.} Its signature contains for every relation symbol ~ such as =, , $=$, or and every natural number k a binary predicate d(.,.)~k taking a set of natural numbers and a single natural number as arguments. The atomic formula d(X,y)~k is true in a timed state sequence if X contains a position smaller than y and the distance (in time) between position y and the last position before y that belongs to X satisfies the condition ~k. The monadic logic of relative distance turns out to have two important properties. First, its satisfiability problem is decidable, for its equivalence to timed automata allows a reduction of the satisfiability problem to the emptiness problem for such automata and this, in turn, is decidable due to Alur and Dill. Second, the monadic logic of relative distance is a powerful logic. One evidence for this is given by showing that the logic is strictly more expressive than the most powerful logic (for the specification of real-time systems) previously known to be decidable, namely the logic MITL^P introduced by Alur and Henzinger. By effectively embedding the latter logic in the former an alternative proof of Alur's and Henzinger's decidability result concerning MITL^P is obtained. Using embedding techniques also the decidability of Manna's and Pnueli's logic TL_Gamma is proved. Timed automata and the languages recognised by them, the so-called timed regular languages, are analysed in more detail. Several aspects are considered. A pumping lemma for timed automata is given, resulting in a formal proof that timed regular languages are not closed under complementation. It is shown that the number of clocks used in timed automata gives rise to an infinite hierarchy of timed regular languages, that the minimal number of clocks required for the recognition of a timed regular language is not computable, and that the property of a two-way timed automaton (Alur & Henzinger) to be reversal bounded is undecidable. Furthermore, unambiguous timed automata are considered, and an inherently ambiguous language is presented. Finally, variations of the emptiness problem for the three types of automata aforementioned and different restrictions concerning the event duration (bounded variation, minimal duration, and unit duration) are discussed. In particular, it is shown that bounded variation leads to a decidable emptiness problem in the case of timed automata, which implies that the full monadic logic of distance is decidable when restricted to timed state sequences of bounded variation. The obtained undecidability results give evidence that the monadic logic of relative distance is a good choice with respect to expressiveness and the requirement of a decidable satisfiability problem

Revisiting timed logics with automata modalities

© 2019 ACM. It is well known that (timed) ω-regular properties such as 'p holds at every even position' and 'p occurs at least three times within the next 10 time units' cannot be expressed in Metric Interval Temporal Logic (MITL) and Event Clock Logic (ECL). A standard remedy to this deficiency is to extend these with modalities defined in terms of automata. In this paper, we show that the logics EMITL0, ∞ (adding non-deterministic finite automata modalities into the fragment of MITL with only lower- and upper-bound constraints) and EECL (adding automata modalities into ECL) are already as expressive as EMITL (full MITL with automata modalities). In particular, the satisfiability and model-checking problems for EMITL0, ∞ and EECL are PSPACE-complete, whereas the same problems for EMITL are EXPSPACE-complete. We also provide a simple translation from EMITL0, ∞ to diagonal-free timed automata, which enables practical satisfiability and model checking based on off-the-shelf tools

Why Liveness for Timed Automata Is Hard, and What We Can Do About It

The liveness problem for timed automata asks if a given automaton has a run passing infinitely often through an accepting state. We show that unless P=NP, the liveness problem is more difficult than the reachability problem; more precisely, we exhibit a family of automata for which solving the reachability problem with the standard algorithm is in P but solving the liveness problem is NP-hard. This leads us to revisit the algorithmics for the liveness problem. We propose a notion of a witness for the fact that a timed automaton violates a liveness property. We give an algorithm for computing such a witness and compare it with the existing solutions

Relating timed and register automata

Timed automata and register automata are well-known models of computation over timed and data words respectively. The former has clocks that allow to test the lapse of time between two events, whilst the latter includes registers that can store data values for later comparison. Although these two models behave in appearance differently, several decision problems have the same (un)decidability and complexity results for both models. As a prominent example, emptiness is decidable for alternating automata with one clock or register, both with non-primitive recursive complexity. This is not by chance. This work confirms that there is indeed a tight relationship between the two models. We show that a run of a timed automaton can be simulated by a register automaton, and conversely that a run of a register automaton can be simulated by a timed automaton. Our results allow to transfer complexity and decidability results back and forth between these two kinds of models. We justify the usefulness of these reductions by obtaining new results on register automata.Comment: In Proceedings EXPRESS'10, arXiv:1011.601

A Unifying Approach to Decide Relations for Timed Automata and their Game Characterization

In this paper we present a unifying approach for deciding various bisimulations, simulation equivalences and preorders between two timed automata states. We propose a zone based method for deciding these relations in which we eliminate an explicit product construction of the region graphs or the zone graphs as in the classical methods. Our method is also generic and can be used to decide several timed relations. We also present a game characterization for these timed relations and show that the game hierarchy reflects the hierarchy of the timed relations. One can obtain an infinite game hierarchy and thus the game characterization further indicates the possibility of defining new timed relations which have not been studied yet. The game characterization also helps us to come up with a formula which encodes the separation between two states that are not timed bisimilar. Such distinguishing formulae can also be generated for many relations other than timed bisimilarity.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690

MTL-Model Checking of One-Clock Parametric Timed Automata is Undecidable

Parametric timed automata extend timed automata (Alur and Dill, 1991) in that they allow the specification of parametric bounds on the clock values. Since their introduction in 1993 by Alur, Henzinger, and Vardi, it is known that the emptiness problem for parametric timed automata with one clock is decidable, whereas it is undecidable if the automaton uses three or more parametric clocks. The problem is open for parametric timed automata with two parametric clocks. Metric temporal logic, MTL for short, is a widely used specification language for real-time systems. MTL-model checking of timed automata is decidable, no matter how many clocks are used in the timed automaton. In this paper, we prove that MTL-model checking for parametric timed automata is undecidable, even if the automaton uses only one clock and one parameter and is deterministic.Comment: In Proceedings SynCoP 2014, arXiv:1403.784

Zenoness for Timed Pushdown Automata

Timed pushdown automata are pushdown automata extended with a finite set of real-valued clocks. Additionaly, each symbol in the stack is equipped with a value representing its age. The enabledness of a transition may depend on the values of the clocks and the age of the topmost symbol. Therefore, dense-timed pushdown automata subsume both pushdown automata and timed automata. We have previously shown that the reachability problem for this model is decidable. In this paper, we study the zenoness problem and show that it is EXPTIME-complete.Comment: In Proceedings INFINITY 2013, arXiv:1402.661

Verification for Timed Automata extended with Unbounded Discrete Data Structures

We study decidability of verification problems for timed automata extended with unbounded discrete data structures. More detailed, we extend timed automata with a pushdown stack. In this way, we obtain a strong model that may for instance be used to model real-time programs with procedure calls. It is long known that the reachability problem for this model is decidable. The goal of this paper is to identify subclasses of timed pushdown automata for which the language inclusion problem and related problems are decidable

Timed Automata Semantics for Analyzing Creol

We give a real-time semantics for the concurrent, object-oriented modeling language Creol, by mapping Creol processes to a network of timed automata. We can use our semantics to verify real time properties of Creol objects, in particular to see whether processes can be scheduled correctly and meet their end-to-end deadlines. Real-time Creol can be useful for analyzing, for instance, abstract models of multi-core embedded systems. We show how analysis can be done in Uppaal.Comment: In Proceedings FOCLASA 2010, arXiv:1007.499