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Robustness and Implementability of Timed Automata Martin De Wulf Laurent Doyen Nicolas Markey Jean-François Raskin Centre Fédéré en Vérification FORMATS-FTRTFT 2004 – Sep 24 th - Grenoble

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Motivation Embedded Controllers … are difficult to develop (concurrency, real- time, continuous environment,...). … are safety critical. Use formal models + Verification: Timed Automata and Reachability Analysis

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Timed Automata closed guardresetLocation Transition Clocks: {a,b}

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Objectives From a verified model, generate (automatically) a correct implementation Using classical formalism (e.g. timed automata) …but interpreting the model in a way that guarantees the transfer of the properties from model to implementation

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Robustness and Implentation Model Perfect continuous clocks Instantaneous synchronisations Reaction time = 0 Digital clocks Delayed synchronisations Reaction time > 0 Implementation vs. Correct model Correct implementation?

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Timed Automata: Semantics Classical Perfect clocks Rate: Guard: Imprecise clocks Rate: Guard: Enlargedvs. Shortcut:

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From Model to Implementation >0 Reach([A´] ) Bad = Timed automaton A is implementable [DDR04] if Reach([A]) Bad = Timed automaton A is correct if which is equivalent to >0 Reach([A´] ) Bad =

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Robustness No ! (see example) Is it always the case that ? >0 Reach([A] ) = Reach([A]) How to compute ? >0 Reach([A] ) [Pur98] gives an algorithm for >0 Reach([A] ) We show that >0 Reach([A] ) = >0 Reach([A] )

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An example showing that Reach([A]) >0 Reach([A] )

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Example

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Classical Semantics Example

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Enlarged Semantics Example

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Example Reach([A] )

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Enlarged Semantics Example >0 Reach([A] ) When

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Enlarged SemanticsClassical Semantics vs. Example >0 Reach([A] ) Reach([A])

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Black cycles are reachable Blue cycles are not ! Classical semantics [A] Enlarged semantics One blue cycle is reachable By repeating this cycle with Δ>0, the entire regions are reachable ! Example [A]

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Cycles in Timed Automata Algortihm [Pur98] is based on this observation: It just adds the cycles to reachable states Until no more cycle is accessible Hence, the implementability problem is decidable ! (and PSPACE-complete) Given a timed automaton A, determine whether there exists Δ>0 such that Reach([A] ) Bad =

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Open questions Maximize Δ such that Decide whether there exists Δ such that Find a practical algorithm for And many others… Reach([A] ) Bad = >0 Reach([A] ) UntimedLang([A] ) = UntimedLang([A])

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References [DDR04] M. De Wulf, L. Doyen, J.-F. Raskin. Almost ASAP Semantics: From Timed Model to Timed Implementation. LNCS 2993, HSCC [Pur98] A. Puri. Dynamical Properties of Timed Automata. FTRTFT 1998.

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Model-based development Make a model of the environment: Env Make clear the control objective: Bad Make a model of the control strategy: ControllerModel Verify: Does Env || ControllerModel avoid Bad ? Good, but after ?

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