Verimag, 2 rue Vignate, 38610 Gi eres, France 2 Timed Systems 2.1 Background Let X b ea set of real-valued variables called clocks de ned on the set of non-negative reals R 0. Clocks will b eused as state variables measuring time progress. The set of the valuations of X isomorphic to R 0 n for some n, is denoted by V. Constant true (resp. f a l s e) denotes the predicate that is true (resp. false) for any valuation v 2 V. For any non-negative real t, we represent by v + t the valuation obtained from v by increasing by t the values of all the clocks. De nition 1. Left-and right-closure A predicate p on X is called left-closed if 8v : :p(v)) 9 > 0 : 8 0 : :p(v + 0) It is called right-closed if it satis es the previous expression where p(v + 0) is replaced by p(v ; 0). Notice that these two de nitions correspond to the usual notions if we consider p as a function of time, where v is a clock valuation. De nition 2. Rising and falling edge Given a predicate p on clocks X, we de ne the rising edge of p, noted p " by: p " (v) = p(v)^9 > 0 : 8 0 2 (0 ] : :p(v ; 0) _ :p(v)^9 > 0 : 8 0 2 (0 ] : p(v + 0) The falling edge of p, noted p #, is de ned by the same formula where v ; 0 and v + 0 are exchanged. De nition 3. Modal operators Given a predicate p on V , we de ne the modal operators 3 k p (\eventually p within k") and 3k p (\once p since k"), for k 2 R 0 f1g. 3 k p (v) i 9t 2 R 0 0 t k :p(v + t) 3k p (v) i 9t 2 R 0 0 t k :9v 0 2 V : v = v 0 + t^p(v 0) We write 3p and 3p for 3 1 p and 3-1 p, respectively, and 2p and 2p for :3:p and :3-:p, respectively. Notice that the operators 3 k and 3k are just a notation for existential quanti cation over time and should not b econfused with temporal logic operators. Expressions with modal or edge operators can b ereduced to predicates on X whenever quanti cation over time can b eeliminated e.g., when the operators are applied to linear constraints on X. For example, 3(1 x 2) is equivalent to x 2 and 3 2 (3 x 5) is equivalent to 1 x 5.
We present a method that allows to guarantee liveness by construction of a class of timed systems. The method is based on the use of a set of structural properties which can be checked locally at low cost. We provide sufficient conditions for liveness preservation by parallel composition and priority choice operators. The latter allow to restrict a system's behavior according to a given priority order on its actions. We present several examples illustrating the use of the results, in particular for the construction of live controllers.
McMillan's unfolding approach to the reachability problem in 1-safe Petri nets and its later improvements by Esparza-Römer-Vogler have proven in practice as a very effective method to avoid stateexplosion. This method computes a complete finite prefix of the infinite branching process of a net. On the other hand, the Local First Search approach (LFS) was recently introduced as a new partial order reduction technique which characterizes a restricted subset of configurations that need to be explored to check local properties. In this paper we amalgamate the two approaches: We combine the reduction criterion of LFS with the notions of an adequate order and cutoff events essential to the unfolding approach. As a result, our new LFS method computes a reduced transition system without the problem of state duplication (present in the original LFS). Since it works for any transition system with an independence relation, this black box partial unfolding remains more general than the unfolding of Petri nets. Experiments show that the combination gives improved reductions compared to the original LFS.
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