An Algebraic Framework for Urgency

Abstract

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.

The Coarsest Congruence for Timed Automata with Deadlines Contained in Bisimulation

“…Contrary to the traditional timed automata setting, bisimulation in the TAD model is not preserved by parallel composition [6]. This is illustrated in the following example.…”

“…Timed automata with deadlines (TAD for short) [24,7,5,6] were introduced to simplify the compositionality problem in timed systems, allowing also the modelling of soft real time systems. At the same time, the TAD model ensures, under reasonable assumption, what is called time reactivity in [6] and time lock freedom in [8], that is, whenever time progress stops there exists at least one transition enabled.…”

“…At the same time, the TAD model ensures, under reasonable assumption, what is called time reactivity in [6] and time lock freedom in [8], that is, whenever time progress stops there exists at least one transition enabled. This model is nowadays embedded in modelling languages such as IF [10] and MoDeST [15], and urgent transitions in Uppaal [4] can be seen as a particular instance of TAD transitions.…”

“…Notice that T 2 shows a similar behaviour since action c cannot be executed: the deadline of a obliges its execution before the guard of c becomes enabled. In fact, T 1 and T 2 are timed bisimilar in the sense of [6]. Suppose now that T 1 is composed in parallel with the automaton stop requiring synchronisation on action a.…”

“…The only exception is what is called strong congruence in [6], which is the usual bisimulation applied directly on TADs. This relation is, however, far too strong as it requires the syntactic equality of guards, deadlines, and clock resets.…”