Timed CTL Model Checking in Real-Time Maude⋆ - IfI

More documents

Recommendations

Info

$G:\MISQ\2006\September 2006\Gregor.wpd - IfI$

30 D. Lepri, E. Ábrahám, and P. Cs. Ölveczky Our implementation of the TCTL model checker is based on the explicit-state CTL model checking approach [8] that, starting with the atomic propositions, recursively computes for each subformula of the desired TCTL formula the set of satisfying reachable states. We implemented specific procedures for a basic set of temporal modal operators and we expressed other formulas into this canonical form. The basic set consists of the CTL modal operators E ϕ1 U ϕ2, E G ϕ, the TCTL≤≥ 10 modal operators E ϕ1 U∼r ϕ2 with ∼∈ {>, ≥}, E ϕ1 U∼r ϕ2 with ∼∈ {0 ϕ2 and the TCTLcb modal operator E ϕ1 U [a,b] ϕ2. The procedures for CTL modalities follow the standard explicit algorithm [8]. For TCTL≤≥ modalities, our implementation adapts the TCTL≤≥ model checking procedure defined in [19] for time-interval structures and to timed Kripke structures with time-diverging paths. We briefly explain the model checking procedure for ϕ = E ϕ1 U [a,b] ϕ2. We first compute the satisfacton set for the CTL formula ˆϕ = E ϕ1 U ϕ2, then we restrict the transition relation (which we here denote by tr) in the timed Kripke structure, to those transitions between states satisfying ˆϕ except transitions from ¬ϕ1-states, obtaining ¯tr. We then recursively compute all the time distances from any state to some ϕ2 state up to b time units. This is computed by the operator computeDistances. The first two arguments of this operator are set of pairs of the kind < r, s > with r a time value and s a state (represented by a natural number in the current implementation), where each pair means that it is possible to reach some ϕ2-state from state s in time r ≤ b. The first set of such pairs is the set of time distances that still have to be “visited”, that is the predecessors of s still have to be visited with the given time stamp r. The second of such sets contains time distances that have already been visited. The third argument is the time bound b and the fourth argument is the restricted transition relation ¯tr. The operator is initially called with computeDistances({< 0, ϕ2-state >},emptyRatNatPairSet,b, ¯tr), since each ϕ2-state can obviously be reached in zero time. vars RNPS0 RNPS1 RNPS RNPS’ : RatNatPairSet . vars TIME0 TIME : Rat . var TRTKS : TransRelTKS . var N0 : Nat . op computeDistances : RatNatPairSet RatNatPairSet Rat TransRelTKS -> RatNatPairSet . ceq computeDistances(( < TIME0, N0 > RNPS), RNPS’, TIME, TRTKS) = computeDistances((RNPS1 RNPS), (< TIME0, N0 > RNPS’), TIME, TRTKS) if RNPS0 := allTimedPredecessors(TRTKS,N0) /\ RNPS1 := addTimeAndFilter(RNPS0, TIME0, TIME) . eq computeDistances(emptyRatNatPairSet, RNPS’, TIME, TRTKS) = RNPS’ . 10 We denote by TCTL≤≥ the restricted TCTL logic with time constraints on the temporal modalities of the form ∼ r, where ∼∈ {},
Timed CTL Model Checking in Real-Time Maude 31 The operator allTimedPredecessors computes the set of < r, s > pairs where s is a predecessor of N0 that can reach N0 in r time in the given transition relation, for each pair of time distances that still have to be visited. Then addTimeAndFilter selects all pairs such that TIME0 + r ≤ b, which are then recursively added to the set of pairs to be “visited”, while < TIME0, N0 > is moved to the “visited pairs”. The computation terminates by returning all the visited pairs when all possible pairs have been “visited”. The satisfaction set of ϕ consists of all states that can reach a ϕ2-state in a time within the interval [a, b]. Notice that this procedure works also for open bounded intervals, and indeed our implementation covers also open bounded intervals, however, the model checked logic is closed under negation only for close bounded intervals. In particular, for modalities with close bounded intervals, we transform each formula of the kind A ϕ1 U [a,b] ϕ2, with a = 0, into its equivalent one A G
Page 1 and 2: Timed CTL Model Checking in Real-Ti
Page 15 and 16: π : πi : π ′ i : πj : π ′
Page 23 and 24: π : πi : π ′ i : πj : π ′
Page 29: Timed CTL Model Checking in Real-Ti

Timed CTL Model Checking in Real-Time Maude⋆ - IfI

Create successful ePaper yourself

Delete template?

Save as template?