Reachability analysis*

1. Reachability analysis* dr. András Pataricza dr. Tamás Bartha BUTE DMIS translated by Zoltán Micskei * Also see the Murata paper

2. Reachability analysis • Reachability problem • Dynamic properties • Representing state space • Simplifying reachability problem

3. Reachability • Analyzing the dynamic behavior • Marking: state • Firing: state transition • Firing sequence: trajectory in the state space, moving from M0 to Mn state • Mn is reachable from M0 iff there exists a firing sequence from M0 to Mn:

4. Reachability analysis In the N Petri net from the M0 starting state • Reachable states • State-based analysis • Possible firing sequences • State-transition (event) based analysis

5. Reachability problem • Reachability problem in Petri nets • Is Mn state reachable from any M0 initial state: • Sub marking reachability problem • finding *, where M’nis any marking whose restriction to a given subset of places agrees with that of a given marking Mn. *N can be omitted to shorten the formula

6. Preliminary: Decidability* • Decidable refers to the existence of an effective method • An effective method (also called a decision procedure) for a class of problems is a method for which each step in the method may be described as a mechanical operation and which, if followed rigorously, and as far as may be necessary, is bound to: • always give some answer rather than ever give no answer; • always give the right answer and never give a wrong answer; • always be completed in a finite number of steps, rather than in an infinite number; • work for all instances of problems of the class. * Wikipedia

7. Solving the reachability problem • The reachability problem is decidable • But it’s space-complexity is exponential in general • The equality problem is undecidable • There is no algorithm for determining if for any two Petri nets • But it is decidable and exponential in time for 1-bounded (safe) Petri nets

8. Dynamic properties

9. Dynamic properties • Properties connected to reachability • Dependon the initial marking • (cf. structural property: independent from the initial marking) • Dynamic properties: • Boundedness • Liveness • Deadlock-free • Reversibility • Home state • Coverability • Persistence • Fairness • Bounded-fair • Globally fair

10. Boundedness • k-bounded (bounded) • In any states reachable M0 in any places the number of tokens does not exceed k • Safe Petri net: 1-bounded • Good for resource and job processing modeling • Is it guaranteed, that the jobs will be processed

11. Liveness • Deadlock-free • In every state at least one transition can be fired • Liveness: can the transition be fired once/more than once/infinity often? • L0-live (dead) if t can never be fired in any firing sequence in L(M0). • L1-live (potentially firable) if t can be fired at least once in some firing sequence in L(M0). • L2-live if, given any positive integer k, t can be fired at least k times in some firing sequence in L(M0). • L3-live if t appears infinitely often in some firing sequence in L(M0). • L4-live or live if t is L3-Iive for every marking M in R(M0). • Simple consequence: from L4 to L1 the liveness properties imply each other

12. Liveness: example t3 t1 t2 t0

13. Liveness: example • t3 transition : L3-live t3 t1 t2 t0

14. Liveness: example • t1 transition : L1-live • t3 transition : L3-live t3 t1 t2 t0

15. Liveness: example • t1 transition : L1-live • t2 transition : L2-live • t3 transition : L3-live t3 t1 t2 t0

16. Liveness: example • t0 transition: L0-live (dead) • t1 transition : L1-live • t2 transition : L2-live • t3 transition : L3-live t3 t1 t2 t0

17. Liveness cont’d • A (P, T, Mo) Petri net is Lx-live • if all tT transition are Lx-live • A (P, T, Mo) Petri net is live • if L4-live, i.e. all tT transition is L4-live • It is deadlock-free in every state trajectory • Every transition can be fired again in the future • Deadlock-free  liveness • Proving liveness is costly in general • In special cases it is simple (e.g. invariants) • Assumes an ideal system

18. Cyclic properties • Reversibility: • The initial state can be reached from each marking • Often models cyclic behavior • Home state: • MN is a home state, if it is reachable from each M in R(M0) • Often models cyclic behavior with initialization

19. Cyclic properties cont’d • Coverability: • Will a marking be reached, which covers marking M • M’ marking covers M marking if • M’M means: • If M is the minimal marking that enables t transition, then • t is not L1-live iff M is not coverable • M is coverable →t is L1-live

20. Dynamic properties: interaction • Persistency: • APetri net (N, MO)is said to be persistent if, for any two enabled transitions, the firing of one transition will not disable the other. • An enabled transition will be enabled until its firing! • Useful for determining: • Do the parallel behaviors affect each other?

21. Dynamic properties: interaction cont’d • Fairness: • Two distinct definition of fairness • Bounded fairness (B-fairness) • Two transitions t1 and t2are said to be in a bounded-fair relation if the maximum number of times that either one can fire while the other is not firing is bounded. • Globally fair: • Afiring sequenceis said to be unconditionally (globally) fair if it is finite or every transition in the net appears infinitely often in it. • Used for determining: • Do the parallel processes hold up each other? • Will all the process be finished?

22. Representing state space

23. Reachability graph • State graph starting from the initial state M0 • Nodes: states  labeled with marking • Arcs: state transition  labeled with firing • For a given node the number of outgoing arcs are at most the number of the enabled transitions in the state represented by the node • Fewer, if the net has priorities • Node with no outgoing arcs represent deadlock • If net is not bounded  infinite number of states • Boundedness  finite state space • Breadth first search from the initial state • Depth first search is not good in a net with infinite state space

24. Coverability tree • There is no correlation between the size/complexity of a Petri net and the size of its state space • Need a representation to handle infinite state space • Coverability tree: also for infinite state space • Similar to reachability graph: nodes represent states, arc represent firings • Critical part: growing of tokens without limits in a place • Use a special symbol for representing infinity:  • If a marking covers an already visited marking in the trajectory, then let  flag the marking of the places, where the number of tokens increased.

25. Generating coverability tree Ltoprocess { M0 } While Ltoprocess Choose the nextM Ltoprocess state ifM was present in the trajectory from the root to the current state then mark M as „old” elseif there is no enabled transition in M then mark M as „dead-end”

26. Generating coverability tree cont’d else // (there is at least one enabled transition in M) for alltT: enabled(t, M) Compute the M’ next state after M: if there is an M’’ in the path from M0 to M, which is covered by M’, i.e. then M” is a covered state, in the node representing M’ replace the marking of the covered places with : else M’ is a new state:Ltoprocess Ltoprocess M’ add an arc from M to M’ labeled with t // return to the start of while loop

27. Example: PN and its coverability tree “dead-end” “old” state “old” state

28. Example: PN and its coverability graph

29. Analyzing the coverability tree Some properties can be determined from the tree: • Petri net is bounded  R(N, M0) reachability graph is finite • Petri is safe  Only 0 and 1 in the labels of the nodes in the coverability tree • A t transition is dead  There is no arc with label t in the coverability tree

30. Simplifying the reachability problem • Hierarchical modeling: • Merging subnets into one single node • Non-determinism of PNs  model abstraction • Limiting the state space in a coarse-grained model • Detailed analysis in a refined model • Compositional verification

31. Simplifying the reachability problem cont’d • Exploiting symmetries: • Analyze the similar subnets only once • Typical applications • Multi processor system, Multitasking, Internet, Client-server • Colored Petri Nets • Well-formed colored Petri nets (WFN)