Petri Nets and Model Checking. Natasa Gkolfi. University of Oslo. March 31, 2017

Transcription

1 University of Oslo March 31, 2017

2 Petri Nets Petri Nets : mathematically founded formalism concurrency synchronization modeling distributed systems

3 Petri Nets Petri Nets : mathematically founded formalism concurrency synchronization modeling distributed systems Invented by C.A.Petri

4 Petri Nets Petri Nets : mathematically founded formalism concurrency synchronization modeling distributed systems Invented by C.A.Petri They are consisting of: places transitions arcs tokens initial marking p 1 t 1 p t 2

5 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

6 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

7 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

8 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

9 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

10 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

11 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

12 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

13 Petri Nets - Mutual Exclusion p 1 p 3 t 1 s t 3 p 2 p 4 t 2 t 4

14 Colored Petri nets High-level Petri nets The extension of Petri nets (called place/transition nets) with abstract data types. Colored Petri nets COLORSET (TYPE) Guard EXPR COLORS(TYPE VALUES)

15 eat, the philosopher needs two chopsticks, and he is only allowed to oexample: which aredining situatedphilosophers next to him (on his left and right side). The chopsticks prevents two neighbours from eating at the same time. cs1 ph1 cs2 ph5 cs5 Rice Dish ph2 cs3 ph4 cs4 ph3 opher system is modelled by the CP-net shown below. The PH represents the philosophers, while the CS colour set represents the

16 colour set represents the philosophers, while the CS colour set represents the chopsticks. The function Chopsticks maps each philosopher into the two chopsticks next to him. Example: Dining Philosophers PH.all() Think PH p val n = 5; color PH = index ph with 1..n; color CS = index cs with 1..n; var p: PH; fun Chopsticks(ph(i)) = 1`cs(i)++1`cs(if i=n then 1 else i+1); Take Chopstick Chopsticks(p) p p Eat p PH CS.all() Unused Chopsticks CS Put Down Chopstick Chopsticks(p)

17 State Space State Space A directed graph having a node for each reachable marking and an arc for each occurring binding element.

18 State Space State Space A directed graph having a node for each reachable marking and an arc for each occurring binding element. There is one to one correspondence between the paths in the state space and the occurrence sequences (where all steps consisting of a single binding element)

19 State Space State Space A directed graph having a node for each reachable marking and an arc for each occurring binding element. There is one to one correspondence between the paths in the state space and the occurrence sequences (where all steps consisting of a single binding element) The strongly-connected-component graph (SCC graph) is the graph derived from the state space where each node is a SCC of the state space. SCC graph is an acyclic graph fewer nodes than the ss mean that there exist infinite occurrence sequences more efficient since often much smaller than the ss

20 marking of the destination node. To improve readability, we have only shown the contents of some of the markings and some of the binding elements. It should Example: be noted thatdining all arcs are double Philosophers arcs (i.e., representsstate two individual Space arcs). Unused: 1`cs(3) Think: 1`ph(2)+ 1`ph(3)+ 1`ph(5) Eat: 1`ph(1)+ 1`ph(4) 10 2:2 Take: {p=ph(1)} Take: {p=ph(4)} 2 3:3 7 2:2 3 3:3 1 5:5 6 3:3 8 2:2 Put: {p=ph(2)} 4 3:3 5 3:3 Put: {p=ph(3)} 9 2:2 11 2:2 Unused: 1`cs(1) Think: 1`ph(1)+ 1`ph(3)+ 1`ph(5) Eat: 1`ph(2)+ 1`ph(4) Unused: 1`cs(5) Think: 1`ph(2)+ 1`ph(4)+ 1`ph(5) Eat: 1`ph(1)+ 1`ph(3) The standard report looks as shown below. To improve the readability of the

21 Behavioral Properties Boundedness properties How many and which tokes a place may hold when all reachable markings are considered. Home Properties A home marking is a marking that can be reached from any reachable marking All the markings in a (single) terminal SCC are home markings

22 Behavioral Properties Liveness Properties A dead marking is a marking in which no binding elements are enabled. Similarly dead transition. A transition is live if, starting from any reachable marking, we can always find an occurrence sequence containing it.

23 Behavioral Properties Liveness Properties A dead marking is a marking in which no binding elements are enabled. Similarly dead transition. A transition is live if, starting from any reachable marking, we can always find an occurrence sequence containing it. Fairness Properties How often transitions occur in infinite occurrence sequences. A transition is impartial if it occurs infinitely often in all infinite occurrence sequences. Removal of this transition implies no infinite occurrence sequences!

24 false) Example: Dining Philosophers ophers system the O-graph grows relatively slow when er of philosophers: PH Nodes Arcs ,364 11,310

25 State Space Reduction Methods Sweep-Line method A progress measure is a function that maps each marking into a progress value. For a given marking, the progress value of any successor marking must be greater or equal to its progress value.

26 State Space Reduction Methods Sweep-Line method A progress measure is a function that maps each marking into a progress value. For a given marking, the progress value of any successor marking must be greater or equal to its progress value. Symmetry method Equivalence classes used for symmetric markings and symmetric binding elements. the ss can be significantly reduced can check all behavioral properties that are invariant under symmetry computing canonical representations of markings and binding elements is computationally expensive

27 A generalization of the symmetry method. Here, no requirement that the equivalence relations should be induced by symmetries. State Space Reduction Methods Sweep-Line method A progress measure is a function that maps each marking into a progress value. For a given marking, the progress value of any successor marking must be greater or equal to its progress value. Symmetry method Equivalence classes used for symmetric markings and symmetric binding elements. the ss can be significantly reduced can check all behavioral properties that are invariant under symmetry computing canonical representations of markings and binding elements is computationally expensive Equivalence method

28 Thank you!