State Space Analysis: Properties, Reachability Graph, and Coverability graph. prof.dr.ir. Wil van der Aalst

Transcription

1 State Space Analysis: Properties, Reachability Graph, and Coverability graph prof.dr.ir. Wil van der Aalst

2 Outline Motivation Formalization Basic properties Reachability graph Coverability graph PAGE 1

3 Motivation

4 Design-time analysis vs run-time analysis validation world business processes people services components organizations models analyzes supports/ controls specifies configures implements analyzes (software) system records events, e.g., messages, transactions, etc. e.g., systems like WebSphere, Oracle, TIBCO/ Staffware, SAP, FLOWer, etc. verification discovery performance analysis (process) model conformance extension event logs design-time analysis e.g. process models represented in BPMN, BPEL, EPCs, Petri nets, UML AD, etc. or other types of models such as social networks, organizational networks, decision trees, etc. run-time analysis e.g., dedicated formats such as IBM s Common Event Infrastructure (CEI) and MXML or proprietary formats stored in flat files or database tables. PAGE 3

5 Analysis of processes X linear algebraic analysis techniques Y Z Petri net Markov chain analysis techniques state-space analysis techniques... PAGE 4

6 Generic questions terminating it has only finite occurrence sequences deadlock-free each reachable marking enables a transition live each reachable marking enables an occurrence sequence containing all transitions bounded each place has an upper bound that holds for all reachable markings 1-safe 1 is a bound for each place s reversible m0 is reachable from each reachable marking, i.e., the initial marking is a so-called home marking. PAGE 5

7 Example p4 t3 p3 p5 terminating it has only finite occurrence sequences deadlock-free each reachable marking enables a transition live each reachable marking enables an occurrence sequence containing all transitions bounded each place has an upper bound that holds for all reachable markings 1-safe 1 is a bound for each place s reversible m0 is reachable from each reachable marking, i.e., the initial marking is a so-called home marking. PAGE 6

8 Specific questions p4 t3 p3 p5 Is it possible to have a token in both and p5? Will t3 always take place? Will t3 always take place assuming "fairness"? Is it possible to execute after? Can both p4 and p5 be empty at the same time? PAGE 7

9 infinite state space state explosion problem PAGE 8

10 Concepts marked net (1,0) (1,1) (1,2)... reachability graph (1,0) (1,ω) coverability graph PAGE 9

11 Relevant material 1. Jörg Desel, Wolfgang Reisig: Place/Transition Petri Nets. Petri Nets 1996: DOI: / _ Tadao Murata, Petri Nets: Properties, Analysis and Applications, Proceedings of the IEEE. 77(4): , April, Wil van der Aalst: Process Mining: Discovery, Conformance and Enhancement of Business Processes, Springer Verlag 2011 (chapters 1 & 5) a) Chapter 1: DOI: / _1 b) Chapter 5: DOI: / _5 c) Events logs: Today's focus is on 1 & 2. PAGE 10

12 Formalization Note: refinement of earlier link between Petri net and transitions system (week 2/3) that is closer to standard literature.

13 Basic Petri net P = {,} T = {,} F = {(,), (,), (,), (,), (,)} PAGE 12

14 Place transition net P = {,} T = {,} F = {(,), (,), (,), (,)} W(,)=2, W(,)=2, W(,)=1, and W(,)=1 PAGE 13

15 Multi-sets M 0 () = 2 M 0 () = 3 PAGE 14

16 Operations on multi-sets PAGE 15

17 Notation M 0 = [,,,,] = [ 2, 3 ] = 2[]+3[] also denoted as (2,3) PAGE 16

18 Preset/postset PAGE 17

19 Examples = [] = [ 2 ] = [ 2 ] = [] = [] = [,] = [] = [] = [ 2 ] = [ 2 ] = [] = [] = [] = [,] = [,] = [ ] PAGE 18

20 Firing rule PAGE 19

21 Notations Murata Desel/Reisig PAGE 20

22 Notations: Firing rule Murata Desel/Reisig PAGE 21

23 Basic Properties

24 Basic properties of a marked Petri net PAGE 23

25 Terminating PAGE 24

26 Deadlock-free PAGE 25

27 Liveness PAGE 26

28 Basic idea of liveness all reachable markings markings where t is enabled PAGE 27

29 Boundedness is 1-bounded, is 3-bounded is 1-bounded, is unbounded PAGE 28

30 Safeness p6 p4 t3 p3 p5 p6 p4 t3 p3 p5 PAGE 29

31 Reversible/home marking. [] is home marking PAGE 30

32 Reachability Graph

33 Definition t M M PAGE 32

34 Reachability graph algorithm 1) Label the initial marking M 0 as the root and tag it "new". 2) While "new" markings exists, do the following: a) Select a new marking M. b) If no transitions are enabled at M, tag M "dead-end". c) While there exist enabled transitions at M, do the following for each enabled transition t at M: i. Obtain the marking M' that results from firing t at M. ii. If M' does not appear in the graph, add M' and tag it "new". iii. Draw an arc with label t from M to M' (if not already present). 3) Output the graph. PAGE 33

35 Example free p4 t3 p3 p5 Step 1: Label the initial marking M0 as the root and tag it "new" (indicated by green color). [,free,p4] PAGE 34

36 Example (continued) free p4 t3 p3 p5 [,free,p4] [,free,p4] [,free,p4] [,free,p5] PAGE 35

37 Example (continued) free p4 t3 p3 p5 [,free,p4] [,free,p4] [,free,p4] [,free,p4] [,p3,p4] [,free,p5] [,free,p5] [,free,p5] PAGE 36

38 Example (continued) free p4 t3 p3 p5 [,free,p4] [,free,p4] [,p3,p4] [,free,p4] [,free,p4] [,p3,p4] [,free,p5] [,free,p5] [,free,p5] [,free,p5] PAGE 37

39 Example (continued) free p4 t3 p3 p5 [,free,p4] [,free,p4] [,p3,p4] [,free,p4] [,free,p4] [,p3,p4] [,free,p5] [,free,p5] [,free,p5] [,free,p5] [,p3,p5] PAGE 38

40 Example (continued) free p4 t3 p3 p5 [,free,p4] [,free,p4] [,p3,p4] [,free,p4] [,free,p4] [,p3,p4] [,p3,p4] [,free,p5] [,free,p5] [,p3,p5] [,free,p5] [,free,p5] [,p3,p5] PAGE 39

41 Example (continued) free p4 t3 p3 p5 [,free,p4] [,free,p5] [,free,p4] [,free,p5] [,p3,p4] [,p3,p5] [,p3,p4] [,free,p4] [,free,p5] [,free,p4] t3 [,free,p5] t3 [,p3,p4] [,p3,p5] [,p3,p4] [,p3,p5] PAGE 40

42 PAGE 41 Example (continued) t3 p5 p3 p4 free [,free,p4] [,free,p4] [,free,p5] [,free,p5] [,p3,p4] [,p3,p4] [,p3,p5] [,p3,p5] t3 t3 [,free,p4] [,free,p4] [,free,p5] [,free,p5] [,p3,p4] [,p3,p4] [,p3,p5] [,p3,p5] t3 t3

43 PAGE 42 Example (continued) t3 p5 p3 p4 free [,free,p4] [,free,p4] [,free,p5] [,free,p5] [,p3,p4] [,p3,p4] [,p3,p5] [,p3,p5] t3 t3 [,free,p4] [,free,p4] [,free,p5] [,free,p5] [,p3,p4] [,p3,p4] [,p3,p5] [,p3,p5] t3 t3

44 Example (complete) p3 free t3 p5 p4 [,free,p4] [,free,p5] [,free,p4] t3 [,free,p5] t3 [,p3,p4] [,p3,p5] [,p3,p4] [,p3,p5] The marked Petri net is: deadlock free live bounded safe reversible all markings are home markings PAGE 43

45 Coverability Graph

46 Problem (1,0) (1,1) (1,2)... ps. (n,m) is a shorthand for [ n, m ] PAGE 45

47 Coverability tree algorithm 1) Label the initial marking M 0 as the root and tag it "new". 2) While "new" markings exists, do the following: a) Select a new marking M and remove the "new" tag. b) If M is identical to a marking on the path from the root to M, then tag M "old" and go to another new marking. c) If no transitions are enabled at M, tag M "dead-end". d) While there exist enabled transitions at M, do the following for each enabled transition t at M: i. Obtain the marking M' that results from firing t at M. ii. If, on the path from the root to M, there exists a marking M'' such that M'(p) M''(p) for each p and M' M'' (i.e., M'' is coverable), then replace M'(p) by ω for each p such that M'(p) > M''(p). iii. Introduce M' as a node, draw an arc with label t from M to M', and tag M' "new". 3) Output the tree. PAGE 46

48 Example Step 1: Label the initial marking M 0 as the root and tag it "new" (indicated by green color). [] PAGE 47

49 Example (continued) Step 2: While "new" markings exists, do the following: Select a new marking M and remove the "new" tag. If M is identical to a marking on the path from the root to M, then tag M "old" and go to another new marking. If no transitions are enabled at M, tag M "dead-end". While there exist enabled transitions at M, do the following for each enabled transition t at M: Obtain the marking M' that results from firing t at M. If, on the path from the root to M, there exists a marking M'' such that M'(p) M''(p) for each p and M' M'' (i.e., M'' is coverable), then replace M'(p) by ω for each p such that M'(p) > M''(p). Introduce M' as a node, draw an arc with label t from M to M', and tag M' "new" [] [] [,] M [, ω ] M' PAGE 48

50 Example (continued) Step 2: While "new" markings exists, do the following: Select a new marking M and remove the "new" tag. If M is identical to a marking on the path from the root to M, then tag M "old" and go to another new marking. If no transitions are enabled at M, tag M "dead-end". While there exist enabled transitions at M, do the following for each enabled transition t at M: Obtain the marking M' that results from firing t at M. If, on the path from the root to M, there exists a marking M'' such that M'(p) M''(p) for each p and M' M'' (i.e., M'' is coverable), then replace M'(p) by ω for each p such that M'(p) > M''(p). Introduce M' as a node, draw an arc with label t from M to M', and tag M' "new" [] [, ω ] [] [, ω ] [, ω ] ω+k = ω-k = ω PAGE 49

51 Example (continued) Step 2: While "new" markings exists, do the following: Select a new marking M and remove the "new" tag. If M is identical to a marking on the path from the root to M, then tag M "old" and go to another new marking. If no transitions are enabled at M, tag M "dead-end". While there exist enabled transitions at M, do the following for each enabled transition t at M: Obtain the marking M' that results from firing t at M. If, on the path from the root to M, there exists a marking M'' such that M'(p) M''(p) for each p and M' M'' (i.e., M'' is coverable), then replace M'(p) by ω for each p such that M'(p) > M''(p). Introduce M' as a node, draw an arc with label t from M to M', and tag M' "new" [] [, ω ] [, ω ] [] [, ω ] [, ω ] PAGE 50

52 Example (complete) Step 3: Output the tree. [] [, ω ] [, ω ] Coverability graph: [] [, ω ] PAGE 51

53 Another example p3 [,p3] Step 1: Label the initial marking M 0 as the root and tag it "new" (indicated by green color). [,p3] [, ω,p3] p4 Step 2... [,p3,p4 ω ] PAGE 52

54 Example (continued) p3 [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] p4 [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3] [, ω,p3] [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] PAGE 53

55 Example (continued) p3 [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] p4 [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3] [, ω,p3] [, ω,p3] [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] PAGE 54

56 Example (continued) p3 [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] p4 [, ω,p3,p4 ω ] [,p3] [,p3,p4 ω ] [, ω,p3] [, ω,p3] [, ω,p3,p4 ω ] [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] PAGE 55

57 Example (continued) p3 Step 3: Output the tree. [,p3] [, ω,p3] [, ω,p3] p4 [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] 4x [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] PAGE 56

58 Example (complete) p3 [,p3] [, ω,p3] p4 [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3] [, ω,p3] [, ω,p3] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] Coverability graph [, ω,p3,p4 ω ] PAGE 57

59 Coverability graph Take the coverability tree and simply merge nodes with identical labels [] [, ω ] [, ω ] [] [, ω ] [,p3] [, ω,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [, ω,p3,p4 ω ] [,p3] [,p3,p4 ω ] [, ω,p3] [, ω,p3,p4 ω ] PAGE 58

60 Another example marked net (1,0) (1,1) (1,2)... reachability graph (0,0) (0,1) (1,0) (1,ω) (1,ω) coverability tree (0,ω) (1,0) (1,ω) coverability graph (0,ω) ps. (n,m) is a shorthand for [ n, m ] PAGE 59

61 ω-markings [,p3] [, ω,p3] [,p3,p4 ω ] [, ω,p3,p4 ω ] PAGE 60

62 Properties The coverability tree/graph is always finite. The marked Petri net is bounded if and only if the corresponding coverability tree/graph contains only ω-free markings. The coverability tree/graph gives an over-approximation. Different Petri nets may have the same coverability tree/graph. PAGE 61

63 Basic relation between reachable markings and coverability tree/graph (1,0) (1,ω) (1,ω) (0,ω) Let n=180. There is a reachable marking with 0 tokens in and at least 180 tokens in. PAGE 62

64 Example (readers and writers) p4 t3 p3 p5 construct coverability graph... PAGE 63

65 Initial part [,p3 ω,p4] [,p4] [,p4] [,p5] [,p5] [,p5] [,p3 ω,p5] [,p3 ω,p5] t3 t3 p4 t3 p3 p5 PAGE 64

66 Coverability tree [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] t3 [,p3 ω,p5] [,p3 ω,p4] t3 [,p3 ω,p4] [,p4] [,p4] [,p3 ω,p4] [,p3 ω,p5] [,p5] [,p5] [,p3 ω,p5] [,p3,p5] [,p3 ω,p5] t3 [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p4] [,p3 ω,p4] [,p3 ω,p5] [,p5] t3 [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p5] t3 t3 [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] t3 [,p3 ω,p4] p4 t3 p3 p5 PAGE 65

67 Coverability graph [,p4] [,p4] [,p5] t3 [,p3 ω,p4] t3 [,p3 ω,p4] [,p3 ω,p5] [,p5] t3 [,p3 ω,p5] [,p3,p5] p4 t3 p3 p5 PAGE 66

68 [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] t3 [,p3 ω,p5] [,p3 ω,p4] t3 [,p3 ω,p4] [,p4] [,p4] [,p3 ω,p4] [,p3 ω,p5] [,p5] [,p5] [,p3 ω,p5] [,p3,p5] [,p3 ω,p5] t3 [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p4] [,p3 ω,p4] [,p3 ω,p5] [,p5] t3 [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p5] t3 t3 [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] [,p3 ω,p4] [,p3 ω,p4] [,p3 ω,p5] [,p3 ω,p5] t3 [,p3 ω,p4] [,p4] [,p4] [,p5] t3 [,p3 ω,p4] t3 [,p3 ω,p4] [,p3 ω,p5] [,p5] t3 [,p3 ω,p5] t3 p4 [,p3,p5] p3 p5 PAGE 67

69 Coverability graph (vector notation) (1,0,0,1,0) (0,1,0,1,0) (1,0,0,0,1) (0,1,0,0,1) t3 (1,0,1,0,1) (1,0,ω,1,0) t3 (0,1,ω,1,0) t3 (0,1,ω,0,1) (1,0,ω,0,1) p4 t3 p3 p5 PAGE 68

70 Analysis results,, p4, p5 are safe p3 is unbounded [,p5] is reachable [,] is not reachable [,p3 180,p5] is coverable p3 t3 p5 p4 [,p4] [,p4] [,p5] [,p5] t3 [,p3 ω,p4] t3 [,p3 ω,p4] [,p3 ω,p5] t3 [,p3 ω,p5] [,p3,p5] PAGE 69

71 Additional properties A transition t is dead if and only if if does not appear in the coverability graph. The coverability graph and reachability graph are identical if the marked Petri net is bounded (i.e., only ω-free markings). The marked Petri net is safe if only 0's and 1's appear in nodes. Any firing sequence of the marked Petri net can be matched by a "walk" through the coverability graph. The reverse is not true!!!! PAGE 70

72 Limitation: Loss of information Two nets with the same coverability graph! [] [, ω ] {[],[, 1 ], [, 2 ], [, 3 ], [, 4 ],...} {[],[, 3 ], [, 6 ], [, 9 ], [, 12 ],...} PAGE 71

73 State-explosion problem (1) 2 n +1 states PAGE 72

74 State-explosion problem (2) place s is 2 n bounded Each round the number of tokens in s can be doubled. PAGE 73

75 Variants Construct the coverability graph on the fly (i.e., do not first construct the coverability tree): the graph may become smaller but process is typically non-deterministic. Several approaches have been proposed to construct "minimal" coverability graphs/sets (see "Alain Finkel: The Minimal Coverability Graph for Petri Nets. Applications and Theory of Petri Nets 1991: ", and "Gilles Geeraerts, Jean-François Raskin, Laurent Van Begin: On the Efficient Computation of the Minimal Coverability Set for Petri Nets. ATVA 2007: ") PAGE 74

76 Conclusion

77 The coverability graph is finite but... some information gets lost in case of unbounded behavior, and it may be huge and impossible to construct. Next: structural methods like invariants, siphons, traps, etc. PAGE 76

78 After this lecture you should be able to: Understand the formalizations, i.e., (P,T,F,W), M, (N,M)[t>(N,M'), etc. Determine whether a concrete marked net is terminating, deadlockfree, live, bounded, safe, and/or reversible, whether a transition is live and/or dead, whether a place is k-bounded, etc. Construct a Petri net that has a set of desirable properties, e.g., a net that is live and bounded but not reversible. Construct the reachability graph of a marked net. Construct the coverability tree of a marked net. Construct the coverability graph of a marked net. Tell which properties can(not) be derived from the coverability tree/graph. Understand the limitations of the coverability tree/graph (loss of information, inability to decide liveness, etc.). Derive conclusions from a concrete coverability tree/graph. PAGE 77

79 Appendix: Formalization of Coverability Graph based on Desel & Reisig

80 Coverability tree & graph Idea: cut-off unbounded behavior using omega (ω) markings (1,0, ω,1, ω,1,2,0) PAGE 79

81 Trivial example (1,0) (1,1) (1,2) (1,0) (1,ω) (1,ω) (1,0) (1,ω)... marked net reachability graph coverability tree coverability graph PAGE 80

82 Extended example marked net (1,0) (1,1) (1,2)... reachability graph (0,0) (0,1) (1,0) (1,ω) (1,ω) coverability tree (0,ω) (1,0) (1,ω) coverability graph (0,ω) PAGE 81

83 Approach 1. Define omega (ω) occurrence sequences. 2. Show that these are finite. 3. Construct coverability tree 4. Construct coverability graph (1,0) (1,ω) (1,0) (1,ω) (1,0) (1,ω) (1,0) (1,ω) (0,ω) (1,0) (1,ω) (0,ω) (0,ω) (1,ω) (1,ω) marked net ω-occurrence sequences coverability tree coverability graph PAGE 82

84 Example of a ω-occurrence sequence ω-occurrence sequence: (1,0) --> (1,ω) --> (1, ω) (1,0) (1,ω) (1,ω) marked net PAGE 83

85 PAGE 84

86 (1) Transitions need to be enabled (1,0) (1,ω) (1,0) (1,ω) (0,ω) (1,0) (1,ω) (1,ω) Only is enabled in (1,0), not. marked net ω-occurrence sequences PAGE 85

87 (2) For non-ω place markings: business as usual (1,0) (0,1) marked net ω-occurrence sequences PAGE 86

88 (3) Introducing omegas marked net (1,0) (1,ω) (0,ω) (1,0) (1,ω) (1,0) (1,ω) (1,ω) ω-occurrence sequences (1,ω) is "reachable" from (1,0) because there is a j (j=0) such that... PAGE 87

89 (4) Stop after second identical marking (1,0) (1,0) marked net (1,0) (1,ω) (0,ω) (1,ω) (1,0) (1,ω) (1,ω) ω-occurrence sequences (1,ω) (1,ω) (0,ω) Marking (0,ω) is dead while (1,ω) markings are not continued after second occurrence. PAGE 88

90 Finite? How long can a ω-occurrence sequence be? How many ω-occurrence sequences are there? Is the coverability tree/graph finite? PAGE 89

91 PAGE 90 Dickson's Lemma ( ) φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 φ10 φ11 φ12 φ13 φ φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 φ10 φ11 φ12 φ13 φ14

92 PAGE 91

93 PAGE 92

94 Coverability tree (1,0) (1,ω) (1,0) (1,0) (1,0) (1,ω) (0,ω) Diagrams are a bit misleading: vertices labeled with a ω-marking are really sequences, e.g., initial node is ε rather than (1,0). (1,ω) (0,ω) (1,ω) (1,ω) (1,ω) ω-occurrence sequences coverability tree PAGE 93

95 Finiteness PAGE 94

96 Example PAGE 95

97 Marking graph (i.e., reachability graph) PAGE 96

98 Coverability tree find the error (also in paper)... PAGE 97

99 Relation ω-markings and normal markings (1,0) (1,ω) (1,ω) (0,ω) Let b=180. There is a marking reachable with 0 tokens in and at least 180 tokens in. PAGE 98

100 Boundedness = "all ω-markings are ω- free" PAGE 99

101 b-boundedness (1,0) (1,ω) (1,ω) (0,ω) is 1-boundned (safe) is unbounded PAGE 100

102 Dead transitions do not appear in cov. tree PAGE 101

103 Coverability graph (versus cov. tree) (1,0) (1,ω) (0,ω) (1,0) (1,ω) (0,ω) (1,ω) coverability tree coverability graph PAGE 102

104 Boundedness implies equivalence PAGE 103

105 Appendix: Examples taken from Murata

106 Coverability tree (same as before) PAGE 105

107 Example PAGE 106

108 Properties PAGE 107

109 Coverability graph PAGE 108

110 PAGE 109