Outline. Petri nets. Introduction Examples Properties Analysis techniques. 1 EE249Fall04

Transcription

1 Outline Petri nets Introduction Examples Properties Analysis techniques 1

2 Petri Nets (PNs) Model introduced by C.A. Petri in 1962 Ph.D. Thesis: Communication with Automata Applications: distributed computing, manufacturing, control, communication networks, transportation PNs describe explicitly and graphically: sequencing/causality conflict/non-deterministic choice concurrency Basic PN model Asynchronous model (partial ordering) Main drawback: no hierarchy 2

3 Example: Synchronization at single track rail segment Preconditions 3

4 Playing the token game 4

5 Conflict for resource track 5

6 Petri Net Graph Bipartite weighted directed graph: Places: circles Transitions: bars or boxes Arcs: arrows labeled with weights Tokens: black dots p2 p1 t1 t2 2 p4 t3 3 6 p3

7 Petri Net A PN (N,M0) is a Petri Net Graph N places: represent distributed state by holding tokens marking (state) M is an n-vector (m1,m2,m3 ), where mi is the non-negative number of tokens in place pi. initial marking (M 0 ) is initial state transitions: represent actions/events enabled transition: enough tokens in predecessors firing transition: modifies marking and an initial marking M0. p1 t1 p2 t2 2 p4 7 Places/Transitions: conditions/events p3 t3 3

8 Transition firing rule A marking is changed according to the following rules: A transition is enabled if there are enough tokens in each input place An enabled transition may or may not fire The firing of a transition modifies marking by consuming tokens from the input places and producing tokens in the output places

9 Concurrency, causality, choice t1 t2 t5 9 t3 t4 t6

10 Concurrency, causality, choice t1 Concurrency t2 t5 10 t3 t4 t6

11 Concurrency, causality, choice t1 t2 Causality, sequencing t5 11 t3 t4 t6

12 Concurrency, causality, choice t1 t2 t5 12 t3 t4 t6 Choice, conflict

13 Concurrency, causality, choice t1 t2 t5 13 t3 t4 t6 Choice, conflict

14 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 14

15 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 15

16 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 16

17 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 17

18 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 18

19 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 19

20 Producer-Consumer Problem Produce Buffer Consume 20

21 Producer-Consumer Problem Produce Buffer Consume 21

22 Producer-Consumer Problem Produce Buffer Consume 22

23 Producer-Consumer Problem Produce Buffer Consume 23

24 Producer-Consumer Problem Produce Buffer Consume 24

25 Producer-Consumer Problem Produce Buffer Consume 25

26 Producer-Consumer Problem Produce Buffer Consume 26

27 Producer-Consumer Problem Produce Buffer Consume 27

28 Producer-Consumer Problem Produce Buffer Consume 28

29 Producer-Consumer Problem Produce Buffer Consume 29

30 Producer-Consumer Problem Produce Buffer Consume 30

31 Producer-Consumer Problem Produce Buffer Consume 31

32 Producer-Consumer Problem Produce Buffer Consume 32

33 Producer-Consumer Problem Produce Buffer Consume 33

34 Producer-Consumer with priority A Consumer B can consume only if buffer A is empty Inhibitor arcs B 34

35 PN properties Behavioral: depend on the initial marking (most interesting) Reachability Boundedness Schedulability Liveness Conservation Structural: do not depend on the initial marking (often too restrictive) Consistency Structural boundedness 35

36 Reachability Marking M is reachable from marking M0 if there exists a sequence of firings σ = M0 t1 M1 t2 M2 M that transforms M0 to M. The reachability problem is decidable. p1 t1 p2 t2 p4 Μ0 = (1,0,1,0) t3 p3 t3 M1 = (1,0,0,1) t2 36 Μ0 = (1,0,1,0) M = (1,1,0,0) M = (1,1,0,0)

37 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Not live 37

38 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Not live 38

39 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Deadlock-free 39

40 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Deadlock-free 40

41 Boundedness Boundedness: the number of tokens in any place cannot grow indefinitely (1-bounded also called safe) Application: places represent buffers and registers (check there is no overflow) Unbounded 41

42 Boundedness Boundedness: the number of tokens in any place cannot grow indefinitely (1-bounded also called safe) Application: places represent buffers and registers (check there is no overflow) Unbounded 42

43 Boundedness Boundedness: the number of tokens in any place cannot grow indefinitely (1-bounded also called safe) Application: places represent buffers and registers (check there is no overflow) Unbounded 43

44 Boundedness Boundedness: the number of tokens in any place cannot grow indefinitely (1-bounded also called safe) Application: places represent buffers and registers (check there is no overflow) Unbounded 44

45 Boundedness Boundedness: the number of tokens in any place cannot grow indefinitely (1-bounded also called safe) Application: places represent buffers and registers (check there is no overflow) Unbounded 45

46 Conservation Conservation: the total number of tokens in the net is constant Not conservative 46

47 Conservation Conservation: the total number of tokens in the net is constant Not conservative 47

48 Conservation Conservation: the total number of tokens in the net is constant Conservative

49 Analysis techniques Structural analysis techniques Incidence matrix T- and S- Invariants State Space Analysis techniques Coverability Tree Reachability Graph 49

50 Incidence Matrix t2 p1 t1 p2 p3 t3 A= t1 t2 t p1 p2 p3 Necessary condition for marking M to be reachable from initial marking M 0 : there exists firing vector v s.t.: M = M 0 + A v 50

51 State equations E.g. reachability of M = T from M 0 = T p1 t1 p2 t2 p3 t3 A= v 1 = = but also v 2 = T or any v k = 1 (k) (k+1) T

52 Necessary Condition only t2 t1 t3 2 2 Firing vector: (1,2,2) Deadlock!! 52

53 State equations and invariants Solutions of Ax = 0 (in M = M0 + Ax, M = M0) T-invariants sequences of transitions that (if fireable) bring back to original marking periodic schedule in SDF e.g. x = T 53 t2 p1 t1 p2 p3 t3 A=

54 Application of T-invariants Scheduling Cyclic schedules: need to return to the initial state i *k2 + o *k1 T-invariant: (1,1,1,1,1) Schedule: i *k2 *k1 + o 54

55 State equations and invariants Solutions of ya = 0 S-invariants sets of places whose weighted total token count does not change after the firing of any transition (y M = y M ) e.g. y = T t2 p1 t1 p2 p3 t3 A T =

56 Application of S-invariants Structural Boundedness: bounded for any finite initial marking M0 Existence of a positive S-invariant is CS for structural boundedness initial marking is finite weighted token count does not change 56

57 Summary of algebraic methods Extremely efficient (polynomial in the size of the net) Generally provide only necessary or sufficient information Excellent for ruling out some deadlocks or otherwise dangerous conditions Can be used to infer structural boundedness 57

58 Coverability Tree Build a (finite) tree representation of the markings Karp-Miller algorithm Label initial marking M0 as the root of the tree and tag it as new While new markings exist do: select a new marking M if M is identical to a marking on the path from the root to M, then tag M as 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: obtain the marking M that results from firing t at M on the path from the root to M if there exists a marking M such that M (p)>=m (p) for each place p and M is different from M, 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 as new. 58

59 Coverability Tree Boundedness is decidable with coverability tree 1000 t2 p1 t1 p2 p3 t3 p4 59

60 Coverability Tree Boundedness is decidable with coverability tree t2 p1 t1 p2 p t t3 p4 60

61 Coverability Tree Boundedness is decidable with coverability tree t2 p1 t1 p2 p t t t3 p4 61

62 Coverability Tree Boundedness is decidable with coverability tree t2 p1 t1 p2 p3 t3 p t t t

63 Coverability Tree Boundedness is decidable with coverability tree t2 p1 t1 p2 p3 t3 p t t t2 010ω 63

64 Coverability Tree Is (1) reachable from (0)? t1 p1 t2 t1 p1 t

65 Coverability Tree Is (1) reachable from (0)? t1 p1 t2 0 t1 ω (0) (1) (2) t1 2 p1 2 t2 65

66 Coverability Tree Is (1) reachable from (0)? t1 p1 t2 0 t1 ω (0) (1) (2) t1 2 p1 2 t2 0 t1 ω (0) (2) (0) 66

67 Coverability Tree Is (1) reachable from (0)? t1 p1 t2 0 t1 ω (0) (1) (2) t1 2 p1 2 t2 0 t1 ω (0) (2) (0) Cannot solve the reachability problem 67

68 Reachability graph t2 p1 t1 p2 p3 100 t3 For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 68

69 Reachability graph t2 p1 t1 p2 p3 t t1 For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 69

70 Reachability graph t2 p1 t1 p2 p3 t t1 t3 For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 70

71 Reachability graph t2 p1 t1 p2 p3 t3 t t1 t3 For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 71

72 Subclasses of Petri nets Reachability analysis is too expensive State equations give only partial information Some properties are preserved by reduction rules e.g. for liveness and safeness Even reduction rules only work in some cases Must restrict class in order to prove stronger results 72

73 Marked Graphs Every place has at most 1 predecessor and 1 successor transition Models only causality and concurrency (no conflict) YES NO 73

74 State Machines Every transition has at most 1 predecessor and 1 successor place Models only causality and conflict (no concurrency, no synchronization of parallel activities) YES NO 74

75 Free-Choice Petri Nets (FCPN) Free-Choice (FC) every transition after choice has exactly 1 predecessor place t1 Confusion (not-free-choice) t2 Extended Free-Choice Free-Choice: the outcome of a choice depends on the value of a token (abstracted non-deterministically) rather than on its arrival time. 75

76 Free-Choice nets Introduced by Hack ( 72) Extensively studied by Best ( 86) and Desel and Esparza ( 95) Can express concurrency, causality and choice without confusion Very strong structural theory necessary and sufficient conditions for liveness and safeness, based on decomposition exploits duality between MG and SM 76

77 MG (& SM) decomposition An Allocation is a control function that chooses which transition fires among several conflicting ones ( A: P T). Eliminate the subnet that would be inactive if we were to use the allocation... Reduction Algorithm Delete all unallocated transitions Delete all places that have all input transitions already deleted Delete all transitions that have at least one input place already deleted Obtain a Reduction (one for each allocation) that is a conflict free subnet 77

78 MG reduction and cover Choose one successor for each conflicting place: 78

79 MG reduction and cover Choose one successor for each conflicting place: 79

80 MG reduction and cover Choose one successor for each conflicting place: 80

81 MG reduction and cover Choose one successor for each conflicting place: 81

82 MG reduction and cover Choose one successor for each conflicting place: 82

83 MG reductions The set of all reductions yields a cover of MG components (Tinvariants) 83

84 MG reductions The set of all reductions yields a cover of MG components (Tinvariants) 84

85 MG reductions The set of all reductions yields a cover of MG components (Tinvariants) 85

86 Hack s theorem ( 72) Let N be a Free-Choice PN: N has a live and safe initial marking (well-formed) if and only if every MG reduction is strongly connected and not empty, and the set of all reductions covers the net every SM reduction is strongly connected and not empty, and the set of all reductions covers the net 86

87 Hack s theorem Example of non-live (but safe) FCN 87

88 Hack s theorem Example of non-live (but safe) FCN 88

89 Hack s theorem Example of non-live (but safe) FCN 89

90 Hack s theorem Example of non-live (but safe) FCN 90

91 Hack s theorem Example of non-live (but safe) FCN 91

92 Hack s theorem Example of non-live (but safe) FCN 92

93 Hack s theorem Example of non-live (but safe) FCN 93

94 Hack s theorem Example of non-live (but safe) FCN 94

95 Hack s theorem Example of non-live (but safe) FCN 95

96 Hack s theorem Example of non-live (but safe) FCN 96

97 Hack s theorem Example of non-live (but safe) FCN 97

98 Hack s theorem Example of non-live (but safe) FCN 98

99 Hack s theorem Example of non-live (but safe) FCN 99

100 Hack s theorem Example of non-live (but safe) FCN 100

101 Hack s theorem Example of non-live (but safe) FCN 101

102 Hack s theorem Example of non-live (but safe) FCN Deadlock 102

103 Summary of LSFC nets Largest class for which structural theory really helps Structural component analysis may be expensive (exponential number of MG and SM components in the worst case) But number of MG components is generally small FC restriction simplifies characterization of behavior 103

104 Petri Net extensions Add interpretation to tokens and transitions Colored nets (tokens have value) Add time Time/timed Petri Nets (deterministic delay) type (duration, delay) where (place, transition) Stochastic PNs (probabilistic delay) Generalized Stochastic PNs (timed and immediate transitions) Add hierarchy Place Charts Nets 104

105 PNs Summary 2 PN Graph: places (buffers), transitions (actions), tokens (data) Firing rule: transition enabled if there are enough tokens in each input place Properties Structural (consistency, structural boundedness ) Behavioral (reachability, boundedness, liveness ) Analysis techniques Structural (only CN or CS): State equations, Invariants Behavioral: coverability tree Reachability Subclasses: Marked Graphs, State Machines, Free-Choice PNs 105

106 References T. Murata Petri Nets: Properties, Analysis and Applications 106