Petri Nets ee249 Fall 2000

Transcription

1 Petri Nets ee249 Fall 2000 Marco Sgroi Most slides borrowed from Luciano Lavagno s lecture ee249 (1998) 1 Models Of Computation for reactive systems Main MOCs: Communicating Finite State Machines Dataflow Process Networks Discrete Event Codesign Finite State Machines Petri Nets Main languages: StateCharts Esterel Dataflow networks 2 1

2 Outline Petri nets Introduction Examples Properties Analysis techniques Scheduling 3 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 Asynchronous model (partial ordering) Main drawback: no hierarchy 4 2

3 Petri Net Graph Bipartite weighted directed graph: Places: circles Transitions: bars or boxes Arcs: arrows labeled with weights Tokens: black dots p1 t1 p2 2 p4 3 p3 5 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 nonnegative 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 2 p4 Places/Transition: conditions/events p

4 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 Concurrency, causality, choice t1 t5 t4 t6 8 4

5 Concurrency, causality, choice t1 Concurrency t5 t4 t6 9 Concurrency, causality, choice t1 Causality, sequencing t5 t4 t6 10 5

6 Concurrency, causality, choice t1 t5 t4 t6 Choice, conflict 11 Concurrency, causality, choice t1 t5 t4 t6 Choice, conflict 12 6

7 Confusion t1 and are concurrent but their firing order is not irrelevant for conflict resolution (not local choice) From (1,1,0,0,0): solving a conflict (t1,) (0,0,0,0,1),(0,0,1,1,0) not solving a conflict (,t1) (0,0,1,1,0) p1 t1 p3 p5 p2 p4 13 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 14 7

8 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 15 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 16 8

9 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 17 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 18 9

10 Communication Protocol Send msg Receive msg P1 P2 Send Ack Receive Ack 19 Producer-Consumer Problem Produce Buffer Consume 20 10

11 Producer-Consumer Problem Produce Buffer Consume 21 Producer-Consumer Problem Produce Buffer Consume 22 11

12 Producer-Consumer Problem Produce Buffer Consume 23 Producer-Consumer Problem Produce Buffer Consume 24 12

13 Producer-Consumer Problem Produce Buffer Consume 25 Producer-Consumer Problem Produce Buffer Consume 26 13

14 Producer-Consumer Problem Produce Buffer Consume 27 Producer-Consumer Problem Produce Buffer Consume 28 14

15 Producer-Consumer Problem Produce Buffer Consume 29 Producer-Consumer Problem Produce Buffer Consume 30 15

16 Producer-Consumer Problem Produce Buffer Consume 31 Producer-Consumer Problem Produce Buffer Consume 32 16

17 Producer-Consumer Problem Produce Buffer Consume 33 Producer-Consumer Problem Produce Buffer Consume 34 17

18 Producer-Consumer with priority Consumer B can consume only if buffer A is empty Inhibitor arcs A B 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 36 18

19 Reachability Marking M is reachable from marking M0 if there exists a sequence of firings s = M0 t1 M1 M2 M that transforms M0 to M. The reachability problem is decidable. p1 t1 p2 p4 Μ0 = (1,0,1,0) p3 M1 = (1,0,0,1) 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 38 19

20 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Not live 39 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Deadlock-free 40 20

21 Liveness Liveness: from any marking any transition can become fireable Liveness implies deadlock freedom, not viceversa Deadlock-free 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 42 21

22 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 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 22

23 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 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 46 23

24 Conservation Conservation: the total number of tokens in the net is constant Not conservative 47 Conservation Conservation: the total number of tokens in the net is constant Not conservative 48 24

25 Conservation Conservation: the total number of tokens in the net is constant Conservative Analysis techniques Structural analysis techniques Incidence matrix T- and S- Invariants State Space Analysis techniques Coverability Tree Reachability Graph 50 25

26 Incidence Matrix p1 t1 p2 p3 t A= 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 51 State equations E.g. reachability of M = T from M 0 = T p1 t1 p2 p3 A= v 1 = = but also v 2 = T or any v k = 1 (k) (k+1) T 52 26

27 Necessary Condition only t1 2 2 Firing vector: (1,2,2) Deadlock!! 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 p1 t1 p2 p3 A=

28 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 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 p1 t1 p2 p3 A T =

29 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 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 58 29

30 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 w 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. 59 Boundedness is decidable with coverability tree Coverability Tree 1000 p1 t1 p2 p3 p

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

32 Coverability Tree Boundedness is decidable with coverability tree p1 t1 p2 p3 p t Coverability Tree Boundedness is decidable with coverability tree p1 t1 p2 p3 p t w Cannot solve the reachability and liveness problems 64 32

33 Coverability Tree Boundedness is decidable with coverability tree p1 t1 p2 p3 p t w Cannot solve the reachability and liveness problems 65 Reachability graph p1 t1 p2 p3 100 For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 66 33

34 Reachability graph p1 t1 p2 p3 100 t1 010 For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 67 Reachability graph p1 t1 p2 p3 100 t For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 68 34

35 Reachability graph p1 t1 p2 p3 100 t For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings 69 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 70 35

36 Subclasses of Petri nets: SMs State machine: every transition has at most 1 predecessor and 1 successor Models only causality and conflict (no concurrency, no synchronization of parallel activities) 71 Subclasses of Petri nets: MGs Marked Graph: every place has at most 1 predecessor and 1 successor Models only causality and concurrency (no conflict) Same as underlying graph of SDF 72 36

37 Subclasses of Petri nets: FC nets Free-Choice net: every transition after choice has exactly 1 predecessor 73 Free-Choice Petri Nets (FCPN) Free-Choice (FC) t1 Confusion (not-free-choice) 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. Easy to analyze 74 37

38 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 concurrency, causality and choice relations are mutually exclusive exploits duality between MG and SM 75 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 76 38

39 MG reduction and cover Choose one successor for each conflicting place: 77 MG reduction and cover Choose one successor for each conflicting place: 78 39

40 MG reduction and cover Choose one successor for each conflicting place: 79 MG reduction and cover Choose one successor for each conflicting place: 80 40

41 MG reduction and cover Choose one successor for each conflicting place: 81 MG reductions The set of all reductions yields a cover of MG components (T-invariants) 82 41

42 MG reductions The set of all reductions yields a cover of MG components (T-invariants) 83 SM reduction and cover Choose one predecessor for each transition: 84 42

43 SM reduction and cover Choose one predecessor for each transition: 85 SM reduction and cover Choose one predecessor for each transition: The set of all reductions yields a cover of SM components (S-invariants) 86 43

44 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 87 Hack s theorem Example of non-live (but safe) FCN 88 44

45 Hack s theorem Example of non-live (but safe) FCN 89 Hack s theorem Example of non-live (but safe) FCN 90 45

46 Hack s theorem Example of non-live (but safe) FCN 91 Hack s theorem Example of non-live (but safe) FCN 92 46

47 Hack s theorem Example of non-live (but safe) FCN 93 Hack s theorem Example of non-live (but safe) FCN 94 47

48 Hack s theorem Example of non-live (but safe) FCN 95 Hack s theorem Example of non-live (but safe) FCN 96 48

49 Hack s theorem Example of non-live (but safe) FCN 97 Hack s theorem Example of non-live (but safe) FCN 98 49

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

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

52 Hack s theorem Example of non-live (but safe) FCN Not live 103 Other results for LSFC nets Let t 1 and t 2 be two transitions of a live and safe Free- Choice net. Then t 1 and t 2 are: sequential if there exists a simple cycle to which both belong concurrent if they are not ordered, and there exists an MG component to which both belong conflicting otherwise

53 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 105 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

54 Summary of Petri Nets Graphical formalism Distributed state (including buffering) Concurrency, sequencing and choice made explicit Structural and behavioral properties Analysis techniques based on linear algebra (only sufficient) structural analysis (necessary and sufficient only for FC)