DES. 4. Petri Nets. Introduction. Different Classes of Petri Net. Petri net properties. Analysis of Petri net models
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1 4. Petri Nets Introduction Different Classes of Petri Net Petri net properties Analysis of Petri net models 1
2 Petri Nets C.A Petri, TU Darmstadt, 1962 A mathematical and graphical modeling method. Describe systems that are: concurrent asynchronous distributed nondeterministic 2
3 Petri Nets Can be used at all stages of system development: modeling analysis simulation/visualization ( playing the token game ) synthesis (Petri net versions of SCT) implementation (Grafcet) 3
4 Application areas communication protocols distributed systems distributed database systems flexible manufacturing systems logical controller design multiprocessor memory systems dataflow computing systems fault tolerant systems... 4
5 Introduction A Petri net is a directed bipartite graph consisting of places P and transitions T. Places are represented by circles. Transitions are represented by bars (or rectangles) Places and transitions are connected by arcs. In a marked Petri net each place contains a cardinal (zero or positive integer) number of tokens of marks. 5
6 Firing rules 1. A transition t is enabled if each input place contains at least one token. 2. An enabled transition may or may not fire. 3. Firing an enabled transition t means removing one token from each input place of t and adding one token to each output place of t. The firing of a transition has zero duration. The firing of a sink transition (only input places) only consumes tokens. The firing of a source transition (only output places) only produces tokens. 6
7 P1 P2 T6 P3 P4 T4 P5 P6 T5 P7 7
8 P1 P2 T6 P3 P4 T4 P5 P6 T5 P7 8
9 P1 P2 T6 P3 P4 T4 P5 P6 T5 P7 9
10 P1 P2 T6 P3 P4 T4 P5 P6 T5 P7 10
11 P1 P2 T6 P3 P4 T4 P5 P6 T5 P7 11
12 P1 P2 T6 P3 P4 T4 P5 P6 T5 P7 12
13 Typical interpretations of places and transitions: Input Places Preconditions Input data Input signals Resources needed Conditions Buffers Transition Event Computation step Signal processor Task or job Clause in logic Processor Output Places Postconditions Output data Output signals Resources needed Conclusions Buffers 13
14 Characteristics Concurrency and divergence Conflict (decision, choice) and convergence or divergence Confusion (conflict + concurrency) t1 t3 t2 or divergence t1 and t2 are concurrent while each of t1 and t2 is in conflict with t3 14
15 State The state of a marked Petri net is its marking M [m(p 1 ),m(p 2 ),,m(p n )]. The marking can be finite or infinite. A Petri net can model an infinite system with a finite number of places. 15
16 Modeling with Petri Nets Tradeoff between Modeling power Formal Analysis Power Petri net specializations: restrictions on the allowed net structures more powerful analytical results and/or simpler algorithms Petri net abbreviations: classes of Petri nets which always can be transformed to ordinary Petri nets PN properties are maintained syntactic sugar 16
17 Petri net extensions (generalizations): classes of Petri nets with additional transition firing rules cannot be transformed back to ordinary PNs PN properties are not always maintained often Turing machine equivalent 17
18 PN Specializations State Graphs (state machines): an unmarked PN is a state graph iff every transition has one input place and one output place a marked state graph is equivalent to a automaton state machine iff it only contains one token can model conflicts cannot model concurrency 18
19 PN Specializations Event graphs (marked graphs, transition graphs): a PN is an event graph iff every place has one input transition and one output transition cannot model conflicts can model concurrency the basis for the max-plus algebra approach Other specializations: conflict-free PNs, free-choice PNs, simple PNs, pure PNs (have no self-loops) 19
20 PN abbreviations Generalized Petri nets Finite Capacity Petri nets High-Level Petri nets 20
21 Generalized Petri Nets P1 P2 2 2 P3 Firing rules: 1. A transition t is enabled if each input place p of t contains at least w(p,t) tokens 2. Firing a transition t means removing w(p,t) tokens from each input place p and adding w(t,q) tokens to each output place q. 21
22 Generalized Petri Nets P1 P2 2 2 P3 Firing rules: 1. A transition t is enabled if each input place p of t contains at least w(p,t) tokens 2. Firing a transition t means removing w(p,t) tokens from each input place p and adding w(t,q) tokens to each output place q. 22
23 Finite-Capacity PN Capacities (strictly positive integers) associated with with places. Transition firing rule: For a transition t to be enabled it is additionally required that the number of tokens in each output place p of t will not exceed its capacity K (p) after firing t. Generalized, finite capacity PNs are sometimes called Place/Transition (P/T) nets. Finite capacity nets where all places have capacity 1 and where input places are interpreted as preconditions, output places interpreted as postconditions, and transitions interpreted as events are called Condition/Event Systems. Separate theory. 23
24 High-Level Petri Nets Predicate/Transition nets Coloured Petri Nets Abstract data types + Petri nets A token has a type Several identical Petri nets modeling, e.g., several identical concurrent activities can be folded into a high-level PN. Can be transformed (unfolded) to an ordinary PN 24
25 PN extensions FIFO nets each place represents a FIFO queue where the tokens are queued Inhibitor arc Petri nets (zero-test PNs) an inhibitor arc connects a place to a transition the inhibitor arc disables the transition when the place contains tokens and enables the transition when the place is empty Priority Petri nets PN + a partial order relation on the transitions 25
26 Petri net properties What can we do with the nets? What properties and problems can be analyzed? Properties can be divided into structural properties marking independent behavioral properties marking dependent 26
27 Reachability A marking M is reachable from a marking M 0 if there exists a sequence of firings that transforms M 0 to M. Denoted M 0 [T 1 T 2 T n > M. M 0 the set of markings reachable from M 0 The Reachability Problem: Decide whether a given marking M M 0 or not. Decidable but takes at least exponential space and time 27
28 Properties Boundedness: Aplacep i is bounded for an initial marking M 0 if for all markings reachable from M 0, m(p i ) k(positive integer) p i is k-bounded. A net is bounded for an initial marking M 0 if all places are bounded for M 0 (similarly for k-bounded) A net is safe iff it is 1-bounded An unmarked net is structurally bounded if the marked PN is bounded for all possible M 0 Boundedness and safeness are important if the places model buffers, inventories, etc. 28
29 Properties Liveness: A transition t i is live for an initial marking M 0 if for every reachable marking M i M 0, a firing sequence S from M i exists, which contains transition t i. APNisliveforM 0 if all its transitions are live for M 0. Interpretation: No matter what marking has been reached from M 0, it is possible to ultimately fire any transition of the net. A transition t is quasi-live for M 0 if there is a firing sequence from M 0 that contains t. A PN is quasi-live for M 0 if all its transitions are quasi-live. A deadlock is a marking such that no transition is enabled. A PN is deadlock-free for M 0 if no reachable marking is a deadlock. 29
30 Properties Reversibility: In a reversible net one can always get back to the initial marking. Persistence: In a persistent net a transition, once it has been enabled, will stay enabled until it fires. 30
31 Invariants Let R be a PN and P the set of places. A marking invariant is obtained if there is a subset P {p 1,,p r } of P and a weighting vector [q 1,,q r ] for which all the weights are positive integers such that q 1 m(p 1 )+q 2 m(p 2 )+ +q r m(p r ) constant for every M reachable from M 0. P is called a conservative component The conservative component property is structural The value of the constant depends on the marking Physical meaning: the system is in one and only one state at a time a number of entities is maintained 31
32 Invariants A firing sequence that causes a return to the initial marking is called repetitive sequence. M 0 [T 1 T 2 T 3 > M 0 The set of transitions involved in the firing sequence is called a repetitive component. A repetitive sequence containing all the transitions is a complete repetitive sequence. 32
33 Analysis Methods How determine if a PN has a certain property? Three types of methods: reachability (coverability) methods linear algebra methods reduction methods 33
34 Reachability tree Create a tree where the nodes represent the reachable markings and the arcs represent the transition firings. Stop the expansion of a node in the tree if the node already contained in the tree. Only for bounded nets. Properties are determined by examining the tree. Exhaustive method, state space explosion. 34
35 P1 P2 P3 P4 P5 P6 T4 T5 ( ) ( ) ( ) ( ) T4 T5 ( ) ( ) 35
36 Reachability Graph Fold all nodes with the same marking in the reachability tree ( ) ( ) T4 T5 ( ) ( ) The finite state automaton equivalent of the Petri net. It is possible to define the language generated by a Petri net. 36
37 Coverability trees Unbounded PNs Introduce a special symbol ω that corresponds to infinitely many tokens. P1 P2 P3 (1 0 w ) (0 1 w ) (1 0 0) (0 1 1) (0 0 0) (1 0 w ) (0 0 w ) 37
38 Coverability trees Unbounded PNs Introduce a special symbol ω that corresponds to infinitely many tokens. P1 P2 P3 (1 0 w ) (0 1 w ) (1 0 0) (0 1 1) (0 0 0) (1 0 w ) (0 0 w ) 38
39 Coverability trees Unbounded PNs Introduce a special symbol ω that corresponds to infinitely many tokens. P1 P2 P3 (1 0 w ) (0 1 w ) (1 0 0) (0 1 1) (0 0 0) (1 0 w ) (0 0 w ) 39
40 Coverability trees Unbounded PNs Introduce a special symbol ω that corresponds to infinitely many tokens. P1 P2 P3 (1 0 w ) (0 1 w ) (1 0 0) (0 1 1) (0 0 0) (1 0 w ) (0 0 w ) 40
41 Coverability trees Unbounded PNs Introduce a special symbol ω that corresponds to infinitely many tokens. P1 P2 P3 (1 0 w ) (0 1 w ) (1 0 0) (0 1 1) (0 0 0) (1 0 w ) (0 0 w ) 41
42 Linear Algebra P1 P2 P3 P4 Incidence matrix: W T 1 T 2 T P 1 P 2 P 3 p 4 Firing sequence: S1 T 1 T 2 T 3 T 1 T 3 Characteristic vector: S1 [212] T Fundamental equation: M i M 0 + WS1 42
43 Linear Algebra P1 P1 P2 P3 P2 P3 T4 P4 P5 P4 P5 T4 T4 M i M 0 + WS
44 Characteristic vectors can be composed through summation S T 1 S T 2 T 3 T 4 T 1 SS T 1 T 2 T 3 T 4 T 1 SS S + S S is a possible characteristic vector if at least one firing sequence S corresponds to it. S (1101) T is not a possible characteristic vector Several firing sequences may correspond to the same characteristic vector, e.g. T 1 T 2 T 3 and T 1 T 2 T 2 both correspond to S (1110) T 44
45 P-invariants Let F be a weighting vector for places (column vector) Let P(F) be the subset of P whose coefficients in F are nonzero. Then: P(F) is a conservative component iff a weighting vector F exists such that F T W 0 F T M i F T M 0 + F T WS The vector F is a P-invariant F T M i is a marking invariant 45
46 T-invariants Consider the characteristic vector S associated with the firing sequence S. T(S) is the support of S, i.e., the set of transitions appearing in S A set Dof transitions is a repetitive component iff a firing sequence S exists such that T(S) Dand WS 0 The vector S is a T-invariant 46
47 P- and T-invariants Only nonnegative P-invariants and T-invariants are considered (P-semi-flows and T-semi-flows). Minimal conservative components exist from which the other can be constructed by composition. Algorithms exists for determining the minimal sets (e.f. Farkas algorithm). Maple or Matlab 47
48 Example Unmarked Net P1 Marked Net P1 P2 P3 P2 P3 P4 P5 P4 P5 T4 T4 W
49 P-invariants (structural property): F (1,1,0,1,0) G (1,0,1,0,1) F + G (2,1,1,1,1) Conservative components (structural property): P(F) {P 1,P 2,P 4 } P(G) {P 1,P 3,P 5 } P(F+G) {P 1,P 2,P 3,P 4,P 5 } Marking invariants (marking dependent): m 1 + m 2 + m 4 1 m 1 + m 3 + m 5 1 2m 1 + m 2 + m 3 + m 4 + m
50 T-invariants (structural property): S (1,1,1,1) Repetitive sequences (marking dependent): S 1 T 1 T 2 T 3 T 4 S 2 T 1 T 3 T 2 T 4 50
51 Reduction Methods Idea: transform the net into a simpler net that preserves the properties of interest by successively applying a set of reduction rules. Different reduction rules preserve different properties boundedness and liveness invariants Semi-automatic method 51
52 Hierarchical Petri nets Substitution places Subnet Substitution transitions Subnet 52
53 Modeling Power Turing machines Extended Petri nets Petri Nets Finite state automata Bounded Petri nets 53
54 Nonautonomous PNs Petri Nets can be divided into: autonomous PNs logical model time is not involved answers the question: HOW nonautonomous PNs transition firing synchronized and/or timed timed models answers the question: WHEN 54
55 Nonautonomous PNs Synchronized Petri Nets events associated with transitions Timed Petri Nets time delays associated with places or transitions Interpreted Petri Nets Synchronized and timed PN + data processing part for computation of variables (actions) and transition conditions very similar to Grafcet Controlled Petri Nets transitions can be enabled and disabled by special control places 55 Supervisory Control Theory for PN
56 Projects Evaluate the analysis possibilities of one of the Petri net tools available in the public-domain. Develop a simple model and see what type of analysis that can be performed. Restrict yourself to standard (noncoloured Petri nets) Lotta?? Evaluate Design/CPN, one of the more widely spread tools for coloured Petri nets. Focus on modeling and simulation. 56
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