Petri Nets Lecturer: Roohollah Abdipour

Agenda Introduction Petri Net Modelling with Petri Net Analysis of Petri net 2

Introduction Techniques for Evaluating SW / HW: – Simulation – Testing – Deductive Verification – Model Checking 3

Introduction (2) Simulation & Testing: – Needs experiments before system deployment – Simulation is performed on an abstraction of a model of the system – Testing is performed on the real system. – It is very diffucult to evaluate all possible interactions / faults / behaviors / states – 4

Introduction (3) Deductive Verification: – Validate a system’s performance based on deductive rules – Applicable to infinite-state systems – A time-consuming process – Needs expert people with good experiments – Usually is used for critical systems (e.g., security protocols) 5

Introduction (4) Model Checking: – An automated technique to evaluate finite-state systems – Perform a thorough search in the state space of the system in order to determine some properties of the system. 6

Introduction (5) The process of Model Checking: Modeling: Transform a design to formalism (e.g., a state-transition graph of the system) Verification: Perform a thorough search in the state space of the system in order to determine whether some properties of the system are true or not. Is performed automatically 7

Petri Net A ‘Petri Net’ was introduced by a German mathematician and used to model a system – Condition: a Boolean description of the state of the system – Event: an action that depends on the state of the system Petri net is a graphical and mathematical tools for the analysis of discrete event dynamic systems. – The system model is represented by a set of conditions and a set of events – A condition is represented by a place and an event by transition. 8

Petri Net: Building Blocks 9 place counter transition Place for user input Basic Elements PN consists of three types of components: places (circles), transitions (rectangles) and arcs (arrows): –Places represent possible states of the system; –Transitions are events or actions which cause the change of state; And –Every arc simply connects a place with a transition or a transition with a place. arcs inhibitor

Petri Net Building Blocks: An example: 10

Petri Net: Marks (Tokens) A marking M is an assignment of tokens (dots) to places of a petri net. Dots are placed in the circle to represent place in which its condition is satisfied –e.g. a resource is available or operation in a process The marking can be represented as M = (1,1,0) 11

Petri Net: Marks (Tokens) A place can have more than one token and therefore can represent a queue – (e.g. a buffer holding several processes). 12

Petri Net : Firing a transition Executing a marked Petri net causes the number and positions of the token to change. The rules for the execution are: – A transition is enabled if all its input places contain at least one token. – Any enabled transition may fire – Firing of a transition results in one token being removed from each of its input places and being deposited of its output places – Execution halts when there are no enabled transitions. – Each time a transitions fires, the marking of the petri net will change. 13

Petri Net : Example 14

Petri Net: Formal Definition A Petri net (PN) is a 5 tuple PN (P,T,IN,OUT,M) where: P = {p1,p2,....,p~} is a finite set of places, T = {t 1, t 2, …,t n } is a finite set of transitions IN:(PxT)S OUT:(TxP)S M:Marking vector 15

Petri Net: Formal Definition (cont’d) IN are input functions defining directed arcs from places to transitions OUT are output functions defining directed arcs from transitions to places S is a set of all nonnegative integers k such that: If k = 1 a directed arc is drawn without a label If k > 1 a directed arc is drawn with label k. If k = 0 no arc is drawn. 16

Petri Net: Formal Definition (cont’d) An example: 17

Petri Net: Firing Rules for Transitions A specific transition t i is said to be firable or enabled if each input place p i is marked with at least w(p i,t i ) tokens where w(p i,t i ) is the weight of the arc from p i to t i. An enabled transition may or may not fire depending on whether or not the event actually takes place. The firing of an enabled transition t i removes w(p i,t i ) tokens from each input place p i of t i, and adds w(p j,t i ) tokens to each output place p j of t i where w(p i,t i ) is the weight of the arc from input place p i to t i, and w(p j,t i ) is the weight of the arc from t i to output place p j 18

Petri Net: Firing Rules for Transitions 19

Petri Net: Firing Rules for Transitions Firing Examples 20

PN: Firing Rules for Transitions Firing Examples 21

PN: Firing Rules for Transitions Firing of a sequence of transitions: 22

PN: Firing Rules for Transitions Firing of a sequence of transitions: 23

PN: Firing Rules for Transitions Firing of a sequence of transitions: 24

PN: Firing Rules for Transitions Firing of a sequence of transitions: 25

PN: Change of States (1) is denoted by a movement of token(s) (black dots) from place(s) to place(s); and is caused by the firing of a transition. The firing represents an occurrence of the event or an action taken. The firing is subject to the input conditions, denoted by token availability. 26

PN: Change of States (2) A transition is firable or enabled when there are sufficient tokens in its input places. After firing, tokens will be transferred from the input places (old state) to the output places, denoting the new state. Note that the examples are Petri nets representation of a finite state machine (FSM). PNs are much more powerful to model systems beyond FSMs. 27

Modelling with petri net Petri net is mainly used for modelling. –Many systems can be modelled by petri net. The system may be of many different kinds like computer hardware, computer software, physical system and so on. –Petri net is used to model the occurrence of various events and activities in the system. 28

Modelling with petri net Sequential Execution: 29

Modelling with petri net – Concurrency: two events could take place in parallel, without interfering with one another. 30

Modelling with petri net Synchronisation: By having two places as input conditions to the same transition, these two conditions become synchronised. 31

Modelling with petri net Example : Concurrency and Synchronisation 32

Modelling with petri net Confilincts: Two transitions t 1 and t 2 are said to be conflict if either t 1 or t 2 can occur but not both of them, –i.e., when there is one place that is acting as input condition to a number of transitions. –Therefore, only one transition can be fire. 33

Modelling with petri net Merging: 34

Modelling with petri net Example: Font Selection 35 Bold on Bold off User presses bold T1 T2 Italic on User presses italic T3 T4 User presses italic

Modelling with petri net Another Example: a finite-state machine – Consider a vending machine It accepts either nickels or dimes Sells 15c or 20c candy bars The vending machine can hold up to 20c Coin return transitions are omitted the next slides are the state diagram of this vending machine which represented by the Petri net Any finite-state machine (or its state diagram) can be modeled with a state machine. 36

Modelling with petri net Another Example: a finite-state machine 37

© LJMU, 2004CY2003- Week 738 Modelling with petri net Mutual exclusion – Recall: Mutual exclusion is a technique of defining entry and exit code so that at most one process is allowed to access the critical region at the same time. The idea is that no process is allowed to enter the critical region, unless it checks that no other process is executing its own critical region. – Mutual exclusion can be represented in Petri net using conflict.

© LJMU, 2004CY2003- Week 739 Mutual exclusion The place s represents the permission to enter the critical region and there must be token in p 1 and p 2. – If both processes want to enter the critical region simultaneously, then transition t 1 and t 2 are in conflict – i.e., only one of them can fire.

© LJMU, 2004CY2003- Week 740 Producer consumer problem bounded buffer

© LJMU, 2004CY2003- Week 741 Analysis of Petri net Petri nets are capable of modelling a large variety of systems and properly representing the interactions between the various actions which can occur. – The strength of Petri net is its capability to model the system. However, modelling by itself is of little use and its necessary to analyse the modelled system. This could lead to important insights into the behaviour of the modelled system.

© LJMU, 2004CY2003- Week 742 Reachibility Given a Petri net, one would like to know which marking M r can be reached from an initial marking M 0. Example: For the Petri net of the bounded buffer consumer producer problem, M 0 = (1, 0, 1, 0, 0, n) and M 1 = (0, 1, 1, 0, 0, n) – is immediately reachable from the marking of M 0.

© LJMU, 2004CY2003- Week 743 Analysis of Petri net : Safeness A place in a Petri net is safe if the number of tokens in that place never exceeds one. – A petri net is same if all its places are safe. what does this Petri net do? Is it safe?

© LJMU, 2004CY2003- Week 744 Analysis of Petri net : Boundedness A place is k-safe or k-bounded if the number of tokens in that place cannot exceed an integer k. – Therefore a place is 1-bounded is simply a safe place. Example: – The Petri net for the producer/consumer problem with a bounded buffer, is it safe? Bounded? – The Petri net for the producer/consumer problem with a unbounded buffer, is it bounded? Why?

© LJMU, 2004CY2003- Week 745 Analysis of Petri net : Conservation A Petri net with an initial marking M 0, is strictly conservative, – if for all the reachable marking, the total number of tokens in each marking is exactly the same as the initial marking. A Petri net can be used to model resource allocation systems. In this systems some tokens may represents the resources. – Therefore, in such Petri nets conservation is an important property such that resources can neither be created nor destroyed.

© LJMU, 2004CY2003- Week 746 Analysis of Petri net : Liveness Another problem that could occur in resource allocation is deadlock. – A deadlock in a Petri net is a transition (or a set of transition) which cannot fire. – A transition is alive if it is not deadlock. – A transition is live in a marking M if it is potentially fireable in every marking in the Petri net.

© LJMU, 2004CY2003- Week 747 Exercises For the above Petri net, indicate whether each is bounded, live, and conservative or not and show why?

© LJMU, 2004CY2003- Week 748 Summary Petri net – concepts – Petri net representation – Firing a transition – Marks (Tokens) Modeling with Petri net – Events and condition