An Introduction to Petri Nets

An Introduction to Petri Nets

Marjan SirjaniMarjan SirjaniFormal Methods LaboratoryFormal Methods Laboratory

University of TehranUniversity of Tehran

An Introduction to Formal Methods

Spring 2006

Petri Net ReferencesPetri Net References

Tadao Murata, Tadao Murata, Petri nets: properties, Petri nets: properties, analysis and applicationsanalysis and applications, , Proceedings of the IEEE, 77(4), 541-Proceedings of the IEEE, 77(4), 541-80, April 1989.80, April 1989.

Wil van der Aalst, Wil van der Aalst, Process Modeling Process Modeling Using Petri NetsUsing Petri Nets, lecture notes., lecture notes.

An Introduction to Formal Methods

Spring 2006

IntroductionIntroduction

Petri Nets: Graphical Petri Nets: Graphical andand Mathematical modeling toolsMathematical modeling tools graphical toolgraphical tool

visual communication aidvisual communication aid

mathematical toolmathematical tool state equations, algebraic equations, etcstate equations, algebraic equations, etc

concurrent, asynchronous, concurrent, asynchronous, distributed, parallel, nondeterministic distributed, parallel, nondeterministic and/or stochastic systemsand/or stochastic systems

An Introduction to Formal Methods

Spring 2006

HistoryHistory 19621962: C.A. Petri’s dissertation (U. : C.A. Petri’s dissertation (U.

Darmstadt, W. Germany)Darmstadt, W. Germany) 19701970: Project MAC Conf. on Concurrent : Project MAC Conf. on Concurrent

Systems and Parallel Computation (MIT, Systems and Parallel Computation (MIT, USA)USA)

19751975: Conf. on Petri Nets and related : Conf. on Petri Nets and related Methods (MIT, USA)Methods (MIT, USA)

19791979: Course on General Net Theory of : Course on General Net Theory of Processes and Systems (Hamburg, W. Processes and Systems (Hamburg, W. Germany)Germany)

19801980: First European Workshop on : First European Workshop on Applications and Theory of Petri Nets Applications and Theory of Petri Nets (Strasbourg, France)(Strasbourg, France)

19851985: First International Workshop on : First International Workshop on Timed Petri Nets (Torino, Italy)Timed Petri Nets (Torino, Italy)

An Introduction to Formal Methods

Spring 2006

Informal Definition

The graphical presentation of a Petri net is a bipartite graph

There are two kinds of nodesPlaces: usually model resources or

partial state of the systemTransitions: model state transition and

synchronization Arcs are directed and always connect

nodes of different types

An Introduction to Formal Methods

Spring 2006

ExampleExample

p2

t1p1

t2

p4

t3

p3

An Introduction to Formal Methods

Spring 2006

StateState

The state of the system is modeled by marking the places with tokens

A place can be marked with a finite number (possibly zero) of tokens

An Introduction to Formal Methods

Spring 2006

FireFire

A transition t is called enabled in a certain marking, if:For every arc from a place p to t, there

exists a distinct token in the marking An enabled transition can fire and

result in a new marking Firing of a transition t in a marking is

an atomic operation

An Introduction to Formal Methods

Spring 2006

Fire (cont.)Fire (cont.)

Firing a transition results in two things:

1. Subtracting one token from the marking of any place p for every arc connecting p to t

2. Adding one token to the marking of any place p for every arc connecting t to p

An Introduction to Formal Methods

Spring 2006

NondeterminismNondeterminism

Execution of Petri nets is Execution of Petri nets is nondeterministicnondeterministic

1.1. Multiple transitions can be enabled at Multiple transitions can be enabled at the same time, any one of which can the same time, any one of which can firefire

2.2. None are required to fire - they fire at None are required to fire - they fire at will, between time 0 and infinity, or not will, between time 0 and infinity, or not at allat all

An Introduction to Formal Methods

Spring 2006

ElevatorElevator

An Introduction to Formal Methods

Spring 2006

One PhilosopherOne Philosopher

An Introduction to Formal Methods

Spring 2006

Four PhilFour Phil

An Introduction to Formal Methods

Spring 2006

Four Phil (cont.)Four Phil (cont.)

An Introduction to Formal Methods

Spring 2006

Formal DefinitionFormal Definition

A classical Petri net is a four-tuple A classical Petri net is a four-tuple (P,T,I,O) satisfying the following (P,T,I,O) satisfying the following conditions:conditions:

1.1. P is a finite set of places, and T is a P is a finite set of places, and T is a finite set of functionsfinite set of functions

2.2. I:P×T→N is a function which I:P×T→N is a function which determines the input multiplicity for determines the input multiplicity for each input arceach input arc

3.3. O:T×P→N is a function which O:T×P→N is a function which determines the output multiplicity determines the output multiplicity for each output arcfor each output arc

An Introduction to Formal Methods

Spring 2006

ExerciseExercise

Model a fair traffic lightModel a fair traffic light