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# se-561 math foundations petri nets - ii - PowerPoint PPT Presentation

SE-561 Math Foundations Petri Nets - I I. Dr. Jiacun Wang Department of Software Engineering Monmouth University. Review: Petri Nets. A Petri net N is a tuple N = { P , T , I , O , M 0 }, where P is a finite set of places, graphically represented by circles

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### SE-561Math FoundationsPetri Nets - II

Dr. Jiacun Wang

Department of Software Engineering

Monmouth University

Jiacun Wang

A Petri net N is a tuple N = {P, T, I, O, M0}, where

• P is a finite set of places, graphically represented by circles

• T is a finite set of transitions, graphically represented by boxes

• Places P and transitions T are disjoint (P ∩T = f),

• I: P × T  N (N = {0, 1, 2, …}) is the pre-incidence function representing input arcs,

• O: T × P  N (N = {0, 1, 2, …}) is the post-incidence function representing output arcs,

• M0 : P N is the initial marking representing the initial distribution of tokens.

Jiacun Wang

p1

p2

t2

t3

Review: Transition Firing

• A transition t is enabled at marking Mi if and only if

Mi≥ I(t)

• Let E(Mi) be the set of all transitions enabled at Mi. Then t E(Mi).

• Suppose that the firing of t takes the Petri net from Mi to Mj. Then

Mj= Mi-I(t) + O(t)

Denoted by Mi[t>Mj

• Example

P = {p1, p2}

T = {t1, t2, t3}

I(t1) = (1, 1), I(t2) = (2, 0), I(t3) = (0, 2)

O(t1) = (1, 0), O(t2) = (0, 1), O(t3) = (0, 1)

M0 = (2, 1)

E(M0) = {t1, t2}

M0[t1> M1 where M1= (3, 1)

Jiacun Wang

• Up to now, we allowed places to be occupied by only “black” tokens.

• Traffic light example: Three lights, one color per light, one place for every light.

• Suppose we have one light that can be either red or green.

• Two attempts at modeling this situation:

Jiacun Wang

• If we had not just black tokens, but colored ones (e.g. red, green), we could construct a more natural model:

• More generally, we could allow arbitrary values as tokens, e.g. to model numeric variables:

green token

red token

1

5

Jiacun Wang

• A general solution is to assign a type to every place, i.e. a set of token values that are permitted on the place:

• In general, a place may contain a multiset of its type.

Jiacun Wang

• In meaningful models, we need transitions to reason about the values of tokens:

Switching the traffic light:

Increasing variable:

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• Suppose we have two processes competing for a common resource. If both try to access the resource simultaneously, there is a ‘referee’ who decides which process should have priority over the other:

Jiacun Wang

• A high-level Petri net (HL-net) is a tuple N = (P, T, I, O, V, S, C, M0), where

• P, T, I, O are as usual;

• V is a set of token values;

• S: P  2Vis a type assignment for places;

• C(t) is the firing condition of transition t (see next slide).

• M0 : P × V  N is the initial marking.

Jiacun Wang

• A firing condition decides which tokens may flow out of the pre-places and into the post-places of a transition.

• Formally, if we let t be the sum of the arc weights leading into and out of t, then the signature of Ct is Ct : Vt ! {false, true}.

• In figures, we place variable names onto the arcs and equip transitions with boolean expressions over these variables, like this:

Jiacun Wang

• If a particular assignment of token values to variables evaluates to true, then the transition may fire under that assignment.

• Firing under some assignment is possible if for every pre-place p, p contains the token values assigned to the variables that are on the arc from p to t.

• Firing removes those tokens and puts corresponding tokens on the post-places.

• The assignment must respect the types; e.g. if v is the variable on the arc from place p to transition t, then v must be assigned to some value from S(p).

Jiacun Wang

• For instance, in the following example, the transition could fire under the assignments

(‘x = 1, x’ = 2), (‘x = 2, x’ = 3), (‘x = 3, x’ = 4), (‘x = 4, x’ = 5).

• In the given marking, we can remove the 2 token and replace it by a 3 token.

Jiacun Wang

• In the common-resource example, suppose the prioritised process is changed after every access:

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• High-level nets allow easier modeling, but they are equally expressive, provided that the set of token values is finite.

• For each high-level place p, create an ordinary place pvfor each v S(p).

• If M0(p, v) = k, then put k initial tokens on the ordinary place pv.

Jiacun Wang

• For each high-level transition t, create an ordinary transition tafor each assignment under which t may fire.

• If (p, t) is a high-level arc with variable x, connect pvto tain the ordinary net, where v is the value of x in a.

• Arcs from transitions to places are treated analogously.

Jiacun Wang

Final Exam Transitions

• Open book, open slides, open …

• No laptop

• Set theory: Venn diagram

• Functions: 1-to-1, onto.

• Graph theory: Graph matrix, Euler paths/circuits, Hamilton paths/circuits

• Petri nets: Given a Petri net, write its formal definition and list all reachable markings.

• Petri nets: Petri net modeling according to specification

Jiacun Wang