SE-561 Math Foundations Petri Nets - I I

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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

Review: Petri Nets

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

t1

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

High-Level Petri Nets: Motivation
• 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

High-Level Petri Nets: Motivation (cont.)
• 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

High-Level Petri Nets: Places
• 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

High-Level Petri Nets: Transitions
• In meaningful models, we need transitions to reason about the values of tokens:

Switching the traffic light:

Increasing variable:

Jiacun Wang

High-Level Petri Nets: Transitions (cont.)
• 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

High-Level Petri Nets: Definition
• 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

Transition Firing Conditions
• 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

Transition Firing Conditions (cont.)
• 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

Example 1
• 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

Example 2
• In the common-resource example, suppose the prioritised process is changed after every access:

Jiacun Wang

From High-Level Petri Nets to Ordinary Petri Nets: Places
• 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
• 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