Inverse petri nets: properties and applications

INVERSE PETRI NETS:PROPERTIES AND APPLICATIONS Yang Shanlin ,Zhuo Jie ,Guo Jun Dept. of Computer Science and Information Engineering Hefei University of Technology ,AnHui, 230009, China

Abstract

conditions and conclusion to be proved, to find the proof process. With the aid of the concept of inverse

On the foundation of basic Petri net theory, the paper

petri nets presented by this paper, it is easy to solve

proposes the definitions of Inverse Petri Nets, discusses

previous problems.

the relations of inverse petri nets and its original petri nets, proves the theorem of transition enablity and the

THE FOUNDATION OF PETRI NET

full- reflection theorem, and outlines the applications of inverse petri nets. We find that the inverse Petri nets are a promising tool for deeling with some appli-

Petri nets are composed of the network elements of

cation problems. Using the concept of inverse Petri

places, transitions, arcs and tokens, etc.

nets and analysis methodology of reachability, it is A general toplogic structure of Petri nets is a 6 - tuple

easy to solve the design problems of initial state and

N= (P, T ,Pre,Post,K, W)

input sequences in discrete event control systems. The in which

concept of inverse Petri nets is meaningful for con-

P = {PI' P2 , ... , Pm} is a finite nonempty set of

structing the Petri net models of describing logic proof

places;

problems.

T = {t 1 , t2 , •.• , t n Keywords:

}

is a finite nonempty set of

transitions; PreCPXT is a set of pre-arcs(input arcs)of

Petri nets, Inverse Petri nets, Original Petri nets

transitions; PostCT X P is a set of post - arcs (output arcs) of transitions.

INTRODUCTION

K

is a m - column vector which is a mapping

Petri nets are a promising tool for describing and

of P upon the set N + of positive integers, and repre-

studying information processing systems that contain

sent the capacity of places. K (PJ is ith element which

parallel process which means the con currency of pro-

denotes the maximun number of tokens that Pi E P can

cess states and independent of transition enablity. Us-

hold.

ing Petri nets, we can analysis the dynamic process

is a mapping of set Pre U Post upon the set

W

N + of positive integer, named as weight function of

and structural properties of the system.

arcs. W ( (PH 1;»

In practice, except for system analysis, there are other

represent the number of tokens

removed from Pi E P ,when 1; E T fires, (p, ,1;) E Pre.

types of problems, such as design of the initial states and input functions (sequences) in control technolo-

W ( (1;, Pi) ) represent the number of tokens

gy. These types of problems are: given system Sand

removed into Pi E P, when transition ti E T fires, (ti ,

destination state X" to find the initial state Xo 1tnd in-

p) E Post.

pnT=0

put function (sequences) u(k) ,and to make the system to arrive the given destination state Xd from the initial state Xo ,with the action of input functions, An-

To describe and study the dynamic of process modelled

other types of problems are about logic proof: given

by petri nets, marking is leaded into Petri net theory ,

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and the rules of transition firing are definited. Mark-

V (Pi,t;)EPre

ing is a m - column vector composed of the token

others

numbers in all places. The strict transition rules with capacity constraint are as follows: where

(1) . A transition t is said to be enabled, if each input

place p of t is marked with at least W ( (p, t»

to-

kens, and each output place p of t will not overflow (no more than capacity limit) when t fires. ( 2). An enabled transition mayor may not fire, de-

Certainly, if the capacity limit of places is considered

pending on whether or not the event actually

,a m - vector K of capacities of places is needed, so

takes place.

that the net structure can be described completely.

(3). A firing of an enabled transition t removes W

« p, t»

Obviously, matrices A + ,A-and vector K correspond

tokens from each input place p of t, and

« t ,p»

tokens to each output place p of

to Petri net N = (P, T , Pre, Post, K, W) ,and ma"trix

( 4 ). A source transition without any input place is

transition fires. If the Petri net to be studied is pure

adds W

A denotes the changed value of tokens in places when

t.

net, then matrix A contains full structure information

unconditionally enabled.

of net, except for place capacity.

( 5). A sink transition without any output place consumes tokens, but does not produce any.

INVERSE PETRI NETS

From given initial marking Mo and transition rule, we can get firing sequences of transitions L (N ,Mo) and

This section presents the definitions of inverse Petri

sets of enabled transitions, reachable marking se-

nets and Original Petri nets at first, and then discusses

quences and sets of reachable marking vector R (N,

the relations between them.

Mo ) ,from which the dynamic properties of process Definition:

can be studied.

A 6-tuple,

By the way, using the complementary place transfor-

N - l= (PN-', T N-' ,PreN-: ,PoStN-' ,K N-', W N- ' )

mation ,a fin it - capacity net can be .transformed into

is a inverse net of a Petri net

an infinit - capacity net, then the weak transition rule

N= (PN, TN ,PreN ,PostN ,K N, W N),

is applicable.

if following equations exit:

To facilit the study of Petri nets theory and computer

PN-' =PN

aid analysis / design, matrices can be used to represent

TN-'=TN

Petri nets. There are three foundamental matrices as-

PreN- ' = { (Pi' t,) I (t; ,P;) E PostN ,V (t;, p;) E PostN}

sociated with Petri nets: token increasing matrix A + ,

PostN-· = {(t; ,Pi) I (Pi' t,) E PreN ' V (Pi ,t;) E PreN}

token reducing matrix A - ,and token changing matrix

KN- ' =KN

A (also named incidence matrix or event matrix).

W N- ' «Pi,t;» E WN=WN( (t; ,p;» E W N

A + , A - and A are n X m matrices, where n is cardi-

V (Pi,t,) EPreN- 'and V (t;,Pi)EPostN

nality of set T ,and m is cardinality of set P. The defi-

W N- ' «t; ,Pi» E WN=WN«P"t,» E W N

nition of their (i, j) elements are respectively as fol-

V (t;, Pi) E PostN-'and V (Pi' t.) E PreN

lows:

v (t;, p;) E Post

Net N is also called an original Petri net of net N - 1 •

others Following we will discuss the relations between the in-

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verse net and its original net.

ken reducing matrix and token changing matrix of Petri net xE {N,N-I} respectively.

An inverse Petri net is also a Petri net.

Equations (1), (2) and (3) are very useful, the If a Petri net N-I is an inverse net of petri net N , then

proof of them are as follows:

net N is also an inverse net of net N-I, which means they are co - in verse.

Proof: equation (1) at-. = (W N-' «4, p;» V (4 ,Pi) E-PostN-· "0 others

On graph, the relations between inverse Petri net and its original Petri net are very obvious: their places ,

... (4)

transitions and capacities of each place are corresponda N =(WN«P;'4» " 0

ing equal; the weight of each arc are corresponding equal; except for arrows of each arc. It is to say, inverse Petri net graph can be obtained from reversing

and

the arrow of each. arc in its original Petri net graph.

t.

... (5)

others

W N-' «4 ,p;» =WN«P; ,4»

V (4,p;) EPosk'and V (p;,4)EPreN ... (6)

An example is shown in Fig. 1f~

V (p;,4)EPreN

1'+ Similarly, we may prove equation (2) ;and from (1) and (2) we may get equation (3). From above discussion, we get the very simple but especially important relations of behaviour properties between inverse Petri net and its original Petri net. Further, we propose theorems and corollaries which are important foundations of inverse Petri net applications. Theorem 1. (enablity theorem of inverse Petri net)

tf

Ps

Given Petri net N and permitted marking M, if transi-

( b ) Fig. 1

tion t; E T N is enabled at marking M ,and after t; fires,

(a) Petri net N

marking M transforms to M' , then in inverse Petri net

(b)inverse Petri net N- 1 of net N

N-Iof net N, marking M' is permitted marking vector, transition 4 E T N is enabled, and after t; fires,

The relations between token increasing matrix, token

marking vector M' transforms to M.

reducing matrix and token changing matrix of inverse

Here, permitted marking vector means a marking vec-

petri net N- ' and its original petri net N are as fol-

tor, in which each element is not more than the ca-

lows: At-'=AN

pacity of corresponding place. (l)

Proof: In Petri net N, becase M is permitted marking vector, and 4 is enabled at M, under marking M, after

(2)

t; fires, marking M transforms to M' , so we get:

(3)

M::;;;K

where At , A; , Ax denotes token increasing matrix, to-

.................. (7)

.................. (8)

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•••..••••.•••••.•. (9i

Corollary 2: Given Petri net (N, Mo) ,if marking M. E R(N ,Mo) then MoE R(N- l ,M.) ,where net N- l is an inverse Petri net of net N. Proof: In Petri net N ,marking M. E R (N ,Mo) ,so that

from equation (9) and (10), we get:

there is at least one transition sequence TS by which .•. ••• .•• ••• ••• ••• ••. (11)

marking can transform from Mo to M.. From corollary 1, we can get: in net N- l ,the inverse net of N ,by the firing of TS- l

from equation (2),we get:

which is inverse transition sequence of TS, the mark•••.••.••..•••..••••• (12)

a~ =aN;-'

ing can transform from th
from equation Ol)and (2) ,we get: Theorem 2: (Full reflection theorem)

In Petri net N ,M. is a given destination marking, SM•

... ••. .•. •.• ••• ... ... (13)

is a set of all initial marking Mo, which satisfy M. E R (N, Mo) , then, in net N- l , the inverse net of N, the reachable marking set of initial marking M. satisfy R from equation (7), (8), (3)and (14) ,we get: In inverse Petri net N - l ,M' is permitted marking vec-

(N- l , M.) =SM,

tor, t; is enabled, under marking M' after t; fires, mark-

Proof: If there is a marking M' 0 E SM. ,that is M. E R

ing M' transforms to M. Where , ai, represent the ith column vector in matrix Ai of net x,xE {N ,N- l } ,y

(N, M ' 0), then from corollary 2 we get : M' 0 ER (N-

M.) ; on the other hand, if there is a marking M" 0 E R (N - l , M.) ,then from the co - inverse of

E{+,-, 0} ·

1,

Petri net and corollary 2, we can get: M. E R (N , Mo) ,that is MoESMo. so we get:R(N'-l,M.)=SM,

From theorem ·1 , we obtain following corollary easily: Corollary 1: Given Petri net N and permitted initial marking Mo, if there is an enabled transition sequence ••• ,td - l , t d ) ,and by which marking vec-

APPLICATION EXAMPLE

TS = {t l ,t Z ,

tor sequence MS

= (Mo. M l , ••. , M.- l' M.) is ob-

For a given Petri net (N ,Mo) ,we can analysis the net

tained. That is : t1

t2

properties using reachable graph(or covered graph) or

td

r' -1

Mo - M l _ · · · - M . - l-M. then, in inverse Petri net N- l of net N ,M., M..l , ... ,

the methods of solving state equation. There are general petri nets analysis problems, but in practice, there

Ml , Mo are all permitted marking vectors; regarding

are another type of problems need to be solved.

M. as initial marking, !here is an enabled transition sequence TS- l = (td , t d - l , •.. ,t Z , tl), by which marking

Problems about: given a petri net N and destination

sequence MS- l = (M. ,M.- l' .•• ,M l ,Mo) is obtained.

marking M., to find initial marking Mo and corre-

That is : td

rt- 1

t2

tl

sponding firing sequences. •

M. - M .-l - · · · - Ml -Mo where d is a limit positive integer which denotes the

In practice, especially in control technology, there are

order of firings.

usually these situations: a system law is known, and a destination state of the system is given, we are re-

The corollary 1 can be proved only by using theorem

quired to design a system's initial state and an input

1 d-times.

function, so as to make the system arrive the destina-

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tion state from the initial state under the action of the

conclusions) ,and use transitions to represent logic

input function. After modelling the system using Petri

proof steps, then the problem becomes: given initial

net, where places represent system states, transitions

marking Mo and destination marking M" , to design an

represent system state transition, and transition firing

appropriate Petri net N , which satisfies M., E (N ,Mo).

is associated with current state and input value, the

As we know, when solving these type of logic proof

problem can be addressed as : given a petri net N and

problems, we usually use analysis inference method,

a destination marking M., , to find permitted initial

from conclusion to conditions, and then from this

marking Mo and transition sequence TS, by which

analysis steps we can find correct proof process eas-

marking can transform to M., from Mo.

ily.

The steps for dealing with these problems are as fol-

In these kind of petri net models, from marking M" ,

lows:

construct a petri net N- 1 , to satisfy Mo E R (N-l,

M" ) ,and then construct inverse net N of petri net First construct the inverse petri net N-l of net N, then using analysis methods of petri net, in net N- 1 ,to get

N-l, thus the net N is the model of proposition proof process.

the reachable set R(N- 1 ,M.,) from initial marking M., and corresponding transition seq uence set TS (N- 1 , M,,). From theorem 2, we may know : all permitted initial :narking are in R (N- 1 ,M,,). From the co - inverse of Nand N- 1 , corollary 1 and corollary 2, se-

CONCLUSION From above discussion, we know inverse petri net is a

lecting MER (N- 1 , M.,) randomly as initial marking

petri net model of inverse process of the process re pe-

of net N ,then with the action of the inverse sequence

sented by its original petri net. Using· the concept of

of transiton sequence TSMETS(N-1,M,,) (TSM may

inverse petri nets and analysis methodology of reacha-

more than one), the marking of N can transform to

bility ,it is easy to solve the design problems of initial

M" from M.

state and input sequences in discrete event control systems. The concept of inverse Petri nets is meaningful

For real problems, it is needed to consider other requirements and some relevant evaluation criteria, so as

for constructing the Petri net models of describing logic proof problems.

to select initial marking and corresponding transition sequence optimally.

But we should pay attention on that the meanings of places and/or transitions in inverse petri net and those

At last, according to processing interpretafion 01 Petri

in its original petri net are properly different. Further

net, from the selected initial marking and transition

discussion of this problem is associated with the rela-

sequence, we can get the system initial state and input

tions of inverse Petri net interpretation and its original

function.

Petri net interpretation.

Problems about: net structure design under given

Reference

[1] Konig Quack, Petri - Netze in der Steuerungs-

marking Mo and destination marking M"

technik, Verlag Technik Berlin, 1988 In real work, especially in logic proof system, there

[2] Taodao Murata, Petri Nets: Properties, Analysis

are these problems: given initial condition and conclu-

and Applications ,Proceedings of the IEEE,vel77 No.

sion of a proposition, we· are required to draw proof

4.1989

process. If use places to represent conditions and conclusions

(including

middle

conditions

and

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