COMPOSITIONAL ANALYSIS FOR REGULARITY, LIVENESS

AND BOUNDEDNESS

Li Jiao

Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

Keywords: Analysis, boundedness, composition, liveness, Petri nets, regularity.

Abstract: By the linear algebraic representation of Petri nets, Desel introduced regularity property (Desel, 1992).

Regularity implies a sufficient condition for a Petri net to be live and bounded. All the conditions checking

the regularity of a Petri net are decidable in polynomial time in the size of a net (Desel and Esparza, 1995).

This paper proves that regularity, liveness and boundedness can be preserved after applying many

compositional operations to Petri nets. This means that, by applying these compositional operations, a

designer can construct complex nets satisfying regularity, liveness and boundedness properties from simpler

ones without forward analysis.

1 INTRODUCTION

As a graphical and mathematical tool, Petri nets

provide a uniform environment for modeling, formal

analysis, and design of discrete event systems. One

of the major advantages of using Petri net models is

that the same model can be used for the analysis of

behavioural properties and performance evaluation,

as well as for systematic construction of discrete-

event simulators and controllers (Zhou and

Venkatesh, 1999).

One net is regular if it satisfies the conditions of

the Rank Theorem described in algebraic methods

(Desel, 1992; Desel and Esparza, 1995). Regularity

is a sufficient condition for ordinary nets to be live

and bounded (Desel and Esparza, 1995). In general,

a system is live if all its operations are eventually

executable, starting not only from its initial state but

also from any reachable state. A system is bounded

if it has a finite number of states. In the terminology

of Petri nets, liveness requires the firability of every

transition starting from any reachable marking,

boundedness implies that the number of tokens

existing in every place will not exceed a certain limit.

For system designs specified in Petri nets, the

major approaches for verification include reach-

ability analysis, direct proving on the basis of

definitions, mathematical programming, characteri-

zation and property-preserving transformation. This

paper relates to the fourth approach, i.e., the

property-preserving transformation. In this approach,

the original net is assumed to satisfy some specific

properties and the transformation is required to

preserve these properties in the transformed net. The

advantage of this approach is that the transformed

net is automatically correct without the need of

forward verification.

Transformations on Petri nets may be roughly

classified into three groups, namely reduction,

refinement and composition. In the literature, there

has been much work related to transformations that

preserve liveness and/or boundedness (Berthelot,

1986; Berthelot, 1987; Esparza, 1994; Koh and

DiCesare, 1991; Suzuki and Murata, 1983; Valette,

1979; Souissi, 1991; Zhou, 1996; Huang, Jiao and

Cheung, 2005). The preservation of regularity has

not been considered. This paper introduces four

kinds of composition operations in terms of places

and transitions. They are: merging two places each

coming from different net; merging two places

coming from the same net; merging two transitions

each coming from different net; merging two

transitions coming from the same net. For each kind

of the four composition operations, this paper proves

that regularity, liveness and boundedness can be

preserved automatically or under some simper

conditions.

This paper is organized as follows: Section 2

presents some basics about Petri nets including

algebraic characterizations. Four compositional

operations in terms of places and transitions are

169

Jiao L. (2005).

COMPOSITIONAL ANALYSIS FOR REGULARITY, LIVENESS AND BOUNDEDNESS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 169-174

DOI: 10.5220/0001178901690174

Copyright

c

SciTePress

introduced and the preservation of regularity is

verified in Sections 3. Section 4 proves that liveness

and boundedness can be preserved under these

operations. In this section, an example is given to

illustrate some results of this paper. Some

concluding remarks are given in Section 5.

2 PRELIMINARIES OF PETRI

NETS

This section outlines the definitions, terminology

and properties as required in the paper.

A net is denoted by N = (P, T, F), where P is a

non-empty finite set of places, T is a non-empty

finite set of transitions with P ∩ T = ∅ and F ⊆ (P ×

T) ∪ (T × P) is a flow relation. The pre-set of x is

defined as

•

x = {y ∈ P ∪ T | (y, x) ∈ F} and the

post-set of x is defined as x

•

= {y ∈ P ∪ T | (x, y) ∈

F}. Similarly, for any subset of Y ⊆ P ∪ T,

•

Y (resp.,

Y

•

) denotes the union of

•

y (resp., y

•

) for all y ∈ Y. A

net N = (P, T, F) is said to be pure or self-loop-free

iff

•

x ∩ x

•

= ∅ ∀x ∈ P ∪ T. We just discuss pure

nets in this paper.

The incidence matrix V of a pure net N is a |P| × |T|

matrix whose element v

ij

at row p

i

and column t

j

is

denoted as follows: v

ij

= 1 if p

i

∈ t

j

•

; v

ij

= -1 if p

i

∈

•

t

j

;

and v

ij

= 0 if p

i

•

•

∪∉

jj

tt

.

A marking of a net N = (P, T, F) is a mapping

M: P → {0, 1, 2, …}. A place p is said to be marked

by M if M(p) > 0. A transition t is enabled or firable

at a marking M if for every p ∈

•

t, M(p) ≥ 1. A

transition t may be fired if it is enabled. Firing

transition t results in changing the marking M to a

new marking M', where M' is obtained by removing

one token from each p ∈

•

t and by putting one token

to every p ∈ t

•

. R(N, M

0

) denotes the set of all

markings reachable from the initial marking M

0

.

A transition t is said to be live in a Petri net (N,

M

0

) iff, for any M ∈ R(N, M

0

), there exists M' ∈

R(N, M) such that t can be fired at M'. (N, M

0

) is said

to be live iff every transition of N is live. A place p

is said to be bounded in (N, M

0

) iff there exists a

constant k such that M(p) ≤ k for all M ∈ R(N, M

0

).

(N, M

0

) is bounded iff every place of N is bounded.

For x ∈ P ∪ T, the cluster of x, denoted as [x],

is the smallest subset of P ∪ T satisfying three

conditions: (1) x ∈ [x]; (2) if p ∈ P ∩ [x] then p

•

⊆

[x]; and (3) if t ∈ T ∩ [x] then

•

t ⊆ [x]. N is said to

satisfy the rank-and-cluster property iff the rank of

its incidence matrix is less than the number of its

clusters by 1.

A net N is said to be connected iff every pair of

nodes (x, y) satisfies (x, y) ∈

∗−

∪ )(

1

FF

. A net N

is said to be strongly connected iff (x, y) ∈

*

F

, i.e.,

there exists a directed path from every node x to

every node y. A P-invariant (resp., T-invariant) of N

is a non-negative integer |P|-vector

α

(resp., |T|-

vector

β

) satisfying the equation

α

V = 0 (resp.,

V

T

β

= 0), where V is the incidence matrix of N. A

P-invariant

α

(resp., T-invariant

β

) of a net is called

semi-positive if

α

≥ 0 and

α

≠ 0 (resp.,

β

≥ 0 and

β

≠

0). The support of a semi-positive P-invariant

α

,

denoted by

〉

〈

α

, is the set of places p satisfying

α

(p)

> 0, and the support of a semi-positive T-invariant

β

,

denoted by

〉

〈

β

, is the set of transition satisfying

β

(t) > 0 (Desel and Esparza, 1995).

A net N is regular (Desel 1992) iff (1) N is

connected, (2) N has a positive P-invariant, (3) N has

a positive T-invariant, and (4) N satisfies the rank-

and-cluster property.

In N, a non-empty set of places D is said to be a

siphon (resp., trap) iff

•

D ⊆ D

•

(resp., D

•

⊆

•

D). A

siphon (resp., trap) is said to be minimal if it does

not properly contain any other siphon (resp., trap).

For more details, please refer to (Recalde,

Teruel and Silva, 1998; Silva, Teruel and Colom,

1998).

3 FOUR COMPOSITIONAL

OPERATIONS AND THE

PRESERVATION OF

REGULARITY

This section considers four compositional operations

in terms of places and transitions. Two of them are

very natural and can be found in the literature. The

other two operations are a little similar to those in

(Berthelot, 1987). Suppose the original nets are

regular. We will prove that, for two of the four

compositional operations, the regularity can be

automatically preserved. For the other two ones,

some simple conditions will be provided under

which the regularity can be preserved.

3.1 Merging a Pair Of Places From

Two Nets

COMPOSITION-BY-PLACE (composition via

merging a pair of places from two different nets):

Consider two disconnected ordinary nets N

1

= (P

1

∪

ICINCO 2005 - ROBOTICS AND AUTOMATION

170

{p

1

}, T

1

, F

1

) and N

2

= (P

2

∪ {p

2

}, T

2

, F

2

), where T

1

∩ T

2

= ∅ and F

1

∩ F

2

= ∅. Let N be composed from

N

1

and N

2

by merging the pair of places p

1

and p

2

into p

12

, that is, N = (P, T, F), where P = P

1

∪ P

2

∪

{p

12

}, T = T

1

∪ T

2

and F = F

1

∪ F

2

.

Theorem 1: Let N, N

1

and N

2

be defined in

COMPOSITION-BY-PLACE. Then, N is regular if

N

1

and N

2

are regular.

Proof: In order to show that N is regular, we

must prove that N is connected, has a positive P-

invariant and a positive T-invariant, and satisfies:

Rank(N) = |C

N

| – 1. The incidence matrices of N, N

1

and N

2

have the forms:

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

=

32

22

31

11

21

12

2

1

0

0

V

V

V

V

TT

p

P

P

V

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

31

11

1

1

11

V

V

T

p

PV

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

32

22

2

2

22

V

V

T

p

PV

(1) Since N

1

and N

2

are connected, it is obvious that

N is also connected.

(2) Since N

1

and N

2

have positive P-invariants,

there exist

α

1

and

α

2

such that

α

1

> 0,

α

2

> 0,

α

1

V

1

=

0 and

α

2

V

2

= 0. Let

α

= (

α

2

(p

2

)

α

1|P1

α

1

(p

1

)

α

2|P2

α

1

(p

1

)

α

2

(p

2

)). Then

α

> 0 and

α

V = (

α

2

(p

2

)

α

1|P1

V

11

+

α

1

(p

1

)

α

2

(p

2

)V

31

α

1

(p

1

)

α

2|P2

V

22

+

α

1

(p

1

)

α

2

(p

2

)V

32

)

=(

α

2

(p

2

)(

α

1|P1

V

11

+

α

1

(p

1

)V

31

)

α

1

(p

1

)(

α

2|P2

V

22

+

α

2

(p

2

)V

32

)) =(

α

2

(p

2

)

α

1

V

1

α

1

(p

1

)

α

2

V

2

)

= 0. This means that

α

is a positive P-invariant of N.

(3) Since N

1

and N

2

have positive T-invariants,

there exist

β

1

and

β

2

such that

β

1

> 0,

β

2

> 0,

V

1

T

1

β

= 0 and V

2

T

2

β

= 0. This means that V

11

T

1

β

=

0, V

31

T

1

β

= 0, V

22

T

2

β

= 0 and V

32

T

2

β

= 0.

Let

β

= (

β

1

β

2

), then V

T

β

=

T

TTTT

VVVV )(

232131222111

ββββ

+

= 0. That

is,

β

is a T-invariant of N.

(4) Since Rank(V

1

) = |C(N

1

)| − 1 and Rank(V

2

) =

|C(N

2

)| − 1 and |C(N

1

)| ≤ |P

1

| and |C(N

2

)| ≤ |P

2

|,

Rank(V

1

) < |P

1

| and Rank(V

2

) < |P

2

|. Hence, the

bottom row V

3i

is a linear combination of the other

rows of V

ii

and Rank(V

i

) = Rank(V

ii

) for i = 1 and 2.

This also implies that the bottom row of V is a linear

combination of the other rows of V and Rank(V) =

Rank(V

11

) + Rank(V

22

) = (|C(N

1

)| − 1) + (|C(N

2

)| − 1).

Since the only change in clustering after

COMPOSITION-BY-PLACE is that the two clusters

([p

1

] in N

1

) and ([p

2

] in N

2

) are merged to one, we

have |C(N)| = |C(N

1

)| + |C(N

2

)| −1. Thus, Rank(V) =

(|C(N

1

)| − 1) + (|C(N

2

)| − 1) = |C(N)| − 1. □

3.2 Merging Two Non-neighboring

Places In A Net

MERGE-N-PLACE (merging two non-neighboring

places in a net): Let a net N = (P

0

∪ {p

1

, p

2

}, T, F)

be a net and p

1

and p

2

satisfy: (

•

p

1

∪ p

1

•

) ∩ (

•

p

2

∪

p

2

•

) = ∅. Let N' = (P

0

∪ {p

12

}, T, F') be obtained

from N by merging the places p

1

and p

2

into a single

place p

12

, where F' = F ∪ {(t, p

12

) | (t, p

1

) ∈ F or (t,

p

2

) ∈ F} ∪ {(p

12

, t) | (p

1

, t) ∈ F or (p

2

, t) ∈ F} − {(t,

p

i

) | (t, p

i

) ∈ F, where i = 1, 2} − {(p

i

, t) | (p

i

, t) ∈ F,

where i = 1, 2}.

Note that since this paper just considers pure

and ordinary nets, we add some conditions to p

1

and

p

2

, i.e., just considering non-neighboring places to

be merged.

Theorem 2 below proposes a simple condition

under which the regularity can be preserved after

applying MERGE-N-PLACE.

Theorem 2: Let N and N' be involved in

MERGE-N-PLACE, where N = (P

0

∪ {p

1

, p

2

}, T, F)

and N' = (P

0

∪ {p

12

}, T', F'). Then, N' is regular if

the following conditions hold:

(1) N is regular.

(2) there exists a positive P-invariant of N α such

that α(p

1

) = α(p

2

).

(3) p

1

and p

2

belong to the same cluster, and

(4) there exists at least one P-invariant α such that

〉

〈

α

∩ {p

1

, p

2

} = {p

i

}, where i = 1 or 2.

Proof: The incidence matrices of N and N' have

the following forms:

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

=

2

1

0

2

1

0

V

V

V

p

p

P

V

T

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

=

21

0

12

0

'

VV

V

p

P

T

V

(1) Since N is regular, N is connected. It is obvious

that N' is also connected.

(2) Since there exists a positive P-invariant of N α

such that α(p

1

) = α(p

2

), let

'

α

= (α

|P0

α(p

1

)) and

''V

α

= α

|P0

V

0

+

α

(p

1

)(V

1

+ V

2

) = α

|P0

V

0

+

α

(p

1

)V

1

+

α

(p

2

)V

2

= 0. This means that

α

is a positive P-

invariant of N'.

(3) Since N is regular, N has a positive T-invariant.

Let

β

be a positive T-invariant of N, then V

T

β

= 0.

It is obvious that

'V

T

β

= 0. That is,

β

also is a T-

invariant of N'.

(4) Since p

1

and p

2

belong to the same cluster,

|C(N')| = |C(N)|. Suppose that there exists one P-

invariant

α

such that

〉

〈

α

∩ {p

1

, p

2

} = {p

2

}. Then,

the corresponding row of p

2

in V can be expressed as

COMPOSITIONAL ANALYSIS FOR REGULARITY, LIVENESS AND BOUNDEDNESS

171

a linear combination of the other rows of P

0

. Hence,

Rank(V') = Rank(V). This means that Rank(V') =

Rank(V) = |C(N)| − 1 = |C(N')| − 1. □

It is obvious from the proof of Theorem 2 that

when p

1

and p

2

belong to different cluster, N' must

be not regular if N is regular and Condition (3) holds.

3.3 Merging A Pair Of Transitions

From Two Nets

COMPOSITION-BY-TRANSITION (composition

via merging a pair of transitions from two different

nets): Consider two disconnected ordinary nets N

1

=

(P

1

, T

1

∪ {t

1

}, F

1

) and N

2

= (P

2

, T

2

∪ {t

2

}, F

2

),

where P

1

∩ P

2

= ∅, (T

1

∪ {t

1

}) ∩ (T

2

∪ {t

2

}) = ∅,

T

1

∩ {t

1

} = ∅, T

2

∩ {t

2

} = ∅ and F

1

∩ F

2

= ∅. Let

N be composed from N

1

and N

2

by merging the pair

of transitions t

1

and t

2

. That is, N = (P, T, F), where

P = P

1

∪ P

2

, T = T

1

∪ T

2

∪ {t

12

} and F = F

1

∪ F

2

.

Theorem 3: Let N, N

1

and N

2

be defined in

COMPOSITION-BY-TRANSITION. Then, N is

regular if N

1

and N

2

are regular.

Proof: Similar to that of Theorem 1.□

3.4 Merging Two Non-neighboring

Transitions In A Net

MERGE-N-TRANSITION (merging two

transitions in a net): Let a net N = (P, T

0

∪ {t

1

, t

2

},

F) be a net and satisfy the following conditions: (

•

t

1

∪ t

1

•

) ∩ (

•

t

2

∪ t

2

•

) = ∅. Let N' = (P, T

0

∪ {t

12

}, F')

be obtained from N by merging the transitions t

1

and

t

2

into a single place t

12

, where F' = F' = F ∪ {(t

12

, p)

| (t

1

, p) ∈ F or (t

2

, p) ∈ F} ∪ {(p, t

12

) | (p, t

1

) ∈ F or

(p, t

2

) ∈ F} − {(t

i

, p) | (t

i

, p) ∈ F, where i = 1, 2} −

{(p, t

i

) | (p, t

i

) ∈ F, where i = 1, 2}.

Note that since this paper just considers pure

and ordinary nets, we add some conditions to t

1

and

t

2

, i.e., just considering non-neighboring transitions

to be merged.

Theorem 4: Let N and N' be involved in

MERGE-N-TRANSITION, where N = (P, T

0

∪ {t

1

,

t

2

}, F) and N' = (P, T

0

∪ {t

12

}, F'). Then, N' is

regular if the following conditions hold:

(1) N is regular,

(2) there exists a positive T-invariant

β

such that

β

(t

1

) =

β

(t

2

),

(3) t

1

and t

2

belong to the same cluster, and

(4) there exists at least one T-invariant

β

such that

〉〈

β

∩ {t

1

, t

2

} = {t

i

}, where i = 1 or 2.

Proof: Similar to that of Theorem 2. □

4 PRESERVING LIVENESS AND

BOUNDEDNESS

Desel and Esparza have shown that the regularity

guarantees the existence of a live and bounded

marking. This section will decide if a given initial

marking M

0

ensures liveness and boundedness of (N,

M

0

), where N is the net obtained by applying the

four compositional operations defined in Section 3.

Lemma 1 below characterizes liveness and

boundedness of regular nets.

Lemma 1: (Desel and Esparza, 1995) Let N be

a regular net. Then, a marking M of N is live and

bounded iff it marks all minimal siphons of N.

In order to check whether the liveness and

boundedness of a marked regular net, it is important

to know whether all siphons are marked after

applying the four compositional operations. The

following propositions state the relationship of

siphons of nets of before and after transformation.

Proposition 1: Suppose that N, N

1

and N

2

are

defined in COMPOSITION-BY-PLACE. Let D be a

siphon of N. Then, D

i

= D ∩ P

i

is a siphon of N

i

if

p

12

∉

D, otherwise D

i

= (D ∩ P

i

) ∪ {p

i

} is a siphon

of N

i

, where i = 1 and 2.

Proof: Since D is a siphon of N,

•

D ⊆ D

•

. Since

P

1

∩ P

2

= ∅, T

1

∩ T

2

= ∅ and D = (D ∩ P

1

) ∪ (D ∩

P

2

) =D

1

∪ D

2

if p

12

∉

D, it is obvious that

•

D

1

⊆ D

1

•

and

•

D

2

⊆ D

2

•

. This means that D

i

= D ∩ P

i

is a

siphon of N

i

if p

12

∉

D. If p

12

∈ D, then

•

p

12

in N =

(

•

p

1

in N

1

) ∪ (

•

p

2

in N

2

) and p

12

•

in N = (p

1

•

in N

1

) ∪

(p

2

•

in N

2

). Since

•

D ⊆ D

•

and T

1

∩ T

2

= ∅,

•

((D ∩

P

i

) ∪ {p

i

}) ⊆ ((D ∩ P

i

) ∪ {p

i

})

•

, this means that (D

∩ P

i

) ∪ {p

i

}

is a siphon of N

i

, where i =1 and 2. □

Proposition 2: Suppose that N and N' are

defined in MERGE-N-PLACE. Let D' be a siphon of

N'. Then, D' is a siphon of N if p

12

∉D', otherwise D

= D' ∪ {p

1

, p

2

} − {p

12

} is a siphon of N.

Proof: Since D' is a siphon of N',

•

D' ⊆ D'

•

in

N'. It is obvious that

•

D' ⊆ D'

•

in N if p

12

∉D'. This

means that D' is a siphon of N. If p

12

∈ D', then

•

p

12

in N' = (

•

p

1

∪

•

p

2

) in N and p

12

•

in N' = (p

1

•

∪ p

2

•

) in

N. Since

•

D' ⊆ D'

•

,

•

(D' ∪ {p

1

, p

2

} − {p

12

}) ⊆ (D' ∪

{p

1

, p

2

} − {p

12

})

•

, this means that D' ∪ {p

1

, p

2

} −

{p

12

}

is a siphon of N. □

Proposition 3: Suppose that N, N

1

and N

2

are

defined in COMPOSITION-BY-TRANSITION. Let

D be a siphon of N. Then, D

i

= D ∩ P

i

is a siphon of

N

i

, where i = 1 or 2.

Proof: Since P

1

∩ P

2

= ∅, T

1

∩ T

2

= ∅ and D

= (D ∩ P

1

) ∪ (D ∩ P

2

) =D

1

∪ D

2

, it is obvious that

•

D

1

⊆ D

1

•

and

•

D

2

⊆ D

2

•

because of

•

D ⊆ D

•

. □

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172

Proposition 4: Suppose that N and N' are

defined in MERGE-N-TRANSITION. Let D' be a

siphon of N'. Then, D' is a siphon of N if {t

12

}

∉

•

D'.

Proof: Since D' is a siphon of N',

•

D' ⊆ D'

•

in

N'. It is obvious that

•

D' ⊆ D'

•

in N if {t

12

}

∉

•

D', i.e.,

D' is a siphon of N if {t

12

}

∉

•

D'. □

Theorem 5: Suppose that N

and N

i

are defined

in COMPOSITION-BY-PLACE. Let M

i

be an initial

marking of N

i

and M be obtained from M

i

such that

M(p) = M

i

(p) if p ∈ P

i

− {p

1

, p

2

} and M(p

12

) =

Max{M

1

(p

1

), M

2

(p

2

)}. Then, (N, M) is live and

bounded if N

i

is regular and (N

i

, M

i

) is live and

bounded, where i = 1 and 2.

Proof: Since N

1

and N

2

are regular, by Theorem

3.1, N is regular. Since (N

1

, M

1

) and (N

2

, M

2

) are live

and bounded, according to Lemma 1, all siphons of

N

1

and N

2

are marked. Let D be a siphon of N. By

Proposition 1, if p

12

∉ D, then D

i

= D ∩ P

i

is a

siphon of N

i

. Hence, D

i

is marked. If p

12

∈ D, then

(D ∩ P

i

) ∪ {p

i

}

is a siphon of N

i

by Property 1 and

thus (D ∩ P

i

) ∪ {p

i

} is marked. In this case, if p

i

is

marked for i = 1 or 2, then p

12

is marked, otherwise,

D ∩ P

i

is marked. This means that all siphons of N

are marked. Thus, (M, N) is live and bounded

according to Lemma 1. □

Theorem 6 Suppose that N and N' are defined

in MERGE-N-PLACE, and M is an initial marking

of N. Let M' be obtained from M such that M'(p) =

M(p) if p ∈ P − {p

1

, p

2

} and M'(p

12

) = Max{M(p

1

),

M(p

2

)}. Then, (N', M') is live and bounded if the

following conditions hold:

(1) N is regular,

(2) p

1

and p

2

belong to the same cluster,

(3) there exists a positive P-invariant of N α such

that α(p

1

) = α(p

2

),

(4) there exists at least one P-invariant α such that

〉〈

α

∩ {p

1

, p

2

} = {p

i

} for i = 1 or 2, and

(5) (N, M) is live and bounded.

Proof: By Conditions (1)-(4) and Theorem 2,

N' is regular. Condition (5) implies that all siphons

are marked according to Lemma 1. Consider any

siphon D' of N'. By Proposition 2, if p

12

∉D', then D

= D' is a siphon of N. Hence, D is marked. If p

12

∈

D', then (D ∩ P) ∪ {p

1

, p

2

}

is a siphon of N and thus

is marked. This means that all siphons of N' are

marked. Thus, (N, M) is live and bounded according

to Lemma 1. □

The example below shows the application of

some results obtained in Section 3 and this section.

Example 1: In Figure 1, both (N

1

, M

1

) and (N

2

,

M

2

) are live and bounded marked graphs. Of course,

N

1

and N

2

are regular. After applying

COMPOSITION-BY-PLACE to them, p

1

and p

2

are

merged into r

1

and (N, M) shown in Figure 2 is

obtained. By Theorem 1 and Theorem 5, N is regular

and (N, M) is live and bounded. After applying

MERGE-N-PLACE to (N, M), p

3

and p

4

are merged

into r

2

and (N', M') shown in Figure 3 is obtained.

Since, in N, p

3

and p

4

belong to the same cluster and

{p

3

, p

12

, p

13

} is a P-component, by Theorem 2 and

Theorem 6, N' is also regular and (N', M') is live and

bounded.

Theorem 7: Suppose that N

and N

i

are defined

in COMPOSITION-BY-TRANSITION. Let M

i

be

an initial marking of N

i

and M be obtained from M

i

such that M(p) = M

i

(p) if p ∈ P

i

. Then, (N, M) is live

and bounded if N

i

is regular and (N

i

, M

i

) is live and

bounded, where i = 1 and 2.

Proof: Similar to the proof of Theorem 5. □

p

11

p

12

p

13

t

11

t

12

t

13

p

3

p

1

(N

1

, M

1

)

p

4

t

21

p

21

p

23

p

22

t

22

t

23

p

2

(N

2

, M

2

)

Figure 1: Two live and bounded Petri nets

p

11

p

12

p

13

t

11

t

12

t

13

r

1

p

3

p

4

t

21

p

21

p

23

p

22

t

22

t

23

Figure 2: Petri net (N, M) obtained from Figure 1

by merging p

1

and p

2

into r

1

.

p

11

p

12

p

14

t

11

t

12

t

13

r

2

r

1

t

21

p

21

p

25

p

23

t

22

t

23

Figure 3: Petri net (N', M') obtained from Figure 2

by merging p

3

and p

4

into r

2

.

COMPOSITIONAL ANALYSIS FOR REGULARITY, LIVENESS AND BOUNDEDNESS

173

Theorem 8: Suppose that N and N' are defined

in MERGE-N-TRANSITION. Let M be an initial

marking of N and M' be obtained from M such that

M'(p) = M(p) for any p ∈ P. Then, (N', M') is live

and bounded if the following conditions hold:

(1) N is regular,

(2) t

1

and t

2

belong to the same cluster,

(3) there exists a positive T-invariant

β

such that

β

(t

1

) =

β

(t

2

),

(4) there exists at least one T-invariant

β

such that

〉〈

β

∩ {t

1

, t

2

} = {t

i

} for i = 1 or 2, and

(5) (N, M) is live and bounded, and

(6) all input places of t

1

and t

2

are marked or every

minimal siphon D' of N' with t

12

∈

•

D' is marked.

Proof: By Conditions (1)-(4) and Theorem 4,

N' is regular. Condition (5) implies that all siphons

are marked according to Lemma 1. Consider any

siphon D' of N'. By Property 4, D' is a siphon of N if

{t

12

} ∉

•

D' and thus D' is marked. By Condition (6),

D' is marked if t

12

∈

•

D'. This means that all siphons

of N' are marked. Hence, (N, M) is live and bounded

according to Lemma 1. □

5 CONCLUSION

This paper studied four compositional operations in

terms of place and transition for pure and ordinary

nets and showed that regularity, liveness and

bounded-ness can be preserved automatically or

under some simper conditions. These compositional

operations are quite natural. COMPOSITION-BY-

PLACE and COMPOSITION-BY-TRANSITION

are usually be used to obtain more complex nets

from some subnets. Liveness and boundedness

preservations on the two operations for different

subclasses of Petri nets have been studied under

different conditions. Our results are based on the

regularity preservation. MERGE-N-PLACE and

MERGE-N-TRANSITION are two operations used

in a net, a little similar to (Berthelot, 1987). Of

course, these results that this paper are contributed

are new and can accommodate the design of

complex systems.

ACKNOWLEDGEMENTS

I would like to thank the anonymous referees for

their helpful comments. The research was funded by

the National Natural Science Foundation of China

under Grants No. 60473007 and No. 60421001.

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