**2.2. Invariants of a Petri Net**

There are two types of invariants in a Petri Net; *Marking Invariants*, also called P-invariant and *Firing Invariant*, also called T-invariant [4].

#### **Definition 1.** (P-invariant)

Marking Invariants illustrate the conservation of the number of tokens in a subset of places of a Petri Net.

A vector, denoted Y, which has a dimension equal to the number of places of a Petri Net is a P-invariant, if and only if it satisfies the following equation:

$$Y^t \times W = \stackrel{\rightarrow}{0}, \qquad Y \neq \stackrel{\rightarrow}{0}. \tag{2}$$

From Equation 1, we deduce that if *Y* is a P-invariant, then for a given marking, denoted *mi*, obtained from an initial marking *m*0, we have:

$$\mathbf{Y}^t \times \mathfrak{m}\_i = \mathbf{Y}^t \times \mathfrak{m}\_0 = k\_\prime \quad k \in \mathbb{N}^\*. \tag{3}$$

This equation represents an invariant marking, it means that if Y is a P-invariant of Petri Net then the transpose of the vector Y multiplied by the marking vector *mi* of the Petri Net is an integer constant regardless of the *mi* marking reachable from the initial marking *m*0. All the places for which the associated component in the P-invariant is nonzero, is called the conservative component of the Petri Net.

#### **Definition 2.** (T-invariant)

A nonzero vector of integers *θ* of dimension | *T* | ×1 is a T-invariant of Petri Net if and only if it satisfies the following equation:

$$W \times \theta = \stackrel{\rightarrow}{0}.\tag{4}$$

From Equation 1, the evolution from a marking *mi* to a sequence whose characteristic vector *θ* back the graph to same marking *mk* = *mi*. The set of transitions for which the associated component in the T-invariant is nonzero is called the support of T-invariant. A T-invariant corresponding to a firing sequence is called feasible repetitive component.

#### **Definition 3.** (Consistent Petri Net)

A Petri Net is said *consistent* if it has a T-invariant *θ* covering all transitions of graph. A Petri Net which has this property is said *repetitive*.

The graph reaches a periodic regime when there is a firing sequence achievable with *θ* as characteristic vector.

#### **Definition 4.** (Conservative Petri Net)

4 Will-be-set-by-IN-TECH

firing sequence from a marking *mi* to the marking *mk*, then a fundamental equation reflecting

*S* is the characteristic vector of the firing sequence *S*. In Figure 1, the firing sequence *S* = {*n*2},

There are two types of invariants in a Petri Net; *Marking Invariants*, also called P-invariant and

Marking Invariants illustrate the conservation of the number of tokens in a subset of places of

A vector, denoted Y, which has a dimension equal to the number of places of a Petri Net is a

From Equation 1, we deduce that if *Y* is a P-invariant, then for a given marking, denoted *mi*,

This equation represents an invariant marking, it means that if Y is a P-invariant of Petri Net then the transpose of the vector Y multiplied by the marking vector *mi* of the Petri Net is an integer constant regardless of the *mi* marking reachable from the initial marking *m*0. All the places for which the associated component in the P-invariant is nonzero, is called the

A nonzero vector of integers *θ* of dimension | *T* | ×1 is a T-invariant of Petri Net if and only if

*<sup>W</sup>* <sup>×</sup> *<sup>θ</sup>* <sup>=</sup> <sup>→</sup>

From Equation 1, the evolution from a marking *mi* to a sequence whose characteristic vector *θ* back the graph to same marking *mk* = *mi*. The set of transitions for which the associated component in the T-invariant is nonzero is called the support of T-invariant. A T-invariant

A Petri Net is said *consistent* if it has a T-invariant *θ* covering all transitions of graph. A Petri

corresponding to a firing sequence is called feasible repetitive component.

0 , *<sup>Y</sup>* �<sup>=</sup> <sup>→</sup>

*<sup>Y</sup><sup>t</sup>* <sup>×</sup> *mi* <sup>=</sup> *<sup>Y</sup><sup>t</sup>* <sup>×</sup> *<sup>m</sup>*<sup>0</sup> <sup>=</sup> *<sup>k</sup>*, *<sup>k</sup>* <sup>∈</sup> **<sup>N</sup>**∗. (3)

*<sup>Y</sup><sup>t</sup>* <sup>×</sup> *<sup>W</sup>* <sup>=</sup> <sup>→</sup>

the characteristic vector is equal to *S<sup>t</sup>* = (0, 1, 0, 0), and from marking *m<sup>t</sup>*

*mk* = *mi* + *W* × *S*. (1)

<sup>1</sup> = (0, 0, 0, 3, 0, 0) by firing of the transition *n*2, after a stay of 2 time

<sup>0</sup> = (0, 3, 0, 0, 3, 0), is

0 . (2)

0 . (4)

the dynamic behavior of Petri Net, is obtained:

reached the marking *m<sup>t</sup>*

units of tokens in the places *P*<sup>2</sup> and *P*5.

*Firing Invariant*, also called T-invariant [4].

obtained from an initial marking *m*0, we have:

conservative component of the Petri Net.

**Definition 2.** (T-invariant)

it satisfies the following equation:

**Definition 3.** (Consistent Petri Net)

Net which has this property is said *repetitive*.

P-invariant, if and only if it satisfies the following equation:

**2.2. Invariants of a Petri Net**

**Definition 1.** (P-invariant)

a Petri Net.

A Petri Net is said conservative if all places in the graph form a conservative component.

The Petri Nets considered here are *consistent* (*i.e.,* there exists a T-invariant *θ* covering all transitions: {*q* ∈ *T*|*θ*(*q*) > 0} = *T*) and *conservative* (*i.e.,* there exists a P-invariant *Y* covering all places: {*p* ∈ *P*|*Y*(*p*) > 0} = *P*). Such graphs verify the next properties [13]:


In the next, we denote by •*q* (resp. *q*•) the set of places upstream (resp. downstream) transition *q*. Similarly, • *p* (resp. *p*•) denotes the set of transitions upstream (resp. downstream) place *p*.
