J. Software Engineering & Applications, 2010, 3, 1118-1124
doi:10.4236/jsea.2010.312130 Published Online December 2010 (http://www.SciRP.org/journal/jsea)
A New Formalism for Modeling a Multi Agent
Systems: Agent Petri Nets
Borhen Marzougui1, Khaled Hassine2, Kamel Barkaoui3
1Higher Institute of Technological Studies of Mednine, Tunisia; 2Faculty of Science, Gabès, Tunisia; 3National Conservatory of Arts
and Crafts, Paris, France.
Received November 11th, 2010; revised November 22nd, 2010; accepted December 1st, 2010
ABSTRACT
In this paper, we present a new formalism for Mod eling Multi Agent Systems (MAS). Our model based a PN is able to
describe not only not the internal state of each agent modeled but also its behavior. Owing to these features, one can
model naturally the dynamic behavior of complex systems and the communication between these entities. For this, we
propose mathematical definitions attached to firing transitions. To validate our contribution, we will deal with real
examples.
Keywords: Multi Agent Systems, Method, Agent Petri Nets, Formalism
1. Introduction
Since the appearance of the Petri Nets [1], the inventors
have not stopped proposing new models, either to im-
prove an already existing formalism or to create a new
model. These formalisms depend from the studied sys-
tem types. They permitted to make the conception more
natural, more intuitive and more familiar by Petri Nets
(PN). Indeed, the Petri Nets can be considered as graphic
and mathematical tools of modeling. To modelize and
analyze the discreet system, particularly the competitive,
parallel and non-determinist ones, it is necessary to
choose the appropriate type of PN to be used. This type
must be capable of modelizing rigorously the systems of
large size as the multi agent systems. Such systems per-
mit to coordinate the intelligent agent behavior interact-
ing and communicating in an environment to achieve
some tasks or to solve some problems [2].
According to [3], the modeling of Multi Agent Systems
(MAS) proves to be applicable to represent the actions of
the agents and their consequences in the environment
that can be complex and of an autonomous evolution. In
fact, the complexity of the system studied is increasing.
The precision, reliability and the hardiness have become
difficult factors to reach. The previous works of the Petri
Nets concentrated on their uses and not on the creation of
the new models as the works of [4-6]. The research of a
new model has been ignored for a long time. However,
there were some works that took into account the exten-
sion of some classic types of the Petri Nets to reach a
more or less generic model to satisfy a need of modeling.
[7] proposed a Petri net model approach to formal speci-
fication of holonic control systems for manufacturing. To
build up models of transition operations robotic, [8] pre-
sented a specific algorithm. Therefore, the integration of
a mathematical tool offers an exact way, in presence of
graphic tools, to succeed the conception of these systems,
particularly the multi agent systems.
The objective of the present paper consists in propos-
ing a new type of Petri Nets based on the agents that help
us understand the functioning of the MAS. In order to
describe the behavior and the interactions of the entities
of the system or the constraints on the variable characte-
ristics of the system, we should make a dynamic model-
ing. This modeling must be achieved by an adequate
formalism that will be presented in our work.
This paper is organized as follows. The second section
describes the Multi Agent Systems and their modeling by
the Petri Nets. Besides, the third section explains the
limits of the classic PN. Our new model titled Agent Pe-
tri Nets is presented and the definitions formulating this
formalism are interpreted in the fourth section. Next, in
the fifth section, we present the correspondence between
the MAS approaches and APN. The sixth section
presents an example of Modeling by Agent Petri Nets.
Section Seven finally concludes the paper and outlines
future works.

A New Formalism for Modeling a Multi Agent Systems: Agent Petri Nets 1119
2. Modeling Multi Agent Systems by Petri
Nets
A multi agent system permits to coordinate the behavior
of agents, interacting and communicating in an environ-
ment, to achieve some tasks or to solve some problems
[9-11]. It allows the decomposition of complex task in
simple sub-tasks which facilitates its development, test
and updating.
The modeling of MAS requires toll to check first cha-
racteristics and properties of agents, then, those of the
system itself. That’s why, several works have been ach-
ieved on the formalization of the MAS by different for-
mal methods as those of work [3,12-15]. In [13] the pro-
posed approach is focused on the formalisms as a lan-
guage for describing the models produced in each phase
of the development process. But this still requires a spe-
cific agent. Also, in [15] an Agents designed thanks to
CLAIM are endowed with cognitive capabilities, are able
to communicate with other agents and are mobile. But
these primitives are limited only to the cognitive agent
and do not address the other classes of agents. So, we
must think to generalize these primitives.
Few formalisms have been defined (as finite state au-
tomata) which are ineffective when must consider as-
pects of parallelism [7]. The algebraic models based on
differential equations are also inefficient to represent
agent’s interactions. So, it is imperative to dispose a
formalism capable of expressing the internal state agents,
their behaviors and the interactions between them. In this
context, the use of the PN to modelize MAS presents a
major contribution. For example, a Colored Petri Net
(CPN) is used in [16-18] to modelize the simultaneous
communications of the agents by using the functions
manipulating colors. Also, to describe the architecture
and the behavior of systems, [19] propose a new model
of transformation for the inversion of the original CPN
and the implementation of the analysis phase. This tech-
nique can provide a formal basis but which requires
enrichment of marking. However, the Object Petri Nets
(OPN) [18] presents a power to modelize the dynamic
aspects of the agents, but use a lot of places and transi-
tions which makes the model rather complex.
3. Limits of the Classic P e tr i N e ts
The classic Petri Nets as those of Place/Transition, Co-
lored, and Object present a deficiency at the level of their
expression if it relates to the system of large size as the
multi agent system. These systems are characterized by
the interactivity of the elements that they compose. For
the Colored PN, for example, the classes of colors cannot
directly express the state of the elements (tokens) of the
system or the relations between them. In addition, the
OPN can describe effectively the internal state of the
tokens but not the relations between them because this
requires additional places and transitions which brings
into play the methods used. Indeed, an Objects Petri Nets
modelizes a multi agent system by a rather high number
of places and transitions. The firing of a token is assured
by the invocation a set of methods. This describes with
difficulty the behaviors of the agents around their envi-
ronments [6].
The multi agent approach can be considered as an
evolution of the object-oriented paradigm. From a con-
ceptual viewpoint, an object is merely a data structure
which is associated with the functions. The agents are
autonomous entities whose behavior does not depend on
an outside expression, contrary to the objects.
The already achieved works concern the modeling of
the MAS by a PN respecting some specific needs. It is
often needed to make a coupling between two types of
Petri Net to satisfy a possibly determinist aspect in the
system specification as the interaction and the commu-
nication between the different entities that compose it. In
[20] a description of the abstract architecture for multi-
agents systems with an indirect interaction has been pre-
sented but with only two agents. Therefore, our idea
consists in benefiting from the properties and characte-
ristics of agents and integrating them in a classic Petri
Net. Thereafter, we propose our approach that consists in
defining a new model of Petri Nets called Agent Petri
Nets. Previous work modeling MAS by classical Petri
nets [8,13,18] does not allow a direct passage to the im-
plementation phase. Because this requires changes to
rules implementing a model of agent oriented languages.
So, we must define a model based agent to simplify its
implementation.
4. Proposed Formalism: Agent Petri Nets
An agent is defined as an autonomous entity capable
of communicating with other agents to partially dis-
cern at least its environment and the objects that sur-
round it, and to have correct or erroneous representa-
tions about the behaviors of a part or the set of the
other agents of the environment [21]. So, contrary to
the objects, an agent possesses an autonomous beha-
vior. It is capable of taking some decisions and estab-
lishing plans of actions to accomplish complex activi-
ties.
4.1. Definition 1: Agent Petri Nets
An Agent Petri Nets is defined as being a directed bipar-
tisan graph that has two types of nodes (place and transi-
tion). The arcs are bonds between these nodes which
indicate the conditions of activation of a transition. Every
transition carries the functions that manipulate the inter-

A New Formalism for Modeling a Multi Agent Systems: Agent Petri Nets 1120
nal state and the behavior of an Agent (Token) in its en-
vironment. The distribution of the tokens in the places at
a given moment is called marking of the Agent Petri Net.
A marking gives the state of the system that depends
on the interaction between the entities that compose it.
The change in internal of the state or the behavior of
every Agent, in the first place or the whole system, in the
second place, is assured by functions.
In a formal manner, Agent Petri Net is defined by the
9-uplet:
jk
Q=P, T, A, Meadow, Post, Pr , F, Ft, Env
where:
 P is a non-empty finished whole of places;
 T a non-empty finished whole of transitions;
 A is a non-empty finished whole of agents;
 Meadow: PxT
N
an application of front inci-
dence;
 Post: PxT
N
 a back application of incidence
corresponds to the arcs;
 Prj: pre condition of firing;
 F (Ai, Aj): agent relation function presenting the
condition of firing;
 Ft: function agent that uses three variables;

tk
Ft=Per,value,Inter;
 Envk: Environment of work that describes multi
agent system.
4.2. Definition 2: Constraints of an Agent Petri
Net
A constraint of an agent Ai is defined as:
Cont (Ai, Pj, TK) = b.
It’s defined as being a firing of a Tk transition des-
cended of a Pj place.
In a formal manner, the constraint on an agent it’s de-
fined as:
, , (, , ).
ijk
iIjJ etkKContAPTb 
Where:
 I: set of numbers of tokens of a place;
 J: set of numbers of places;
 K: set of numbers of Transitions;
 i: index of an Agent;
 j: index of a Place;
 k: index of a Transition;
 b: Boolean (0 or 1).
4.3. Definition 3: Pre-condition Function: Prj
That is to say Cont (Ai, Pj, TK) = b, for a number ni of
agents which is engaged in an environment, we get:
i=ni
jijk
i=1
Pr=Cont (A, P , T)=b

where Prj it’s the pre-condition function descended of a
place Pj.
By hypothesis, an agent Ai must only be engaged in
only one environment. It is defined as:



i
CardEnvA=1
The Boolean value sent back by Prj gives the starting
point to an action (transition). The engagement of an
agent in a well-defined environment will be preceded by
the control made by this function. The subset ni of the
agents has an equitable environment cardinality that is
equal to 1 or 0.
1) If Prj = 0 then the pre-condition is not valid and in
this case at least exists an agent that is already en-
gaged in another environment.
2) If Prj = 1 then the pre-condition is valid and in this
case it is guaranteed that all agents are not already
engaged in another environment.
Illustration (Figure 1): It is supposed that:
Workshop1 contains the Machine M1 and M2 but
Workshop2 contains the Machine M3, M4 and M5:
1) Case 1: the two machines M1and M2 belong to the
same workshop (Environment): Workshop1. In this
case their use is permitted: pre-condition Pr0 = 1.
2) Case 2: the Machine M1, M3, M4 and M5, cannot
belong to the same workshop (Environment: Work-
shop2) because the Machine M1 is already engaged
in another environment. In this case one cannot
cross the transition T1.
4.4. Definition 4: Functi on of Adhere nce
(Relative to an Agent)
This function gives rise to a relation between an agent
and its environment. The engagement of an Agent Ai in
an environment Envk describes initially an adherence
criterion then the number of times that this agent has
been engaged in Envk.
This offers more mechanisms of explanations and mi-
nimizes the difficulties with the tasks that require know-
ledge of the world (Env) which can be obtained only by
memorizing or reasoning and not by perception. The
definition of Ferber [2] that showed that a cognitive
agent has the capacity to reason on representations of the
world, to memorize some situations, to analyze them, to
foresee some reactions possible for any action, to draw
the conducts of the future events and therefore to plan its
own behavior is used.
In a formal manner, the adherence function of an agent
Ai, in an environment Envk noted Apai is defined by:
A
Aand EnvEnv,
ik

(, ,,)
iik
A
paApaAEnvb d

A New Formalism for Modeling a Multi Agent Systems: Agent Petri Nets 1121
dZ


where:
a) b: constraint = Prj (b = 0 or b = 1):the engagement
of Ai in Enk.
b) d: adherence degree: altogether gives the number of
times that the agent Ai has been engaged in Envk.
Importance of the adherence function
Constantly, this function gives a description of the re-
lation between the agent and a well defined environment.
It guarantees the update of the knowledge base of an
agent. An agent's reaction depends on its environment.
The evolution of Agent Petri Nets depends on the system
to study what implies, implicitly, that each agent seeks
the criterion of competence of another. That’s why it
must interpret the value of d.
This value d can enrich the knowledge base of an
agent.
Illustration: (Figure 2) let’s take the example of the
Figure 1 with some modifications:
c) Before the firing of T0 and T1, agent M1 admits d=
0 for the environment of Workshop1 and Work-
shop2 respectively:
 ApM1 = Apa (M1, Workshop1, 1, 0);
 ApM1 = Apa (M1, Workshop2, 1, 0).
After the firing of T0, agent M1 will have d = 1 for the
Workshop1 environment and 0 for Workshop2:
 ApM1 = Apa (M1, Workshop1, 1, 1);
 ApM1 = Apa (M1, Workshop2, 1, 0).
Agent M1 leaves his Workshop1 environment: firing
of the T2 transition. It will allow it to be free.
After the shooting of T1, agent M1 will have d = 1 for
the environment of Workshop1 and Workshop2 respec-
tively:
 ApM1 = Apa (M1, Workshop1, 1, 1)
 ApM1 = Apa(M1, Workshop2, 1, 1)
4.5. Definition 5: Adherence Function (relative to
an environment)
The creation of the adherence function Apai of an A
i
agent in an environment Envk allows us to find an adhe-
rence function Apei conversely. This new function de-
scribes the set of the agents which belong to the same
environment j with certain degree of adherence di.
The function of adherence related to an Envk envi-
ronment is defined as:

1
,,
ini
jkii
i
A
peApeEnvA d






where:
 ni: number of agents in the environment;
 Ai: Index Agent I;
 di: degree of adherence of agent of i index.
Thus, this function can be simplified as:

1
,ini
jki
i
A
peApe Envd






Illustration: From the APN of Figure 2 the degree of
adherence of every environment can be deduced:

5
1
1 1,i
i
i
A
peWorkshopApe Workshopd







5
1
2 2,i
i
i
A
peWorkshopApe Workshopd






with 5 is the number of machinery (Agent) used. The
following adherence matrix can be deduced:
Figure 1. Illustration of the pre-condition function Pr.
Figure 2. Illustration of the adherence function.

A New Formalism for Modeling a Multi Agent Systems: Agent Petri Nets 1122
4.6. Definition 6: Moderator Agent
An agent is said to be moderator if it is priority than
another. This indicates that the moderator dominates at
the time of a communication, or it possesses a hierar-
chical degree (dh) less elevated (dh = 2 dominates dh =
3). An agent is said to be total moderator if it is impor-
tant by contribution to all agents of its environment.
Thus, it possesses a hierarchical degree dh that is equal
1.
4.7. Definition 7: Function Relationship F:
This function is defined as a function admitting two en-
tries E1 and E2 and in exit only one boolean value S. The
entry E1 is imperatively moderator. This function
presents a pre-condition of firing of a transition (Figure
3).
Thus, this relation is defined by the function
F (E1, E2) = S.
Let Ai and Aj two agents in the same environment
Envk.

, , ,
ij ij
A
AAAFAA S  
where:
 A: set of the agents of the environment;
 Ai: is an agent moderator;
 Aj: is not an agent moderator;
 S: Boolean (1 or 0).
Thus, this function can be generalized to get a function
of relation agent of n order: set of n agents a function F
as:

12
,,,
n
F
AAA S
1) If S = 0 then no relation exists between the two
agents communicate between each other. In this
case, the non-moderator agent cannot enter in rela-
tion with the moderator agent whether voluntarily or
obligatorily by the moderator agent of total order,
or because it is already occupied.
Figure 3. Function relationship.
2) If S = 1 then a relation between the two agents
concerned is established. In this case, the agent
moderator asks for the establishment of a commu-
nication with another agent that is called non-
moderator and which accepts this demand.
4.8. Definition 8: Function Agent Ft
The function agent describes the relation between two
agents, the data interchange and the behavior of each of
them. It modifies the values descended directly of an
agent. These define the capacity to discern and to react to
the modifications occurred in its environment. Generally,
it is written as follows:


,,
tk
F
tPer ValueInter
1) Initially,


0, , 0
tk
Ft , it implies that there is
not any interaction between the agents. If the value
of Per = 0 then directly we have Inter = 0. Never
can we have Per = 0 nor and Inter = 1. Value = Φ, in
this case no action is triggered and the previous sit-
uation of the agents is maintained.
2) In the course of the firing of the transition k
t, there
will be change of values between the agents. In this
case Per takes the value 1, Inter takes the value 0
and Value defined the action or the task to achieve.
The relation of order already defined gives the sense
of transfer of the information.


So :1,,0
tk
F
t Value
3) After the firing of the transition k t, Inter takes the
value 1; it indicates that the action has been
achieved successfully.


So :1,,1
tk
F
tValue
5. From the Multi Agent Systems Toward the
Agent Petri Nets
In the form of a table, a kind of correspondence between
the two approaches according to well defined characte-
ristics will be given.
Table 1 summarizes the various properties of multi-
agents systems and that of our formalism. Indeed, the
ability to express our model is the benefit of two impor-
tant paradigms: Petri nets and MAS. The ability of ex-
pression is seen from that carried:
 The agent is presented as a single token. The rela-
tionship between the agent and its environment is
defined by a function. RdPA model describes the
behavior of an agent clearly. The firing of a transi-
tion gives a newstate for the agent and therefore for
the whole system.
 The control of behavior of agent and its analysis are
done easily by interpreting the transition carrying a
function.
The underlying idea of Agent Petri Nets is to properly
represent the agent and its autonomy in communication
with other agents while maintaining a simple graphical re-
presentation and understandable. This means they com-
bine clear denotation which can be displayed in the gra-

A New Formalism for Modeling a Multi Agent Systems: Agent Petri Nets 1123
Table 1. Correspondence between MAS and APN.
Characteristic Multi agent
Systems Agent Petri Nets
Name
Agent Token
State of the
system Place
Set of rules Meadow condition (Prj)
Set of relations
of actions Transition
Agent Admin-
istrator Agent Total Moderator
Class
Agent ReactiveAgent not Moderator
Agent Cogni-
tive Agent Moderator
Agents HydrideAgent Total Moderator
Autonomy Interaction
between agents
Function relation relation-
ship
Reactivity Agent - Agents Function Agent
Ft = <Per, Valeur, Inter>
Heterogeneity Agent – Envi-
ronment
Related to adherence Func-
tion an agent
Sociability Environment –
Agents
Related to adherence func-
tion an agent (Apai) and her
environment:

,
1
j
ini
Env d
ji
i
Ape Ape



 

Intelligence
Comportment,
capacity of
interaction
Exploitation of the values
possible of Per, Inter and
Value.
phical representation with an operational semantics that
can be executed. The model can be performed for the sim-
ulation. This is an advantage for understanding the inte-
raction and ensures its successful implementation. How-
ever, Agent Petri Nets can also be used for abstract mod-
eling that can display previews or details (sophistication)
of the system.
6. Example: Modeling Of print Job
The system is composed by two agents CPU and Printer.
We propose to model this system by two types of PN
namely PT PN and APN. Figure 4 present a modeling of
Print Job by PTPN.
where:
 P1: CPU pending;
 P2: data ready for printing;
 P3: printer pending;
 P4: print data;
 T1: preparation of data to print;
 T2: begin print;
 T3: end print.
Figure 5 present a modeling of Print Job by APN.
There is reduction (folding) at waist level Petri net.
Indeed, the number of places becomes 2 instead of 4 and
the transition becomes 2 instead of 3. The initial marking
M0 is the first case M0 = (1, 0, 1, 0). For the second case,
M0 = (2, 0). This also implies that we can get a tree re-
duced marking. Thus, the APN model describes an intel-
Figure 4. Modeling of print Job by a classic PN.
Figure 5. Modeling of print Job by APN.

A New Formalism for Modeling a Multi Agent Systems: Agent Petri Nets 1124
ligent manner the state and behavior of the system with a
simple graphic presentation.
7. Conclusion
We defined the framework of our work by expressing
our position compared to works which treated the formal
methods for modeling of the systems multi agent. Be-
sides, a formalism that combines the Petri Nets and the
MAS is proposed. This combination gives birth to a new
formalism called “Agent Petri Nets”. Actually all defini-
tions in relation to this type of network are given. It ben-
efits from the characteristics of the agents and multi
agent systems. Indeed, each token of a place represents
an agent and the transition is equipped with a set of func-
tions that describes particularly the condition of its firing
and the relations between the agents. The major contri-
butions of an Agent Petri Net are firstly the power of
expression, secondly the modeling of the interactions
size of network and finally the gain at the level of mod-
eling time. The definition of this model helps in model-
zing the internal state and the dynamic behavior of an
agent in MAS efficiently. We will explore more deeply
the modeling of other real cases of different complex
systems. Thus, we must add some functions to model the
mobility of an Agent and its migration from one envi-
ronment to another.
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