Towards an Abstract Recursive Agent
Adriana Giret, Vicente Botti *
Dpto. Sistemas Inform´ aticos y Computaci´ on.
Universidad Polit´ ecnica de Valencia, Valencia, Spain.
agiret,vbotti {@dsic.upv.es}
* Corresponding Author: Vicente Botti.
Dpto. Sistemas Inform´aticos y Computaci´on.
Universidad Polit´ecnica de Valencia, Camino de Vera s/n.46071
Valencia, Spain.
Phone: 34 96 3877000. Fax: 34 96 3877359.
Abstract
Current business trends, policy markets, production requirements, etc., have created the need for integrating pre-existing Multi Agent Systems (MAS). In the agent-specialized literature, we have found very little work about agent architectures and methodologies that allow us to carry out recursive and dynamic analysis, design, and implementation of MASs.
Several difficult challenges for automated systems can be tackled by giv- ing full meaning to the agent concept: adopting a recursive definition of agents and allowing for the dynamic creation of agents by the agents themselves. In this work, we propose a definition of an abstract recur- sive agent and an initial formalization of its behaviour in terms of the behaviour of its constituent agents.
1 Introduction
Nowadays arises the need to integrate pre-existent Multi Agent Systems (MAS) in domains where these integration and/or cooperation are imposed by business trends, policy markets, production requirements, etc.
The need for some kind of hierarchical aggregation in real world systems has been recognized in the intelligent manufacturing field. These systems have to remain readable while they are expanded in a wide range of temporal and spatial scales. For example, a modern automobile factory incorporates hun- dreds of thousands of individual mechanisms (each of which can be an agent) in hundreds of machines which are grouped into dozens or more production lines.
Engineers can design, build, and operate such complex systems by shifting from the mechanism to the machine or to the production line (depending on the problem at hand) and by recognizing the agents of higher levels as aggregations of lower-level agents. Also, in e-commerce applications, an enterprize is a legal entity which is independent of the individual people who are its employees and directors.
The question arises as to whether an agent can be a collection of several in- teracting agents, a hierarchy, or some other type of organization. In [4], Gasser pointed out that almost all the proposals for agent architectures have not ad- dressed the general problem of how to treat collections of agents as higher-order entities -e.g, how to treat organizations as agents.
In the agent-specialized literature, we have found very little work about agent architectures and methodologies which allow us to carry out recursive and dy- namic analysis, design, and implementation of Multi Agent Systems. Most of the current approaches start from an atomic agent definition such as an indivisi- ble entity and build Multi Agent Systems as compositions of interacting agents.
Most of the approaches do not deal with systems in which their components may be Multi Agent Systems themselves. The only work we have found about recursive agent model is by Occello. In [13], he proposes a recursive approach to build hybrid Multi Agent Systems. From a given set of elementary agents, Oc- cello proposes a recursive agent structure definition and two recursive functions,
to build a higher level agent. His work is based in a rigorous analysis of recursive properties in Multi Agent Systems structures, such as agent and environment, and two functions defined on them, interaction and organization. A recursive agent is a MAS, that is, a set of (recursive) agents and (recursive) environment objects. The interaction function allows us to model all the communication acts which can occur either with other agents or with the environment (perception, action and cognitive interaction). The organization function is modelled as a set of relations between agents. These relations can be of three types: acquain- tance, communication and subordination. However, in Occello’s work, there is no formal definition of the recurrence property to define the behaviour of one level of recursion with regard to another. In our work, as in Occello’s work, a MAS can be viewed as a set of agents at a given level and as a whole agent at an upper level. We call it an abstract recursive agent. It is abstract because it exists only at the analysis and design phases and it is not a real executing agent (at the coding stage it is replaced by its constituent elementary agents). Unlike Occello, we will not build an engine to manage the interaction for a MAS at run time, because this interaction is already managed by its constituent agents. To define the behaviour of the abstract recursive agent, instead of using a interac- tion function (as in Occello’s work), we do it with the reactive and intentional behaviour of its constituent agents. We believe that these definitions will make the formal definitions of the recurrence property straightforward.
It is important to point out that recursive modeling in the context of our work is different from the work done by Gmytrasiewics and Durfee [6] and by Tambe [17]. Gmytrasiewics and Durfee proposed a Recursive Modeling Method as a theoretical framework for representing and using the knowledge that an agent has about its expected payoffs and those of other agents. That is, a representation of the benefits an agent expects to get given the combination of actions chosen by all the agents. On the other hand, Tambe proposed a different approach for an agent’s models of other agents’ behaviors. He proposed the combination of architectural features that enable an agent to generate flexible and reactive behaviors of other agents. Note that in both of these works, the
recursive model is not a modeling artifact for representing Multi Agent Systems.
In this work, we try to define a set of concepts to help in the construction of Multi Agent Systems. The aim of this paper is to introduce the idea of an ab- stract recursive agent (AAgent) as a modeling artifact to represent Multi Agent Systems and a formalization of its behaviour. Figure 1 shows our objective.
Intuitively, an AAgent is made up of interacting autonomous entities, each of which may be an agent or in turn an AAgent. Therefore, to define each of the AAgent agenthood characteristics, we propose the definition of a relation or pseudo-function that maps each constituent agent characteristic into one or more corresponding AAgent characteristics. In Figure 1, we assume agents with mental attitudes, such as beliefs, goals and intentions, and with perceptions and actions components. The AAgents interacts with each other at different levels of abstraction. In Figure 1, we can see, at the lowest abstraction level, interacting agents. On the other hand, as the abstraction levels go up, we can see interact- ing AAgents (Multi Agent Systems or agents). We have to define the following relations: RB - Recurrence Relation for Beliefs; RG - Recurrence Relation for Goals; RI - Recurrence Relation for Intentions; RP - Recurrence Relation for Perceptions; andRA - Recurrence Relation for Actions. In this work, we present a study ofRA, RP and RG.
We are convinced that several difficult challenges for automated systems may be tackled by giving full meaning to the agent concept: adopting a definition of abstract recursive agents (AAgents) and allowing for the dynamic creation of AAgents (organizations of agents) by the AAgents and the agents themselves.
To this end, we propose a definition for abstract recursive agents in section 2.
In section 3, we try to formalize Multi Agent System behaviour in terms of its constituent agents’ behaviour. With a definition of the Multi Agent Sys- tem behaviour it will be possible to abstract away from the individual agents of the system and shift from different levels of abstraction (depending on the problem at hand). At a given abstraction level, we will be able to consider lower-level entities as individual AAgents (see Figure 1) without having to take the structure of these AAgents into account, that is, without worrying whether
they are Multi Agent Systems or agents. This will allow us to focus only on the AAgents’ interactions at the current abstraction level. In section 4, we extend this definition to the level n. Finally, we state our conclusions in section 5.
2 Abstract Recursive Agent - AAgent
With a recursive approach for developing Multi Agent Systems (MAS) as sys- tems in which their components may be MASs themselves, the idea is as follows.
When we begin to analyze a group of agents (MAS) A, we identify the agents {a1, a2, ..., an} which execute certain functions. These agents may encapsulate individual persons, physical, or software entities (atomic agents). They may also be other groups of agents (MAS), say B, so we can have ai = Bi, which we treat as black boxes. We can take this perspective as long as our analysis can ignore the internal structure of the member groups (MAS). However, subse- quent analysis generally needs to ’open’ these black boxes and look inside them to see the agent components and their corresponding functions. For example, when analyzing B, B ={b1, b2, ..., bm}. At this point, we insist on identifying which of B’s member agents is actually responsible for filling B’s role in A.
To support these ideas it seems appropriate to provide an abstract recursive agent (AAgent) definition which will allow us to build Multi Agent Systems.
This definition is based on the widely known agent definition of Wooldridge and Jennings [18].
Definition 1. An AAgent is a software system with a unique entity, which is located in some environment, which as a whole, perceives its environment (en- vironment sensitive inputs). From these perceptions, it determines and executes actions in an autonomous and flexible way - reactive and proactive. These ac- tions allow the AAgent to reach its goals and to change its environment. From a structural point of view, an AAgent can be an agent (atomic entity); or it can be a Multi Agent System (with a unique entity) made up of AAgents which are not necessarily homogeneous.
An AAgent is in a higher conceptual abstraction level than an agent. An
AAgent can be seen as a MAS, an organization, a federation or an institution with the added value that it can also be a composition of all these abstraction models. Furthermore, when we define two interacting AAgents, we could also be modelling two interacting organizations, federations, MASs or institutions.
An AAgent will exist only at modelling stages, in the end (at coding stages) it may be replaced by a group of agents or also by a single agent.
Definition 1 provides a functional and structural AAgent perspective. The functional perspective is based on the widely known agent definition of [18], in which an agent is an autonomous, reactive and proactive entity. On the other hand, the structural perspective introduces an indirect recursion when indicating that an AAgent may be a MAS, which in turn is made up of AAgents, each one of which may be a MAS or an agent.
Definition 2. A Multi Agent System is made up of two or more AAgents which interact to solve problems that are beyond the individual capabilities and indi- vidual knowledge of each AAgent.
Definition 2 extends the traditional notion of Multi Agent Systems when indicating that a MAS is made up of AAgents. This could be a very useful property because with this we could have a MAS made up of interacting MASs.
Figure 2 shows the graphical representation of the AAgent structure in UML.
An AAgent is a generalization of a MAS and an agent. A MAS is made up of two or more AAgents which can be agents or, in turn, MASs. This representation is similar to the one presented in [15] for the holonic perspective for agent-oriented software engineering. In fact, our AAgent structural definition is inspired by holonic concepts [8, 11].
There are two levels in an AAgent. The abstraction level and the recursion level. The abstraction level is used in the analysis and design phases. When we begin to analyze a MAS A we identify the group of agents {a1, a2, ..., an}.
Agents {a1, a2, ..., an} are said to be in a lower abstraction level than A. Let m be the abstraction level of A, then {a1, a2, ..., an} are in m − 1 abstraction level. Subsequent analysis will ”open” these agents, for example when analyzing a1, we could have that a1 ={a11, a12, a13}. Then the abstraction level of each
agent in{a11, a12, a13} will be m − 1, and so on. The recursion level is defined as follows:
Definition 3. Let a be an agent and A and Ai be AAgents. The recursion level of an AAgent is:
LevelR(A) =
0 A = a,
max{LevelR(Ai)} + 1 A = {A1, A2, ..., Ak}, 1 ≤ i ≤ k.
From definition 3 we have:
• AAgent of recursion level 0 is an agent.
• AAgent of recursion level 1 is a MAS made up of interacting agents.
• AAgent of recursion level n > 1 is a MAS made up of interacting AAgent of recursion level < n.
The designer point of view will determine the nature of what is being ob- served at each moment. From the outside, a system can be considered as an AAgent since it has agenthood characteristics. On the other hand, from the in- side, that is, from the internal structure, the AAgent can be considered as being composed of a group of interrelated AAgents (MAS). When there are no more subdivisions, the AAgent can be considered as being a simple agent. The end of the recursion is defined by the designer since the subdivision exists whenever it is useful for the definition of the problem being modeled. In the end, at the lowest abstraction level, only the agents which make up the global MAS will be apparent, but as the abstraction levels go up, there will be some agents and some AAgents that are refined as MASs.
2.1 Why recursiveness?
One of the most difficult challenges for automated systems is scalability and adaptation. In life systems there are many useful concepts, including examples on how to scale up, evolve, adapt, interoperate, organize, and so on. Complex
and adaptive life systems are large and intricate and require active autonomous entities. Life systems are recursive and they enable the construction of very complex systems from simpler entities. What about agents?. Are agent agents arranged in clusters, a hierarchy, or some other type of organization?. In this section, we present two examples in which the usefulness of our definition can be observed to describe complex problems with multiple levels of abstraction.
Let us suppose a Multinational company, called AG, which has different National companies distributed among different countries. The objective is to model the multinational as a MAS.
Each National company can be an AAgent since it has agenthood character- istics. The National company is autonomous in its national environment; it acts in the national market with its own market and production rules. At the same time, it must be able to interact with other National companies to exchange materials, personnel, knowledge, etc. The National company, is also governed by the rules and norms of the Multinational for its international relations (other National companies).
The international companies’ relationships define the rules, norms and poli- cies of the multinational. In Figure 3(a), geographical areas can be observed in which the relationships among the national companies are narrower. In addition the commercial agreements among the different countries define new interrela- tion rules among the national companies of these zones. For example, in Europe, the European Union countries are governed by certain standards and norms of the community; and in South America they are governed by the Southern Cone Common Market - MERCOSUR (Paraguay, Argentina, Chile, Brazil, Uruguay and Bolivia) and by the Andean Community (Bolivia, Colombia, Ecuador, Peru and Venezuela). The relationships of the countries of these markets with other countries or regional markets are managed by their local market rules. Each market can be modeled as an AAgent. This generalization is shown in Figure 3(b). It is important to note that Bolivia, as a National Company, belongs to two Regional Companies (MERCOSUR and the Andean Community).
Up to this point, we have identified 4 levels of abstraction (Figure 4(a)): the
Multinational company, the Regional companies, and the National companies.
We have been able to model the Multinational as a MAS, which is composed by AAgents that are related to each other with certain behavior patterns that define the Multinational company. If the National companies are made up of agents (AAgents of recursion level 0), we can think of a National company as a traditional MAS (AAgent of recursion level 1), the Regional companies as AAgents of recursion level 2 and the Multinational as an AAgent of recursion level 3.
Apart from modeling the outside relationships, if the designer’s interest is also to model the internal structure of each National company, the National company should be observed from inside. Inside each National company there would be new companies located in different cities or with autonomy for certain activities. In turn, each Local company is subordinated to the National company and each National one to the Multinational. Thus, we have a new level of abstraction, the Local company as an AAgent of recursion level 1, the National company as an AAgent of recursion level 2, the Regional company as an AAgent of recursion level 3 and the Multinational as an AAgent of recursion level 4 Figure 4(b).
If the National company is not subdivided into city companies or autonomous companies, then the National company is a traditional MAS composed of na- tional domain-specific agents (AAgents of recursion level 0), which are interre- lated agents and carry out specific functions. These national domain-specific agents define the services provided by the National company inside the country and outside the country. This very same analysis should be made for each Local company until we reach the agents, which define and implement the activities of the company as a whole. In summary, the final result of the analysis should be similar to Figure 4(c). In Figure 4(c), it can be observed that the National com- pany is composed of zero or more Local companies, and each Local company, in turn, is an AAgent of recursion level 1.
Again, the Multinational can be considered from the outside as an AAgent, since it is located in an environment, the world market; it is autonomous; it has
its own economic and market policies; it is social, i.e. it interacts with other entities for purchasing, selling, recruiting, leasing, etc.; it is pro-active, since, for example, according to the world market trends it is able to modify its current market policies, etc.
Another interesting example is the domain of mobile physical cooperating robots. We can model this domain as a traditional MAS made up of agents which are robots that have to cooperate to fulfill a given MAS goal. Each robot has its own agenthood characteristics. It is autonomous (it acts without the direct intervention of humans or other robots), it is social (it cooperate with other robots), it is reactive (it responds to changes in its environment), it is proactive (it is able to try to fulfill it’s own plan or goals). Let’s suppose a postal service system in which the robots are the postmen. Mail can arrive at any moment, each piece of mail has an addressee and a priority (normal, urgent, etc.). The team of postmen has to determine a good way to deliver the mail on a building floor according to the priority, the addressee, the team members, their current positions, their skills and environment obstacles (a closed door, a barrier in their path, etc.). However, current robot architectures are very complex. In such an architecture, a robot is made up of multiple processing units, each of which is an autonomous entity. Therefore, we have to model the robot as a system made up of cooperating agents. We have to add a new level of abstraction into our initial postal service model. However, thanks to our definition, a traditional MAS is an AAgent which has the same agent functional properties, so we do not need to modify the postal service level. We can still have cooperating robots, but they are now AAgents of recursion level 1, and the robot constituent agents are AAgents of recursion level 0.
3 MAS behaviour
In this section, we will try to formalize the behaviour of a MAS. This will be a starting point towards a formal definition of an AAgent behaviour. An AAgent is autonomous, reactive and proactive so we have to define these behaviours in
terms of its constituent AAgents behaviour. The first step in this definition is the definition of MAS behaviour in terms of its constituent agent behaviour, that is, the definition of AAgent of recursion level 1 (AAgent/MAS made up of interacting agents) in terms of AAgents of recursion level 0 (agent). Then, in a similar way, we will be able to define AAgents of recursion level n in terms of AAgents of recursion level < n. First of all, we will introduce some notations and definitions which will be used throughout this section.
The set of all MASs of the universe is denoted by A.
Let N be the number of all the agents of the universe.
Let a MAS be represented by Am,n, where m represents the abstraction level and n represents the recursion level. We will use A when there is no place to confusion.
The constituent agents of the MAS A are denoted by ai, where i denotes the i-th element of the set of n agents of A (nis the total number of agents in A).
A primitive action which can be executed by some agent ai∈ A, is denoted by γij, where j denotes the j-th element of the set Γi, of ri primitive actions of the agent ai.
A group action which can be executed by a group of agents{a1, a2, ..., ak} (ai∈ A) is denoted by s. Let SA be the set of group actions of A.
A perception of an agent ai ∈ A is denoted by pij, where j denotes the j-th element of the set Pi, of hi perceptions of the agent ai.
Let oij be a goal of an agent ai∈ A, where j denotes the j-th element of the set Oi, of mi goals of the agent ai.
Let O = N
i=1Oi be the union set of all goals of the agents of the universe.
In the majority of cases, agents exist in the context of MAS, whose global behaviour derives from the interaction among the constituent agents. In these
cases, agents exhibit social behaviour; they interact with one another by either cooperating to achieve a common objective or helping each other to achieve their individual objectives.
A MAS has an internal structure, which is derived from the interrelationships among its member agents and from the purpose of the MAS. From the outside, a MAS has a behaviour that emerges from its internal structure. This behaviour could be reactive and/or intentional.
3.1 Reactive
The behaviour of a reactive agent is based on the basic principle of action/reaction.
A MAS is reactive because, when it (as a whole) perceives events or changes in its environment, it reacts by making changes in that environment.
The set of perceptions of a MAS A, in terms of the perceptions of its con- stituent agents, may be defined as follows:
Definition 4. The set of perceptions PA of a MAS A is:
PA= n
i=1
Pi , Pi is the set of perceptions of the agent ai∈ A.
In the same way, the set of actions of a MAS A, in terms of the actions of its constituent agents, may be defined as follows:
Definition 5. The set of actions ∆A of a MAS A is:
∆A= n
i=1Γi ∪ SA, Γi is the set of primitive actions of the agent ai, and SA is the set of group actions of A.
Recall the Multinational AG of section 2.1. Let’s suppose AG is an automo- bile factory. There are two types of companies in AG. Manufacturing companies which manufacture automobile parts, and Assembling companies which assem- ble manufactured parts.
Let’s focus on the Local companies of AG (MAS/AAgent of recursion level 1).All the agents in a Local company can act and perceive their local environ- ments. That is, they operate at their local area of influence.
A Manufacturing Local company, is made up of one Director Agent, a few Manager Agents, and several Manufacturer Agents. The Director Agent has the following set of actions: to accept manufacturing orders, to schedule man- ufacturing orders and to assign manufacturing orders to the Managers of the company. The Director Agent also has the following perception: manufacturing orders. The Manager Agent has the set of actions: to accept schedule orders, to schedule manufacturing task, and to assign tasks to Manufacturer Agents of the company. The Manager Agent also has the perceptions: manufacturing tasks and manufacturing orders. The Manufacturer Agents are task-specific agents so each one of them has specific skills. Let’s suppose there are the following set of actions distributed among the Manufacturer Agents: to weld, to rivet, to screw and to transport. Let’s also suppose the following set of perceptions: screws, materials and products.
From definition 5, a Manufacturing Local company has the actions: to accept manufacturing orders, to schedule manufacturing orders, to assign manufactur- ing orders to Managers of the company, to accept schedule orders, to schedule manufacturing task, to assign tasks to Manufacturer Agents of the company, to weld, to rivet, to screw and to transport. It also has the following set of per- ceptions, from defintion 4: manufacturing orders, manufacturing tasks, screws, materials and products.
In a similar way, each Assembling Local company is made up of one Director Agent, a few Manager Agents, and several Assembler Agents. The Director Agent has the following set of actions: to accept assembling orders, to schedule assembling orders and to assign assembling orders to Managers of the company.
It also has the following perception: assembling orders. The Manager Agent, has the set of actions: to accept schedule orders, to schedule assembling tasks, and to assign tasks to Assembler Agents of the company. It also has the perceptions:
assembling tasks and assembling orders. The Assembler Agents are task-specific agents so each one of them has specific skills. Let’s suppose there are the following set of actions distributed among the Manufacturer Agents: to assemble each type of part and to transport them. Let’s also suppose the following set
of perceptions: parts, materials and products. The Assembling Local company as a MAS has the set of actions: to accept assembling orders, to schedule assembling orders, to assign assembling orders to Managers of the company, to accept schedule orders, to schedule assembling tasks, and to assign tasks to Assembler Agents of the company, to assemble each type of part and to transport them. It also has the set of perceptions: assembling orders, assembling tasks, parts, materials and products.
3.2 Intentional
The behaviour of an intentional agent is guided by deliberative processes which are based on mental attitudes such as beliefs, knowledge, desires, intentions, commitments, etc. In this work, we focus on MAS goals.
The definition of the intentions of a single agent can be simple and evident by using any of the well-known deliberative [7, 10] or hybrid agent architectures [2, 3, 5]. However, the question of how to define the MAS intentions in terms of its members’ intentions, no matter if they are agents or MASs, comes up.
This definition seems more difficult. Furthermore, there is no MAS architecture which can support it.
The definition of MAS intentions is crucial when specifying MAS behaviours and when defining the interactions among different MASs.
MAS intentions can be very complex to define, because the behaviour of the system as a whole is not simply the sum or composition of its agents’ behaviour, but something more. The behaviour of a MAS not only depends on the indi- vidual behaviours of each constituent agent, but also on the interactions among each other. A very clear example is an emergent system, in which the global behaviour results from the interaction among the constituent entities.
In order to define the intentional behaviour of the system, it is necessary to determine the possible types of relations which can exist among the constituent agents of a MAS; that is, we have to identify ’what agents do together ’.
Parunak et. al. [14] describe agent group behaviour in terms of the follow- ings types of relations: correlation, coordination, cooperation and contention.
Congruence and coherence are the concepts which link any type of correlation (coordination, cooperation, contention and competition) to the system objec- tives and the individual agent goals [14].
Two-level goals are shown in Figure 5, MAS-level goals and individual agent- level goals. Coherence defines relation patterns among agents (correlation, coop- eration, contention and competition) that yield congruence; whereas congruence expresses alignment between the system behaviour and the system goal. To il- lustrate these two concepts, let’s suppose a simple example. Consider a group of agents, in a Local company of the Multinational AG of section 2.1. The Local company has its rules and norms of organization. There is a Director Agent, a Production Manager Agent, and a number of Worker Agents. The Local company has the goal “to minimize delivery time”. To achieve this goal, the Local company has established the following rules of communication and delegation. The Director Agent is responsible for receiving product orders and for negotiating the delivery date with the customer. After this, the Director Agent communicates with the Production Manager Agent to schedule the prod- uct order. The Production Manager Agent makes a first proposal for delivery time, trying to minimize the time for delivery and taking into account the avail- ability of the Worker Agents. The Director Agent and the Production Manager Agent negotiate about the first proposal. They continue with this negotiation until they reach an agreement which fulfills the company goal. When they have an agreement the production plan is scheduled and the Worker Agents are re- sponsible for accomplishing it. A congruent behaviour is, for example, when the Director Agent, the Production Manager Agent and the Worker Agents fulfil their own responsibility in the scenario described above. A coherent relation pattern is, for instance, the relation among the Production Manager Agent and the Director Agent for negotiating a schedule to minimize the delivery time.
Up to this point, we have indicated the possible types of relations that can exist among the constituent agents of a MAS. Below, we will present our definition of MAS goals.
The basic idea is as follows. We have to identify coherent relation patterns,
that is, group behaviours that are congruent with some goal. To do this, the first step is to identify all the coordination patterns, cooperation patterns, contention patterns or competition patterns in the MAS. Once we have identified these patterns, the second step is to determine which goals are satisfied by these relation patterns. Finally, the MAS goals will be all the goals identified in the second step.
As pointed out by Parunak et. al [14], coordination, cooperation, competi- tion and contention are reflected in the interaction patterns among the agents.
They are ordered sequences of actions which at the same time can be modeled as plans to achieve the system’s global goals. A formal definition of ordered se- quences of actions (relation patterns) is presented in the following paragraphs.
Definition 6. An instance of a primitive action is defined as < ai, γij > such that the agent ai∈ A is involved in the primitive action γij ∈ Γi.
Definition 7. An instance of a group action is defined as < {a1, a2, ..., ak}, s >
such that the set of agents{a1, a2, ..., ak} ⊂ A, 1 < k < n are involved in group action s∈ SA.
Primitive and group actions may be combined into finite sequences to specify more complex interactions. Sequences are composed of at least one action and may contain a mixture of primitive/group actions. A sequence is denoted by Σ. In intentional systems, actions are carried out in order to attain goals. Our interest is in identifying which goals are attained by Σ.
An instance sequence of primitive and group actions specifies the actions and the agents that will perform them. If Σ is an action sequence, its instantiation is denoted by Σ. Let ΣA be the set of all action sequence instances of a MAS A.
Definition 8. Let Σ =
A∈A
ΣAbe the union set of all the sets of action sequence instances of all the MASs inA.
We have to define a relation among the action sequences instances of the universe Σ and all the goals of the universe O 1. That is, given an action sequence instance, this relation will match it with the set of goals it satisfies.
We call this relation f . It is defined as follows:
Definition 9. Let f be the function f : Σ −→ O. f(Σ) is the set of goals that are achieved by Σ.
f is a function because for every Σthere is at least one goal o∈ O achieved by Σ. f is a non-injective function because a goal or set of goals can be achieved by more than one action sequence instance. For example, consider the “blocks world” problem. A given goal can have infinite action sequence instances which achieve it. f is a non-surjective function because it can be a goal in O for which there is no action sequence instance in Σ which achieves it. For example, consider the goal ”to solve a given NP problem in a polynomial time”, there is no action sequence instance (algorithm) which achieves it.
Function f defines the goals of a MAS in a bottom-up fashion. From the behaviour of the constituent agents it defines MAS goals. We can also go in the top-down fashion. We can define the constituent agents’ behaviour from the MAS goals. To do this, we define g as follows:
Definition 10. Let g be the relation g : O −→ Σ. g(o) is the set of action sequence instances which achieve the goal o.
g can be seen as the reciprocal of f. g is a relation but not a function because there could be a goal o∈ O for which there is no action sequence instance in Σ which achieves it. However, we can build a function from g, as follows:
Definition 11. Let g be the function g : O −→ (Σ ∪ {ˆσ}), where ˆσ is the blank action which does not achieve any goal.
g(o) =
Σ f(Σ) = o, σˆ otherwise
1It can be proved thatO exists and it is infinite and numerable.
Relation g will allow us to translate a given MAS goal into a group of agents’
behaviours. That is, g will be a guideline for a designer when he is about to develop a MAS to fulfill some given goals. Moreover, when there is a goal for which there is no action sequence instance in g, then there will be no possible MAS in the universe to achieve it. In other words, the MAS is unfeasible.
The steps in the operational definition of MAS behaviour are the following:
When there is a group of interacting agents, and the objective is to define the emergent goals of the MAS A, the following bottom-up approach has to be applied:
• Step 1: Build the set ΣA with all the action sequence instances (coor- dination, cooperation, competition and contention patterns) observed in the MAS A.
• Step 2: Build OSA, the set of goals of the MAS A, applying f to each action sequence instance identified in the previous step. That is:
Definition 12. OSA=
|ΣA|
i=1f(σi), σi∈ ΣA.
When OSA, the set of goals of the MAS A is already defined and the objective is to find out the group of agents (with certain skills) to fulfill these goals, the following top-down approach has to be applied:
• Build ΣA, the set of action sequence instance of the MAS A, applying g to each goal of the set OSA. That is:
Definition 13. ΣA=|OSA|
i=1 g(oi), oi∈ OSA.
Let us suppose a Manufacturing Local company of the Multinational AG. To define the Manufacturing Local company, in a bottom-up approach, we have to focus on the relation patterns observed in the company. Let’s suppose, among others, the following relation patterns: negotiation between the Director and the Managers, for scheduling an order; communication between the Manager and a
Manufacturer for assigning a task; communication between a Manager and the Director about order status; communication between the Manager and a Man- ufacturer about task status; cooperation among Manufacturers for transporting materials; coordination among Manufacturers to build parts. Let’s suppose we have defined O2. Using f on the relations patterns of the Manufacturing Local company, we can obtain the following goals: to minimize delivery time, to man- ufacture parts, to cooperate with the manufacturer partners, to negotiate for manufacturing orders, to receive manufacturing orders, and many more goals.
As you may imagine, this set could be very large, although we can augment f with domain specific information to remove redundant and/or impossible goals.
We are now working on the recursive definition of domain specific information such as beliefs and knowledge.
4 Behaviour of an AAgent of recursion level n
The extension of the definitions provided in the previous section to AAgent of recursion level n > 1 is straightforward. We simply have to consider every AAgent of recursion level≥ 1 as a simple agent. An AAgent of recursion level n is denoted by Am,n, where m denotes abstraction level.
• Actions of an AAgent of recursion level n > 0 is defined recursively as:
∆Am,n =
∆A n = 1
Ax,j∈Am,n
∆Ax,j n > 1
• Perceptions of an AAgent of recursion level n > 0 is defined recursively as:
PAm,n =
PA n = 1
Ax,j∈Am,n
PAx,j n > 1
2Instead of having O we can have a subset, O. O could be the set of manufacturing- specific goals. That is, we can build O taking into account all the possible goals in the manufacturing domain.
Recall the example presented in section 3.1. Let us consider a National company (AAgent of recursion level 2) made up of one Manufacturing Local company and two Assembling Local companies. The set of actions of the Na- tional company is the union of the set of actions of one Manufacturing company and two Assembling companies. The set of perceptions is defined in a similar way. The National company knows how to manufacture and how to assemble in its area of influence(the influence areas of its Local companies). A Regional company (AAgent of recursion level 3) made up of National companies will have a broader influence area and further skills.
• Goals of an AAgent of recursion level n > 0 are defined as follows. We can build OSAnby using function f on the action sequence instances obtained from the observations of relation patterns among the constituent members (recursion level < n). The action sequence instances are derived from the above recursive action definition of an AAgent.
The goals of the National company could be: to minimize delivery time, to manufacture automobile parts, to assemble automobile parts, to cooperate with the manufacturer partners, to cooperate with assembler partners, to negotiate for manufacturing orders, to negotiate for assembling orders, to receive manu- facturing orders, to receive assembling orders, to receive aggregated part orders, to build aggregated parts, to negotiate for aggregated part orders, to sell aggre- gated parts in the country, to build automobiles, to receive automobile orders, to negotiate for automobile orders, to sell automobiles in the country, among other goals. Note that when the abstraction levels go up, new goals emerge.
This is so because new composed skills are present and new relation patterns can be defined. It is important to point out that the new goals and relation patterns of every new abstraction level are supported by the initial basic skills of simple agents of level 0.
5 Conclusion
In this work, we have proposed an abstract recursive agent (AAgent) definition.
This definition provides a functional and structural AAgent perspective. The functional perspective is the well-known agent definition of [18], in which an agent is an autonomous, reactive and proactive entity. On the other hand, the structural perspective introduces an indirect recursion when indicating that an AAgent may be a MAS, which is at the same time made up of social AAgents, each one of which in turn may be a MAS or an agent. This definition allows us to carry out a dynamic and recursive analysis and design of a Multi Agent System.
In section 3, we have proposed a formalization of MAS behaviours in terms of its constituent agent behaviour. In summary, the reactive behaviour of a MAS is determined by its perception which is defined as the union of the set of perceptions of its agents. It is also defined by its actions, which in turn are defined as the union of the group actions executed by its member agents and the union set of the primitive actions carried out by each of its constituent agents.
The intentional behaviour of a MAS, considering a BDI agent architecture, is determined by its goals, desires and intentions. In this work, we have focused on defining goals. We have proposed an operational definition of MAS goals. The goals of a MAS can be defined using two different approaches depending on the problem at hand. The top-down approach is defined for situations where the set of goals of the MAS is given and the objective is to determine which group of agents could reach it. To this end, we have proposed a function g(o) which from a given MAS goal o obtains the set of action sequence instances (relation patterns) which fulfil the goal, if there is some and the null action otherwise.
That is, g will be a guideline for a designer when he is about to develop a MAS to fulfill given goals. On the other hand, the bottom-up approach is defined for situations where there is a group of interacting agents and the objective is to find out what the goals of the emergent MAS are. The first step is to identify all the observed relation patterns among the interacting agents. Then we build the set of MAS goals using f on every action sequence instance identified in the
previous step. Function f (Σ) obtains the set of goals that are fulfilled by the action sequence instance Σ. In section 4 we have extended the definitions of section 3 to an AAgent of recursion level n > 1.
This paper is a preliminary report of our research. In this work, we have presented some definitions and formalization of concepts which are the basis for a new MAS methodology. We are now working on this new MAS methodology for analysis, design, and implementation of very complex systems following a dynamic and recursive approach. The idea behind this methodology is as fol- lows. At a given abstraction level, we will be able to consider lower-level entities as individual AAgents without having to take the structure of these AAgents into account, that is, without worrying whether they are Multi Agent Systems or agents. This will allow us to focus only on the AAgents’ interactions of the current abstraction level. It will be possible to abstract away from the individ- ual agents of the system and shift from different levels of abstraction (depending on the problem at hand).
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Figure 1: Abstraction Levels.
Figure 2: Abstract Recursive Agent.
Figure 3: Regional and National Companies of a Multinational Company.
Figure 4: Different levels of abstraction identified in an agent-oriented mod- eling of a Multinational Company.
Figure 5: Congruence and Coherence.
G
B I
PA
P B G
IA P B G
IA P B GIA
P B G
IA P B G
IA
P B GI
AP B G
IA P B G
IA P B GIA G
B I
PAP B G
IA G
B I
PA
P B G
IA P B G
IA
P B G
IA A-Agentk (Level m,n)
A-Agent1 (Level m-x,1) A-Agent2 (Level m-x,1) agent1 agent2agent3 agent5 agent6agent4agent7
agent8 agent10 agent9
agent11 agent7agent1
Figure1:AbstractionLevels
26
Multiagent System Agent +MAS
1..*
+Agent 2..n
AAgent
Figure 2: Abstract Recursive Agent
Canada
USA
Peru Mexico
Bolivia
Brazil
Paraguay
Argentina
England
Spain Portugal
Italy France
South Africa
Australia Russia
China
Japan
Canada
USA
Mexico
European Community
South Africa
Australia Russia
China
Japan Andean
Community
MERCOSUR a
b
Figure 3: Regional and National Companies of a Multinational Company.
Multinational
Regional National
11
1
AAgent (Level 1) AAgent (Level 3)
AAgent (Level 2)
a
Multinational
Regional National
1 1
1
Local 1
0..*
AAgent (Level 2)
AAgent (Level 1) AAgent (Level 4)
AAgent (Level 3)
b
Multinational
Regional National
11
1
Local 1
0..*
Agent
Agent
1
*
1 *
AAgent (Level 2)
AAgent (Level 1) AAgent (Level 4)
AAgent (Level 3) AAgent (Level 0)
AAgent (Level 0)
c
Figure 4: Different levels of abstraction identified in an agent-oriented modeling of a Multinational Company.
System goals
Individual goals
a1
Individual goals
a2
Individual goals
a4
Individual goals
an
Individual goals
a3
Congruence
Coherence
Figure 5: Congruence and Coherence
