A Logic of Agent Organizations

# A Logic of Agent Organizations

Virginia Dignum Delft University of Technology - Dept. Technology, Policy and Management
The Netherlands
email: m.v.dignum@tudelft.nl
Frank Dignum Utrecht University - Institute of Information and Computing Sciences
The Netherlands
email: dignum@cs.uu.nl
###### Abstract

Organization concepts and models are increasingly being adopted for the design and specification of multi-agent systems. Agent organizations can be seen as mechanisms of social order, created to achieve global (or organizational) objectives by more or less autonomous agents. In order to develop a theory on the relation between organizational structures, organizational objectives and the actions of agents fulfilling roles in the organization a theoretical framework is needed to describe organizational structures and actions of (groups of) agents. Current logical formalisms focus on specific aspects of organizations (e.g. power, delegation, agent actions, or normative issues) but a framework that integrates and relates different aspects is missing. Given the amount of aspects involved and the subsequent complexity of a formalism encompassing them all, it is difficult to realize. In this paper, a first step is taken to solve this problem. We present a generic formal model that enables to specify and relate the main concepts of an organization (including, activity, structure, environment and others) so that organizations can be analyzed at a high level of abstraction. However, for some aspects we use a simplified model in order to avoid the complexity of combining many different types of (modal) operators.

## 1 Introduction

The growing complexity of (information) systems, characterized by distribution, heterogeneity, openness and dynamicity has lead to an increasing interest in organizational concepts by MAS researchers. As in human organizations, the specification of an explicit organization for a MAS helps coordinating the agents’ autonomous behavior [30, 2]. Traditional design of MAS is agent-centered, that is, mainly concerned with the representation of an agent’s internal knowledge or behavior. Recent research has pointed out that an agent-centered perspective on MAS has several problems. According to Jennings [31]: “the patterns and the outcomes of the interactions are inherently unpredictable, and predicting the behavior of the overall system based on its constituent components is extremely difficult (sometimes impossible) because of the high likelihood of emergent (and unwanted) behavior”.

A possible approach to organization in MAS is to look at how human organizations are organized and try to apply results from many decades of research in Organizational Theory (OT) to agent systems. Researchers in computer science and artificial intelligence are increasingly adopting concepts from the OT field to design more efficient and flexible distributed systems ([43, 6, 20]). In some of this work organizational structures are used in the design phase of the MAS and agents built as a kind of refinement of these structures. Thus agents and organizations are not independent entities. In earlier work, we have propagated to view organizations as separate entities from agents that also exist after the MAS is created. This is especially important in cases of open agent systems, where agents created by third parties can enter and leave. Taking this point of view, agents are not just fulfilling roles of an organization and following the patterns of interaction set out by the organizational structures. They have an independent existence (like in human organizations people have an existence independent from the organization). Following this approach, it becomes very important to describe the exact relation between the organizational entities on the one hand and the agents on the other hand.
In earlier work, we have taken an OT view to define organizations as a set of entities regulated by mechanisms of social order and created by more or less autonomous actors to achieve common objectives [11]. However, due to its nature, OT research tends to be not very formal from a computational perspective, which makes it difficult to move from its use as a concept or paradigm towards applying social and organizational concepts for the formalization of MAS social concepts. On the other hand, existing formal models are often limited to a specific domain and are difficult to validate [27]. That is, in general, logic lends itself better for the detailed analysis of one particular aspect of reality.

The above considerations provide two different motivations for this work. On the one hand, the need for a formal representation of organizations, with their environment, objectives and agents in a way that enables to analyze their partial contributions to the performance of the organization in a changing environment. On the other hand, the need for such a model to be realistic enough to incorporate the more ‘pragmatic’ considerations faced by real organizations. Most existing formal models lack this realism, e.g. either by ignoring temporal issues, or by taking a very restrictive view on the controllability of agents, or by assuming complete control and knowledge within the system (cf. [46, 42]).

Much of our own work in the last decade, has also focused on providing detailed formalisms of a specific aspect of social aspects of agent systems (e.g. norms, responsibility, delegation, counts-as,…). However, to date there are no comprehensive models that integrate all these issues. One of the reasons for this lack, is that a complete model is necessarily too complex to be done at once. In this paper, we take the middle ground by incorporating many aspects that are required for realism, but sometimes take simple formalizations in order to keep the logic manageable, where we refer to other work that could be used to extend some of the aspects wherever possible. The main goal of this article is rather an analysis of organizational structures and their function on a high level, that enables to represent the foundations of agent organizations without considering all the details involved. This framework includes all main concepts involved in organizational modeling, and describes a kind of placeholders where later on more complete theories for detailed concepts can be inserted. In a sense, the work presented in this paper simplifies many of the concepts for which powerful logics have been developed, but this simplification enables the integration of all aspects in one formalism. E.g. we will reduce the concept of responsibility to the concept of (organizational) initiative. Initiative can be seen as part of responsibility. It indicates that a certain role in the organization is made responsible to perform a task and the initiative to do something about it lays with that role.

Organizational analysis and planning aim to capture the functioning and development of organizations that account for the ways in which organizations respond to and bring about changes, and to design an organization’s structure accordingly to bring about greater efficiency. Different approaches to organizational analysis and planning have evolved from the rational view propagated by Taylor at the beginning of the 20th century [44], to cognitive models of organizations, and include also holistic and sociotechnical system views. These views bring forward the need to identify structures, components, objectives and environment as separate but interelated aspects of organization [28]. This implies that formal models for organizations must meet the following requirements:

1. represent notions of ability and activity of agents, without requiring knowledge about the specific actions available to a specific agent (open environments),

2. accept limitedness of agent capability,

3. represent the ability and activity of a group of agents,

4. deal with temporal issues, especially the fact that activity takes time,

5. represent the concept of ’being responsible’ for the achievement of a given state of affairs,

6. represent global (organizational) objectives and their dependency on agents’ activity, by relating activity and organizational structure,

7. represent organizations in terms of organizational roles

8. relate roles and agents (role enacting agents)

10. represent organizational dynamics (evolution of the organization over time, changes of the agent population),

11. deal with resource limitedness and the dependency of activity on resources (e.g. costs),

12. deal with normative issues (representation of boundaries for action and the violation thereof)

In the remainder of this paper, we will describe a thoroughly revised version of the Logic for Agent Organizations (LAO) introduced in [14] that will be increasingly extended to include most of these requirements. The introduction of LAO, based on CTL* is done incrementally starting with the first requirements and extend it successively to incorporate more complex requirements. The formalism presented in this paper does not yet include requirements 9 to 12. These issues have been dealt with in some of our other work, e.g. we dealt with normative issues extensively in e.g. [25]. It is a matter for future work to connect the work reported in this article to that other research. In the same way, more in-depth characterizations of other agency and social concepts can be given by incorporating existing work in, e.g., power and delegation [32, 5]. Where appropriate we will indicate the place where these more elaborate theories can be substituted into the framework for the basic relations that we adopted for the present paper.

This article is organized as follows. Section 2 gives the background for this research and discusses related work. The following sections build the formal logical framework of LAO for organizations and reorganization. In Section 3 we define the basis for organizations through abilities, activities and related concepts (which we all together call achievement modalities) of agents and groups. In Section 4, the formal model for organizations including structural and interaction properties is presented. Section 5 proposes an axiomatization for LAO. Section 6 shows how LAO can be applied to provide a formalization of different organization types. Finally, Section 8 presents our conclusions and directions for future work.

## 2 Organization Theory

We use concepts from Organization Theory (OT) as basis for this work. Even though it can be said that OT lacks formality, its concepts have been studied and used in practice for many years with successful results. OT defines Organization as an entity that allows and supports an individual (be it a person, a computer system, or an institution) to recognize its role, and the roles of others, in accomplishing collective goals. Furthermore, OT recognizes that organizations are instruments of purpose, that is they are seen as coordinated by intentions and objectives [34].

Furthermore, organizations are assumed to have both a structural and a strategic component [16]. These two components are linked in the following way. The organizational strategy influences contingencies such as size, innovation, diversification and geographic distribution of the organization. These characteristics result in different coordination mechanisms. E.g. if the whole organization resides at the same physical location informal communication can be used to react quick to any situation, but this is more difficult when the organization is spread all over the globe. Coordination mechanisms are also dependent on task uncertainty and task dependency. Different organizational structures are more appropriate for different types of tasks. E.g. factories usually have more hierarchical organization structures than consultancy companies. A change in the strategy of the organization might change both the tasks and the contingency factors and thus will have consequences for the organizational structure.

OT has for many decades investigated the issue of organizational structure. Organizational structure has essentially two purposes [17]: (1) it facilitates the flow of information within the organization in order to reduce the uncertainty of decision making; and (2) it integrates organizational behavior across the parts of the organization so that it is coordinated. Because the organizational structure is defined over the roles within an organization and not the individual agents it also lends stability over time to the organization. The organizational structure raises two challenges: division of labor and coordination [36]. The design of organizational structure determines the allocation of resources and people to specified tasks or purposes (through the roles those people fulfill), and the coordination of these resources to achieve organizational objectives [22]. Ideally, the organization is designed to fit its environment and to provide the information and coordination needed for its strategic objectives.

### 2.1 Elements of organization

Inspired by [43], we classify components of organizations into three broad classes. The first are (task) environmental factors, which are the components and features of the task (such as size, time constraints, uncertainty). The second are the structural factors, which are the components and features of the organization (such as roles, dependencies, constraints, norms and regulations). The third class of factors are agent factors, which are the characteristics of the individual agents concerning task capability, intelligence (including decision making and reasoning capabilities), social awareness, etc. These three classes of factors jointly determine the performance of the organization. To sum up, the three main issues that must be represented in any model aimed at understanding or specifying organizational performance or behavior are:

1. Environment/world: this is the space in which organizations exist. This space is not completely controllable by the agents and results of agent activity are not guaranteed.

2. Structure: describes the roles and relationships holding in the organization, and the strategy indicating the intentions of the organization in terms of objective(s).

3. Agents: are entities with the capability to act, that is to control the state of some element in the environment. Agents are defined by their capabilities.

In accordance to our previous work (e.g. [11], organizational roles, like agents, are first class concepts in our model that provide an explicit separation between organization and the agents that act on it, and enable to separate organizational objectives from agent goals. We therefore assume that agents do not necessarily share the objectives of the organization.

### 2.2 A Scenario for Organization

The study of organizations is particularly interesting in understanding and managing the congruence of organizations with their environment. That is, why is it that some organization thrive under given conditions, while others fail? Due to their characteristics of autonomy, social conformance, flexibility, collaborative problem solving, mobility, and distributed architecture, agent based modelling and/or simulation are widely used in the area of organization analysis. In particular, agent systems, have been advocated for modelling supply-chain applications [21, 37]. In this section we present a scenario to illustrate the formal modelling of organizations proposed in this paper. The scenario is based on the work described by Pelletier et al. [38] on the analysis of reorganization of the Dutch gas pipeline transport market. Economic reforms from the last decade have impacted a large reorganization of supply chain processes, such as the opening of traditional monopolist markets to competition.

The classic model for the gas market was a simple linear value chain. In this situation organization is characterized by a fairly stable environment, where parties and interactions are known and tasks are mostly fixed. In this situation, hierarchies are proven to be efficient and robust [36]. The stakeholders in this situation are the state acting as a (local) monopolist, who organizes and directs the transmission of gas from producer to consumer by controlling and directing other parties involved, namely the trader (responsible for exchanging gas from wellhead producer to a local geographic area), the shipper (responsible for transport of high-pressure gas from origin to destination) and the local transport manager (responsible for the overall capacity and flow of gas).

Following the political decision on the liberalization of the gas market in the Netherlands, the role of the monopolist disappears and the possibility for the other partners to directly contract each other and for multiple parties to enter the market is created. The environment became much more dynamic, with different parties having the possibility to create and manage supply chains and new services for the costumer. When the monopolist role is removed, the existing hierarchical structure collapses and steps must be taken to assure that the overall organization will still be able to achieve its objectives. Typically, such situation will demand extra capabilities from the parties involved in order to coordinate their activities without control by the monopolist.

## 3 Formal Core of Agent Organization

In this section we present the basis of the Logic of Agent Organization (LAO) that includes environment, agents and organization structure. We start by formally defining the environment as a set of temporally ordered set of states of affairs, or worlds. Secondly, we define the agents (and groups) in terms of their capabilities, abilities and activity possibilities. Finally, we will formally define organizational structure and activity, based on responsibilities and objectives.

### 3.1 Environment

In OT, the environment is commonly defined as the forces outside the organization that can impact it. The environment changes over time is not fully controlled by the organizations and individuals that populate it. For a formal representation of the environment, we use Kripke semantics to describe the environment and its changes. In the following, a world describes an actual state of the environment, and transitions indicate the effects of possible changes (the opportunities and threats in terms of organizational theory concepts). The set of propositional variables describes the vocabulary (or ontology) of the organization domain.

We use the well-known branching time temporal logic CTL* [18] to describe basic organization and environment concepts. For a set of propositional variables, the language, consists of two subsets: the set of state formulae and the set of path formulae, defined as follows, where we use the usual CTL* notation: : all, : there is, : next, : future, : always in future, and : until.

A simple semantic structure over which formulae of are interpreted is a tuple

 M=(W,Rt,π) (1)

where:

• is a non empty set of states,

• is a partial ordered set of (temporal) transitions between two elements of , ,

• is a valuation function which associates each with the set of atomic propositions from that are true in that world,

Each world describes the propositions of that are true in that world, and, each proposition in corresponds to a set of worlds where it is true. A transition between worlds represents an update of the truth value of (some) propositions in . The semantics of CTL* [18] distinguish between path and state formulae. A state formula is interpreted wrt a state and a path formula is interpreted wrt a path through the branching time structure given by . A path (or trace) in is a (possibly infinite) sequence , where and . We use the convention that denotes a path, and denotes state in path . We write (resp. ) to denote that state formula (resp. path formula ) is true in structure at state (resp. path ).

###### Definition 1 (World Semantics)

The rules for the satisfaction relation for state and path formulae are defined as:
iff , where iff not iff or iff , if then iff , and iff , where iff not iff or iff iff iff iff such that and

Because there are two constructs that we will use very frequently we define some abbreviations for them:

This semantic structure enables the specification of the organization environment, with the affairs holding in a state and the possible changes of state.

### 3.2 Agents and groups

The semantic structure described in the previous section does not consider the agents in the system. Intuitively, the idea is that, in organizations, changes are for some part the result of the intervention of (specific) agents. The language for LAO is defined as an extension of the language described above with semantics given as an extension and slight modification to CTL*. The semantics of includes the rules listed in Definition 1, and is extended with rules for the other modalities described in the remainder of this section. Before we give a formal definition of we first change the semantic structure over which formulae of will be interpreted and which will form the basis on which the new modalities can be given an intuitive semantics.

 MO=(Ψ,A,W,Rt,T,π,c) (2)

where:

• is a finite, non-empty set of domain facts

• is a non-empty set of states. Each state indicates a set of domain facts that hold in that state.

• is as in (1),

• is an interpretation function that associates an element of with an element of .
We overload the meaning of by also using it to associate sets of domain facts with sets of atomic propositions (). This allows us to use again as an evaluation function that associates with each world the set of atomic propositions from that are true in that world.

• is a finite, non-empty set of agents,

• is a function that associates elements of and with the facts in

• is the set of agent labels on elements of ,

The first thing to notice is that we introduced a set of domain facts next to the set of atomic propositions. This is done for two reasons. First of all the capabilities of agents will be defined over the domain facts. I.e. agents can change the truth values of atomic propositions that are linked to the domain facts (and formulas derived from them). However, agents are not capable to control all possible atomic propositions. Secondly, unlike the set of atomic propositions, the set of domain facts is per definition finite. Although this limits the expressiveness of the logic, this restriction still maintains the requirement of realism for organization models as usually the set of facts considered important for an organization is captured in finite databases. This assumption simplifies a number of definitions later on (allowing us to use operations on finite sets and using universal quantification as just an abbreviation of enumeration of facts) and is not too restrictive for the domain facts, but would be for the set of all atomic propositions.

The set of worlds is built on the set of domain facts such that we can easily connect atomic propositions to worlds using the domain facts.

This model also contains a set of agents and two relations that link the agents to the CTL* structure. The first relation that links agents to the original CTL* structure is . This relation indicates the capabilities of agents in a particular state. If then agent has the capability to change the truth value of when in state . This relation thus forms the basis of what sort of things agents are capable to influence.
The relation links sets of agents to particular transitions. These sets of agents indicate the agents that influence the changes on that transition. That is, for a transition , indicates the set of agents that indeed contribute to the changes indicated by that transition. Moreover, for each world and each agent we can indicate the set of transitions starting from for which has influence.

###### Definition 2 (Transition influence)

For a model 111Note that in the remainder, we will drop the reference to the semantic model whenever it is clear from the context. , a world and an agent , the transition influence of in , is defined by: and

Note that, at any moment, unexpected change can also occur, which is not a result of the action of any of the agents in that world (i.e. unlabeled or partially labeled transitions are possible).

#### 3.2.1 Agent Activity.

The notions of agent capability and action have been widely discussed in MAS. In this section, we draw from work in the area of the well known logical theory for agency and organized interaction introduced by Kanger-Lindahl-Pörn, more specifically from the work of Santos et al. [42] and Governatori et al. [23]. In short, their works assume that in realistic situations not all capabilities are always conductive of successful action - one can attempt to achieve something but without success. Three modal operators , and are used to described these differences. The first one, , expresses direct and successful actions: i.e. a formula like means that the agent brings it about that , that is, is a necessary result of an action by . The second one, corresponds to indirect and successful actions, i.e., means that ensures that , that is, is a necessary result of an action of some agent following an action by . Finally, means that attempts to make it the case that . The idea is that is not necessarily successful. An axiomatic definition of and is given in [42, 23].

One of the main reasons to start from these abstract operators for actions and attempts is that they allow us to conceive agents to be a kind of black boxes. We do not have to specify which precise actions and procedures the agents actually have to execute in order to reach a certain situation. From an organizational perspective, agents are autonomous and have different decision mechanisms (and even architectures and/or implementation). Only their outward behavior can be observed (i.e. agents are black boxes for the organization) and an attempt to perform an action can only be described if it is ‘obvious’ that the agent tried to perform an action. It means that we abstract away from any intentional and/or volitional elements of attempts which are completely internal to the agents. It also means that we can only state that an agent attempts to perform an action if it at least has the capability to perform the action. This is again justified by the organizational perspective on actions. If e.g. a university would let a sociology lecturer try to teach a class on nuclear physics and he failed we would not say that this was a serious attempt, because it is obvious that this person does not have the capability to teach this class (even though at an individual level the sociology lecturer might genuinely have intended to teach the class, formed intentions, etc.). This aspect is even more obvious in agent organizations. An agent should only attempt to see to it that a situation holds if it knows which actions it has to perform and it has these capabilities (designed in its program, action or plan base). So, although we have used variants of dynamic logic to characterize attempts of actions before [47] and recently a new logic of attempt is described in [33], these are very useful to describe attempts from an individual’s point of view, but not from an organizational perspective.

Abstract logics of agency start by defining the modal operator for direct, successful action, and introduce the other operators, for capability, ability, attempt or indirect action, based on the definition of . This results in axioms such as , informally meaning that if sees to it that then is (cap)able of . From a realistic perspective, such a definition is pretty uninteresting. Obviously, if you indeed do something, you must have had the ability to do it, but the interesting issue is to, given one’s capabilities, determine under which circumstances one can reach a certain state of affairs. That is, to determine in which situations it can be said that leads to . For instance, if agent is capable of achieving and also responsible and no other agent is interfering then agent will actually achieve .

Our approach is thus to start by giving the definition of agent capability and use this definition to progressively introduce the definitions for ability, attempt and activity. We furthermore provide a semantic definition of the modal operators instead of the usual axiomatic definition. The idea is that a semantic definition helps to precisely identify all the wanted characteristics, while a sound axiomatic system can be given on top of these definitions. Of course, it is also possible to start with an axiomatic system and construct a (canonical) semantic framework. However, in this case the semantics will add little to our understanding of the fundamental properties of the modal operators. Finally, we also agree with [23] that the assumption taken in [42] that indirect action always implies an impossibility for direct action is rather strong, and will not use it.

Intuitively, the ability of an agent to realize a state of affairs in a world , depends not only on the capabilities of the agent but also on the status of that world. Therefore, we define the ability of , to represent the case in which the agent has not only the potential capability to establish but is currently in a state in which it has influence over some of the possible transitions that lead to a state where holds. Thus the agent also has an actual possibility to use its capability. The attempt by agent to realize is represented by . An agent attempts to realize if holds after all states that can be reached by a transition that is influenced by . In our definition of attempt, an attempt only fails in case another agent interferes and tries achieve something which prevents to be achieved. So, we do not consider that the environment might just prevent to be achieved with some probability. We also assume an agent only attempts to achieve things it is capable of achieving (which does not necessarily mean the agent knows it is capable of achieving them). In the special case in which all next possible states from a given state are influenced by an agent , we say that is in-control in , represented by . Finally, the stit operator, (’agent sees to it that ) represents the result of successful action (that is, holds in all worlds following the current one). This notion of agent activity is based on that introduced by Pörn [41] to represent the externally ‘observable’ consequences of an action instead of the action itself, and as such abstracts from internal motivations of the agents. Stit can be seen as an abstract representation of the family of all possible actions that result in .

Formally, the agent operators are defined as follows, where all definitions are introduced as an extension to CTL*. Given a set of agents, we start by defining the language for LAO as an extension of as follows:

###### Definition 3

Given a set of agents, is such that

Semantics to all of these operators are build over the semantic notion of capabilities. As discussed above, the intuition is that an agent possesses capabilities that make action possible. That is, in order to talk about agent activity, or, that agent possesses the ability to make proposition hold in some next state in a path from the current world, we need to establish the control of the agent over the (truth) value of . For each agent , we partition the set of atomic facts in any world of M in two classes: the set of atomic facts that agent can control, , and the set of atomic facts that cannot control, . Given a set of propositional capability of , we define inductively as follows:
, , , , , iff and
Agents can control the state of in a world, represented by iff the agent controls the set of atomic facts that yield true. Furthermore, LAO is defined such that no agent can control the obvious (tautologies), and if an agent controls a fact it also controls its negation .

###### Definition 4 (Agent Capability)

Given an agent , agent capability, is defined as: iff and

###### Definition 5 (Agent Ability, Attempt and Activity)

Given an agent , agent ability, , agent attempt, , agent control, , and agent activity, , are defined as:
: iff and : iff and : iff : iff

Note that these definitions include the notion that action takes time by representing the result of action in a following world.

As an example, consider model , where , as depicted in figure 1. We furthermore define that . In this model, it holds that:
(because and ) if (because )

#### 3.2.2 Group Activity

Agents are, by definition, limited in their capabilities, that is, the set of facts in the world that they can control. This implies that certain states of affairs can only be reached if two or more agents cooperate to bring that state into existence. We define control and action of a group of agents based on the combined atomic capabilities of the agent in the group, using the normal set theoretic notions. I.e. and . In the same way as for single agents, we define composed capability of a group as follows:
, , , iff and

Table 1 gives an overview of the semantics of the and operators for groups of agents. We refer the reader to [14] for a more extensive formal specification.

## 4 Organization Structure in LAO

The idea behind organizations is that there are global objectives that can only be achieved through combined agent action. One of the main reasons for creating organizations is to provide the means for coordination that enables the achievement of organizational objectives in an efficient manner. From the definition of group capabilities, it easily follows that groups of agents have at least as many capabilities as each of their members and thus groups are more “powerful”. In order to achieve its objectives, it is necessary that besides employing the relevant agents, the organization assures that their interactions and responsibilities enable an efficient realization of its objectives.

According to this view of organization, even if the agents in the organization have group control over all organizational objectives, they still need to coordinate their activities in order to efficiently achieve those objectives. Furthermore, in most cases, the objectives of the organization are only known to a few of the agents in the organization, who may have no control over those objectives. It is therefore necessary to organize the agents in a way that enables objectives to be passed to those agents that can effectively realize them. We need therefore to extend our groups to include coordinating and task allocation concepts.

Formally this is achieved by adding a set of organizational structures to the model that can be used to evaluate constructs that describe them. Each organizational structure is a tuple

 Oi=(Asi,Ri,reai,≤i,Di,Obji,Ki) (3)

describing the elements of an organization, as follows:

• : indicates in each state the set of agents belonging to organization

• : indicates in each state the set of roles belonging to organization where stands for the (finite) set of all possible roles

• : indicates in each state which agents play which role(s)

• : defines the dependency relation between roles in an organization

• : indicates the desired state of the organization in each world

• : indicates which role in the organization needs to realize a certain objective of the organization

• with:
: indicates the positive “knowledge” of the organization
: indicates the negative “knowledge” of the organization

The function defines the agents that participate in the organization at each point in time. It determines the workforce and thus also the capabilities available to the organization.

The function defines the roles of the organization. Although we assume that the set of roles is more stable than the set of agents participating in the organization, we keep open the possibility for reorganization by having the set of roles depend on the state.

The function relates the agents to the roles they play at each moment in time. Note that, at any time, agents can play more than one role and roles can be played by more than one agent.

The relation indicates the structural dependency between the set of roles of organization . This relation can be seen as a kind of power relation between the agents playing the roles in an organization222LAO abstracts from the different types of power that exist in real organizations and furthermore does not elaborate further the consequences of power in communication between agents. More on this aspect can be found in [13].. Distribution of tasks is done following these dependency structures. is a poset satisfying the following properties:

1. (reflexivity)

2. : if and then (transitivity)

The reflexivity property makes sure that each role has power over itself and the transitivity property makes sure that an agent can distribute tasks to agents that are somewhere below it in the dependency structure333In reality, this transitivity is not always automatic, i.e. the chain of command must be followed to delegate tasks downwards. By this formalization, is meant the final distribution of tasks resulting from this chain of commands.. Because we also want to be able to talk about distribution of tasks by an agent to a group of other agents we generalize the dependency relation to groups of roles, as follows:
iff iff

Intuitively, the ordering relation in the set of roles determines the interaction possibilities between agents playing the roles. Organizational structures influence the way that agents in the organization can interact. The relation indicates that an agent playing is able to interact with an agent playing in order to request or demand some result. means that agents playing role can demand a result from agents playing any of the roles in . E.g. the CEO of a company can order both the sales manager as well as the production manager to achieve some goal (such as increase number of products sold). Finally, means that for an agent fulfilling a role from one can find a role in such that an agent fulfilling that role can demand a result from . E.g. for every person working in a construction company (carpenter, plumber, etc.) there is a manager that supervises that person. This construction makes it possible to say that the board of managers is managing the workers, while not every manager can direct all workers. In this paper, we will not further detail the types of interactions between agents (delegation, request, bid, …) but assume that the relationship will achieve some result, through a more or less complex interaction process. More on this issue can be found in [13].

The set of domain facts given by indicates the state that the organization tries to achieve. If the organization has achieved its objective.
The relation indicates which role is responsible (through the agents playing it) to initiate actions to achieve a certain objective of the organization (given by a set of domain facts, which cannot include ). We say that an organization is well-defined iff , i.e. for all desired objectives of the organization there is a role in charge of reaching that objective.
The last relation denotes the “knowledge” of the organization. In this paper, we assume this knowledge to be information that is available to all agents in the organization. It can be seen as a central database. We mainly introduce this element because we want to be able to describe situations where agents might have certain capabilities, but this is not known in the organization and thus a task is not delegated to the right agent. In [13] we discuss how the structural dependencies in an organization can influence the flow of information in that organization. As before, we take here the incremental approach, using as a ‘placeholder’ for more complex treatment of organizational information. We also explicitly do not model this operator as a traditional modal epistemic relation which would require a thorough discussion of all possible connections between this modal operator and the action related operators. This is also left for future research. We do, however, distinguish and in order to be able to distinguish between the fact that something is known to be true, known to be false or not known. We also require that which means that all facts that are known are also true and which means that things that are known to be false do not hold in the world.

Given the above organizational concepts, we can now extend the language . First we introduce some predicates that describe various organizational relations.

###### Definition 6

Given an organization , the organization language introduced in Definition 3, is extended to include the following predicates, where , and :
member, role, play, dep, know, incharge with not containing negations, desire with not containing negations
with the following semantics:
iff , iff , iff , iff , iff for , iff for , iff and , iff for , iff for , iff and iff and

We do not allow negations within the scope of and in order to keep the semantics of these concepts relatively simple. 444If we would allow negations we could e.g. not use simple sets of desires to define the semantics, because this does not provide a means to express a difference between and . In most practical cases it also appears sufficient to model all that we want as most desires and objectives are formulated as positive facts that are to be established. The same facts can be defined for groups of roles. In particular, given an organization structure and the definition of the dependency relation between roles we can also define a similar relation of dependency on a group of roles as follows:

###### Definition 7 (Group Dependency)

For , and group of roles , dep(,r,U) is defined as
.

Besides the above extensions we also want to extend the language with organizational actions, i.e, the actions performed by agents playing a role in the organization.

###### Definition 8

The organization language in Definition 6, is extended, for organization , to include the following predicates. , , and :
, , , ,

Where e.g. stands for the fact that is part of the capabilities of the role enacted by agent . In order to be able to give semantics to these types of actions we will change the semantic model with respect to actions slightly. Mainly we couple the agent labels in the model to roles. A restriction is that agents can only be coupled to roles they are playing. We assume that everything an agent does is done in its capacity of enactor of a certain role. So, agents do not act without having a role attached to it. We can still talk about an agent performing an action if the role is not important. This is made possible through the second clause in definition 9. We could also facilitate this by having a special role (something like ”agent”) which has no necessary capabilities and can be played by any agent in any state and denotes that the agent is not acting as playing an organizational role. We do not pursue this further in this paper, because we are mainly interested in agents functioning completely within the organizational setting.
Within we change the relation such that it relates tuples of agents and roles to transitions and the capabilities are now also defined for roles and tuples of agents playing roles:

• associates agents with facts in , i.e. indicates the capabilities of

• associates roles with facts in , indicating the necessary capabilites that an agent should have to play role .

• is a function that associates tuples (agent,role) in a world w with the facts in

• is the set of (agent,role) labels on elements of ,

The function restricts the relation such that can only hold if . I.e. an agent can only play a role if it has the necessary capabilities to play that role. Furthermore, is only defined iff . We assume that agents can get extra capabilities when playing roles and thus that . The set is defined inductively over in the same way as was defined over .
Roles only appear in the transition influence relation when played by an agent: can only be an element of iff holds.

We also redefine the notion of transition influence as follows:

###### Definition 9 (Transition influence)

Given a model , a world and an agent , role such that , the transition influence of () in , and are defined by:
and and :

Given the definitions above we can keep all the definitions on agent capability, ability, attempt and activity. But we now have additional definitions for agents playing roles.

###### Definition 10 (Role enacting Agent Capability, Ability, Attempt, Control and Activity)

Given agent ,, role enacting agent capability, , role enacting agent ability, , role enacting agent attempt, , role enacting agent control, , and role enacting agent activity, , are defined as:
: iff and such that : iff and : iff and : iff : iff

A few comments are due concerning the above definitions. For the purpose of this paper, we see agents as actors of roles in a organization, that is, the agent’s objectives are those of the roles they enact. We assume that, by acting according to their capabilities, agents work towards organizational objectives. We assume that agents have their own internal motivations to enact a role [7], but that motivation is not discussed further in this paper. In fact, the notion of agent in LAO is that of role-enacting agent as described in [11]. Furthermore, organizational knowledge is treated, for the moment, as a label marking facts that are explicitly known by the organization. That is, we do not provide epistemic capabilities in order to reason about knowledge, rather than asserting the fact that something is known. Future extensions to LAO will improve these initial assumptions.

Like with actions of agents we can extend the above definitions to groups of role enacting agents:

###### Definition 11 (Groups of Role enacting Agents Capability, Ability, Attempt, Control and Activity)

Given a set of agents and a set of roles , role enacting agent capability, , role enacting agent ability, , role enacting agent attempt, , role enacting agent control, , and role enacting agent activity, , are defined as:
: iff and such that : iff and : iff and if then : iff and ) : iff

### 4.1 Organization Capability

Based on the definitions given in the previous sections, we now define organization capability (or scope of control). In fact, an organization is only as good as its agents. An organization is said to be capable of achieving if there is a subset of agents in that organization that has the capability to achieve . Of course, whether an organization actually is able to achieve also depends on whether the task of achieving this objective arrives at the agents that are capable of achieving it. We will treat that aspect later on. Formally the capability of an organization is defined as follows:

###### Definition 12 (Organization Capability)

Given a model , a world and organization organizational capability is defined as:
iff

### 4.2 Responsibility and Initiative

To further refine the concept of organization, we need to be able to describe and reason about the responsibilities that are linked to the roles within the organization. Responsibilities within an organization enable agents to make decisions about what each member of the organization is expected to do, according to the organization specification, and to anticipate the tasks of others [26]. Informally, by responsibility we mean that an agent or group has to make sure that a certain state of affairs is achieved, either by realizing it itself or by delegating that result to someone else. Extensive work on responsibility has been done, amongst others, by Jones and Sergot [32] and by Castelfranchi [5], which provides a richer description of the concepts of responsibility and power. In line with our aim to provide a comprehensive but minimal formal framework to reason about organization and change, in the remainder of this section we focus on the issue of initiative taking as part of responsibility. I.e. we concentrate on the case that, if the organization has defined the responsibilities of to include some activity, agents playing that role have to initiate some action towards that activity. They cannot wait until they are asked by others to perform some task. On the other hand, the organization assumes that the agents playing the role will eventually do something about the activity that the role is responsible for555In this paper, we don’t develop further on the issue of eventual action..
We introduce the operator to represent the case that a certain role is in charge (has the initiative) of accomplishing a certain state. In order to describe this notion of initiative from the role perspective, we introduce a new operator, , such that means that has the initiative to achieve , which means that an agent playing should perform some action to achieve . The initiative operator is also defined for a group of roles, , in a similar way.

###### Definition 13

Formally, we extend the language as follows:

Before we give a semantic definition of Initiative we first show how initiative is related to incharge() previously defined facts:

 ⊨incharge(Oi,r,φ)→Irφ (4)

The validity expresses that if a role defined by the organization to be responsible to achieve it will automatically take the initiative to act such that will be achieved. Because roles do not act themselves, this means that an agent playing the role will act in order to achieve . In a richer model this would of course also involve the decision process of the agent(s), including forming the decision of delegating or performing itself. In that case, the definition below would be extended by a weaker version that states that if the role is in charge of than an agent enacting that role is obliged to take initiative to achieve . This normative aspect of the responsibility is quite important, but also very complex in combination with delegations in organizations. See e.g. [9, 10] for more details on this matter. Again, we leave the combination of that theory with the organizational framework for future work.

Note that when an agent will take initiative to achieve a state of affairs this does not guarantee successful achievement of that state of affairs. In fact, initiative can be defined informallu as: role has initiative to achieve iff an agent playing will eventually attempt to achieve or attempt to put another role in charge of . Formally initiative is defined in terms of attempt, as follows:

###### Definition 14 (Initiative)

Given model and organization , , initiative , resp. , for role , resp. group , is defined as:
iff , for some iff , for some

Note that the above definition requires that if a role has the initiative to achieve a certain state there also is at least one agent playing that role (which can then try to achieve ). From this agent it is required that eventually it attempts to achieve or that it attempts to put another role in charge of achieving . Whether the agent can attempt to ”delegate” an initiative to another role depends on the relations there exists between the roles. In the next section we will define the notion of delegation that indicates when such an attempt is certainly possible. Different properties can be defined for the initiative operator, which identify different types of organizations. For example, a well-defined organization is one where there is someone in charge for each of the organizational objectives.

###### Definition 15 (Well-Defined Organization)

Given an organization in a model , , we say that is a well-defined organization if it satisfies the following requirement:

 MO,w⊨desire(Oi,φ)→∃r:(role% (r,Oi)∧Irφ) (5)

We call an organization successful if the organization also has the capabilities to achieve each objective. Formally,

###### Definition 16 (Successful Organization)

Given an organization in a model , , we say that is a successful organization if it satisfies the following requirement:

 MO,w⊨desire(Oi,φ)→COiφ∧∃r:(role(r,Oi)∧Irφ) (6)

Note that, due to the definition of , this definition of success is dependent on the specific agents that play the roles in the organization.

### 4.3 Delegation

Delegation of tasks is defined as the capability to put an agent, or group, in charge for that task (through the roles they play). In an organization, the power of delegation is associated with structural dependencies. That is, by nature of their dependencies, some agents are capable of delegating their tasks to other agents, through the respective roles they play in the organization. Formally,

###### Definition 17 (Power of delegation)

Given an organization in a model , , the power of delegation for between two roles is defined as the following constraint in the model:
if then

Given the notions of initiative and structural dependency introduced above, we can define a good organization as follows:

###### Definition 18 (Good Organization)

Given an organization in a model , , we say that is a good organization if it satisfies the following requirement:
if then and and

where represent a group of roles in and is defined as the set of agents playing one of the roles of the set .

That is, a good organization is an organization such that if the organization has the capability to achieve and there is a group of roles in the organization responsible for realizing it, then the roles being in charge have a chain of delegation to roles that are played by agents in that are actually capable of achieving it. Note that it is possible that . Good organizations satisfy the following property:

 Irφ→◊HOiφ (7)

which informally says that if there is a role in charge of a given state of affairs, then eventually the state will be attempted (of course, the success of such attempt is dependent on possible external interferences). This follows immediately from the previous definition. We define an efficient organization as:

###### Definition 19 (Efficient Organization)

Given an organization in a model , , we say that is a efficient organization if it satisfies the following requirement:

This states that if a role is in charge to achieve but none of the agents playing role is capable, but it is known that an agent playing role does have the capability to achieve and also is a subordinate role of then some agent playing role will delegate the responsibility to role . Thus hands responsibility of tasks to those agents of which it is known that they can achieve them (if it cannot achieve them itself).

### 4.4 Supervision

Related to the notion of a good organization is the idea that agents should supervise each other’s work. I.e. if role is in charge that agent attempts to achieve a certain objective then role becomes responsible for that objective again if fails in his attempt. For instance, when a project leader delegates the task of implementing a module of the system to a certain person and that person fails to implement the module (maybe because he becomes ill) then the project leader should take back the task and give it to someone else (or do it himself). Note that we talk about the project leader being a role in the organization. In principle it does not matter which person plays that role of project leader. The person always has to act when the programmer fails. We can define this property as follows:

###### Definition 20 (Supervising Duty)

Given an organization in a model , , and group of roles , and a group of agents playing the roles , the supervising duty of roles with respect to the group agents to realize is defined as:
.

The definition states that if attempts to realize with a certain degree of uncertainty while under the responsibility of , then roles becomes directly in charge of achieving every time fails to realize .

## 5 On the axiomatization of LAO

In the previous sections we have introduced both syntax and semantics of LAO. In order to provide a logical characterization of LAO, in this section we discuss an axiomatization of LAO and give a number of theorems. The soundness of the axioms follows directly from the definitions and thus we abstain from giving formal proofs, and only indicate proof sketches where necessary. We also do not pretend that the given axiom system is complete, which would require a separate paper due to the intricate relations between the modal operators introduced.
Moreover, we give most axioms on the operators applied to single agents. These axioms can also be formulated for groups of agents. We leave these out as they would just duplicate all axioms without adding any new intuition.

We give the axioms in sets that are related to the same intuition. The first important intuition is that we do not want agents to have capabilities to achieve tautologies. I.e. achievements should be related to state of affairs whose truth value can be influenced and thus can be either true or false. This leads to the following axioms.

##### Axioms
(A1)

(A2)

(A3)

The soundness of follows directly from the definitions. The other achievement operators are defined over the capability operator and thus they “inherit” the property directly. This can be seen after we presented the axioms relating the operators a bit further below.

Given these axioms the next question obviously is whether the achievement modalities are “normal” modalities in the sense that the K axiom holds. Having the K axiom allows to at least combine achievements or attempts in order to reason about the combinations. Fortunately we still have that for most of them:

(A4)

(A4r)

(A5)

(A5r)

(A6)

(A7)