Semantics of organizational actions
In Sect. 2 we have informally touched upon a number of activities that typically take place within an organized group of agents: delegation, information, monitor. In what follows we give a formal semantics of these activities, aiming at capturing some of their essential features. We do not consider our analysis, however, to exhaust all the aspects involved in the notions of delegation, control and monitor within organizations. Our aim is rather to capture those aspects that look more relevant in relation with the notions of responsibility which have been introduced in Sect. 2 and which are formally investigated in Sect. 5.
Essentially, the semantics we propose formalizes the connection between the organizational structures of power, coordination and control and the organizational activities of delegation, information and monitor. The existence of structural links between roles guarantees the successful performance of those organizational actions. In other words, in order for a group of agents to act in an organized way, that is, to be able to manage their collective endeavors, specific structures between the roles are necessary in order to guarantee the effectiveness of organizational activities.
If a power relation holds between roles r and s, all delegation acts performed by an agent a enacting role r on agents enacting role s succeed in creating an obligation for these agents. Analogously, if a coordination relation holds between roles r and s, all information acts performed by agents enacting role r to agents enacting role s are successful in the sense that they create knowledge in these agents. Finally, if a control relation holds between roles r and s, all monitoring acts performed by agents enacting role r on agents enacting role s do not only create knowledge in the controller about the relevant state of affairs, but they also determine an obligation for the controller in case the controlled agent did not perform the action that is monitored.
Definition 4.1
(Semantic constraints for
a: delegate(b,α)) For any step st and s-set S such that \(a: delegate(b,\alpha) \in act(S)\) and \(S \in st\) (with \(a,b \in Ag\) and \(\alpha \in Act\)) and any \(w \in W\):
$$ reach(st,w) = \left\{ \begin{array}{ll} \{ w^\prime \ | \ w^\prime \in reach(st^\prime,w) \; \hbox{and} \; M,w^\prime \models O (b: \alpha) \} \\ \;\hbox{if}\; M,w \models K_a O(a: \alpha) \; \hbox{and} \; M,w \models Power(r,s) \\ \ \ \ \wedge rea(a,r) \wedge rea(b,s) \\ reach(st^\prime,w), \ \ \hbox{otherwise} \end{array} \right. $$
where st′ is the step obtained from st removing all occurrences of \(a: delegate(b,\alpha)\) from each of its s-sets.
Intuitively, if a power relation exists between roles that are enacted by two agents and the delegating agent about knows the to-be-delegated obligation, then a delegate action has as effect an obligation for the recipient. A delegate action implements therefore, given an appropriate power link, a form of “your wish is my command” principle.
More technically, the definition states that, provided that the preconditions in the first clause hold, the set of state-transitions generated by a step st where event \(a: delegate(b,\alpha)\) is performed, is the subset of the state-transitions generated by the step st′ where no \(a: delegate(b,\alpha)\) takes place in which all transitions end up satisfying O(b: α). In other words what delegate actions do, with respect to all the other actions being performed in the step, is just creating obligations, given that the necessary preconditions hold. If the preconditions do not hold, then it is as if the action was never performed (this is the intuitive meaning of the “otherwise” clause).
Definition 4.2
(Semantic constraints for
\(a: inform(b,\phi)\)) For any step st and s-set S such that \(a: inform(b,\phi) \in S\) and \(S \in st\) (with \(a,b \in Ag\) and \(\alpha \in Act\)) and any \(w \in W\):
$$ reach(st,w) = \left\{ \begin{array}{ll} \{ w^\prime \ | \ w^\prime \in reach(st^\prime,w) \;\hbox{and}\; M,w^\prime \models K_b \phi \} \\ \;\hbox{if}\; M,w \models K_a @_{+1}\phi \;\hbox{and}\; M,w \models Coord(r,s) \\ \ \ \ \wedge rea(a,r) \wedge rea(b,s) \\ reach(st^\prime,w), \ \ \hbox{otherwise} \end{array} \right. $$
where st′ is the step obtained from st removing all occurrences of \(a: inform(b,\phi)\) from each of its s-sets.
Intuitively, if there exists a coordination link between the role enacted by the informing agent and the role enacted by the recipient, and provided that the informing agents knows that the to-be-communicated content is going to be the case in the next state reached by the system \((K_a @_{+1}\phi)\), then an inform action always results in the creation of the corresponding epistemic state in the recipient. A coordination relation enables thus agents with reliable and trustworthy information channels. Typical to-be-communicated contents are obligations to perform some action in the next state, or information about actions just undertaken.
In analogy with the definition of delegation it is stated that, provided that the preconditions in the first clause hold, the set of state-transitions generated by a step st where event \(a: inform(b,\phi)\) is performed, is the subset of the state-transitions generated by the step st′ where no \(a: inform(b,\phi)\) takes place in which all transitions end up satisfying \(K_b \phi\).
Definition 4.3
(Semantic constraints for
\(a: monitor(b,\alpha)\)) For any step st and s-set S such that \(a: monitor(b,\alpha) \in S\) and \(S \in st\) (with \(a,b \in Ag\) and \(\alpha \in Act\)) and any \(w \in W\):
$$ reach(st,w) = \left\{ \begin{array}{ll} \{ w^\prime \ | \ w^\prime \in reach(st^\prime,w) \\ \;\hbox{and}\; M,w^\prime \models K_a DONE(b: \alpha) \} \\ \;\hbox{if}\; M,w \models DO(b: \alpha) \wedge K_a O(b: \alpha) \\ \;\hbox{and}\; M,w \models Control(r,s) \wedge rea(a,r) \wedge rea(b,s) \\ \{ w^\prime \ | \ w^\prime \in reach(st^\prime,w) \\ \;\hbox{and}\; M,w^\prime \models K_a DONE(b: \overline{\alpha}) \wedge K_a O(a: \alpha) \} \\ \;\hbox{if}\; M,w \models DO(b: \overline{\alpha}) \wedge K_a O(b: \alpha) \\ \;\hbox{and}\; M,w \models Control(r,s) \wedge rea(a,r) \wedge rea(b,s) \\ reach(st^\prime,w), \ \ \hbox{otherwise} \end{array} \right. $$
where st′ is the step obtained from st removing all occurrences of \(a: monitor(b,\alpha)\) from each of its s-sets.
Intuitively, if a control relation exists between the role enacted by the monitoring agent and the one enacted by the monitored agent, and the monitoring agent knows a certain action ought to be performed by the monitored agent, then the monitor action is:
-
an informative action (Meyer and Van der Hoek 1995), i.e., after the performance of \(a: monitor(b,\alpha)\) either \(K_a DONE(b:\alpha) \vee K_a DONE(b: \overline{\alpha})\);
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an action generating a recovery obligation on the monitoring agent, in case the monitored agent did not performed the action whose performance is checked.
This semantics constraint models therefore the idea that a control link between two roles on the one hand enables the monitoring agents with the necessary tools and capabilities for being always able to ascertain whether the action to be checked was actually performed or not, and on the other it attributes to the monitoring agent tasks of a recovery kind. Notice that monitor actions are performed in parallel with the to-be-monitored actions.
Looking at the definition from a more technical point of view, we see again the same patterns used in Definitions 4.1 and 4.2. If the preconditions in the first clause hold true, then the set of state-transitions generated by a step s where event \(a: monitor(b,\alpha)\) is performed, is the subset of the state-transitions generated by the step s′ where no \(a: monitor(b,\alpha)\) takes place in which all transitions end up satisfying \(K_a DONE(b:\phi)\). If the preconditions in the second clause hold true, then the set of transitions leads to worlds all satisfying \(\models K_a DONE(b: \overline{\alpha}) \wedge K_b O_b \alpha\). If none of the preconditions hold, the monitor action does not influence the transition generated by the step.
It is easy to see that Definitions 4.1, 4.2 and 4.3 make Formulae 9, 10 and 11 in Table 2 valid in our models.
Table 2 Validities concerning organizational actions
Organizational actions and knowledge
We consider organizational actions to be such that their necessary effects are always known to the agents performing them. In other words, the actions of delegating, informing, and monitoring, once executed by an agent, always determine the knowledge about their necessary effects which the agent expects. To use the terminology of Meyer and Van der Hoek (1995), these actions are always accordant to plan.
Definition 4.4
(Knowing the effects of organizational actions) If \(a \in OrgEvt\), then for any step st s.t. \(a \in st\):
$$ \begin{aligned} & \forall w_1, w_2: w_2 \in R(st,w_1) \Rightarrow (\forall w_3: (w_2 {\mathcal{K}}_{i} w_3 \Rightarrow (\exists w_4: (w_1 {\mathcal{K}}_{i} w_4 \\ & \& \ w_3 \in R(st,w_4))))). \end{aligned} $$
with i being the agent of event a.
Intuitively, the constraint guarantees that any world reachable via a concatenation of the transitions of step st and \({\mathcal{K}}_i\), is also reachable via a concatenation of \({\mathcal{K}}_i\) and the transitions generated by s. It can be proven (see Meyer and Van der Hoek 1995) that such constraint validates Formulae 12, 13 and 14 in Table 2.
Organizational actions, knowledge, and deontics
Organizational actions are activities by means of which any collective agency can be managed. As we have seen they are dependent on the organizational structure of a group. When an agent, given a role-based plan and an enactment configuration, is appointed to perform a certain organizational action we consider reasonable to assume that it also knows about this appointment. In fact, we consider such knowledge to follow from the role enactment itself: if an agent enacts a role it acquires knowledge about its tasks. This motivates the following semantic constraint.
Definition 4.5
(Knowing about organizational tasks) If \(a \in OrgEvt\), then for any step st s.t. \(a \not\in s\):
$$ \begin{aligned} & \forall w_1, w_2: ((w_2 \in R(st,w_1) \Rightarrow M,w_2 \models V) \Rightarrow (\exists w_3: w_1 {\mathcal{K}}_i w_3 \\ & \& \ w_2 \in R(st, w_3))) \end{aligned} $$
with i being the agent of event a.
The constraint states that if all the worlds reachable via st satisfy the violation constant, then the same worlds are reachable via a concatenation of \({\mathcal{K}}_i\) and transitions generated by s. It is easy to see that such a constraint makes Formulae 15, 16 and 17 in Table 2 valid.
Role-based and agent-based plans
Organizations “represent rationally ordered instruments for the achievement of stated goals” (Selznick 1948), that is, organizations arise in order to achieve specific objectives, and these objectives are pursued defining a number of subgoals contributing to the overall purpose of the organization. These subgoals identify the roles that are played in the organization. The relation between subgoals and overall objectives of the organization, i.e., the primitive decomposition of tasks within the organization, defines the essential form of organizational structure: “viewed in this light, formal organization is the structural expression of rational action” (Selznick 1948). Roles are the basic units over which this structure ranges determining the source of the “rational order” holding in the organization. The above quotes consider then the decomposition of tasks as the central source of structure within organizations: structure is necessary for each organization to pursue its objectives.
In order for the objectives of an organization to be realized the organization needs to “translate” them in concrete sub-goals to be systematically reached following specific plans, i.e., via complex collective actions which, once performed, guarantee the achievement of those objectives. Normally, this “translation” of objectives into plans is described via the two steps of task division and task allocation.
Through the task division process, goals are reduced to complex actions. Notice that task division consists of two steps. First a raw plan is found, which consists only of the atomic actions necessary for carrying out the organizational goal at issue:
$$ \alpha_1 \bullet \cdots \bullet \alpha_n $$
where • stands for one of the event composition operators (so, • ∈{;,&,+}), for all \(1 \leq i \leq n\;\alpha_i \in {\mathcal{A}}\) (we consider thus plans to be spelled out in terms of atomic actions). In terms of our running example, suppose that the program committee has selected the following task division for the notification of acceptance: the chairman collects the submitted papers and divides the papers among the other PC members; the PC members review the papers they have received from the chairman and send their results to the chairman; the chairman makes the final decision which papers are selected for the workshop and informs the authors about the decision. Such a row plan does not include all the organizational actions necessary for the program committee to manage the performance of the plan itself. Coordination actions are required between the chairman and the PC members (PC members should know they have to review the papers appointed to them by the chairman) as well as monitoring actions (the chairman should control all reviewers do their job and possibly take appropriate counter measures in case of failure). We call raw plans including all the necessary organizational actions proper plans or simply plans.
Through the task allocation process, each atomic component of the complex action, which is intended to realize a specific objective of the organization, is allocated to one agent. Within groups displaying an explicit organizational structure, the task allocation process consists of two essential steps. First, given a plan, each action component of the plan is linked to a role of the organization. In this view, roles are therefore placeholders within a plan description. A plan in which the atomic action components are indexed with roles identifiers is called role-based plan and it looks like this:
$$ r_1: \alpha_1 \bullet \cdots \bullet r_n: \alpha_n $$
where • stands for one of the event composition operators (so, • ∈{;,&,+}), for all \(1 \leq i \leq n\;\alpha_i \in {\mathcal{A}}\) and \(r_i \in AR\). Notice that, obviously, different actions can be indexed with a same role.
The second step in a task allocation consists in the so-called role enactment specifying which agent of the organization plays which role. Again, different roles can be enacted by a same agent.Footnote 8 In this work, agents playing a role in an organization are called role enacting agents or rea’s. We have already introduced the notion of role enactment configuration in Definition 3.1 formalized by the relation Rea in OS structures and representable in \({\mathcal{L}}^{ORG}\) via finite conjunctions of the form:
$$ rea(a_1, r_1) \wedge \cdots \wedge rea(a_i, r_i) \wedge \cdots \wedge rea(a_n, r_n) $$
such that \(\forall 1 \leq i \leq n, a_i \in Ag\) and \(r_i \in AR\).
Given a role-based plan and a role-enactment configuration a corresponding agent-based plan can be obtained which specifies which agent of the organization has to play which role in the plan. In other words a role-based plan \(Plan(AR, \tau)\) and a role enactment configuration Rea univocally determine an agent-based plan, i.e., a complex event description.
Definition 4.6
(Agent-based plan) An agent-based plan \(Plan(Ag,\tau)\) for a task τ within the set of agents Ag is a structure:
$$ Plan(Ag, \tau) = \left\langle Plan(AR, \tau), Rea\right\rangle $$
As such, agent-based plans be represented in \({\mathcal{L}}^{ORG}\) as an event expression of the form:
$$ a_1: \alpha_1 \bullet \cdots \bullet a_n: \alpha_n $$
where • ∈{;,&,+}, for all \(1 \leq i \leq n \; \alpha_i \in {\mathcal{A}}\), \(a_i \in Ag\) and such that:
-
\(Plan(Ag, \tau)\) is obtained from \(Plan(AR, \tau)\) by substitution of the role indexes r
i
with the agent indexes a
i
according to Rea,
-
\(M, w \models [a_1: \alpha_1 \bullet \cdots \bullet a_n: \alpha_n] \tau\).
The definition makes explicit the translation step of the organizations’ objective into concrete plans for groups of agents: from an organizational level (roles) to a collective agency level (agents). Complex event expressions can be seen as the result of an instantiation process of role-based plans via role enactment configurations.
Plans and structure
In the previous section we distinguished between raw plans, i.e., complex action descriptions not including any organizational action, and proper plans, i.e., complex action descriptions which include instead organizational actions. The step from raw plans to proper plans is the most typical feature of planning a collective activity with respect to planning an individual one. When a plan concerns only the performance of a single agent, organizational activities such as delegating, informing and monitoring loose their meaning since those activities just happens within the single mind of one individual agent. Groups have, instead, no single mind even though they can act as if they had one precisely by undertaking appropriate organizational activities. Given a raw plan, an organization always needs to elaborate a corresponding proper plan which can accordingly manage the knowledge flow and the control issue within the group.
As we have seen in Sect. 4.1, organizational actions require, in order to be successful, specific structural constraints among the roles of the organization and specific enactment configurations. So, if an agent-based plan requires a certain agent a to inform agent b about \(\phi\) then a suitable coordination link between the roles enacted by a and b should be put in place, or otherwise the information action could fail in transferring the necessary knowledge to b. Analogously, if an agent-based plan requires a to monitor the performance of b with respect to action α, a suitable control link between the roles enacted by a and b should be effective, or otherwise the monitoring action could fail not creating the necessary knowledge in a. These observations have precise formal counterparts. In fact, it can easily be seen that if the suitable structural links and enactment configurations do not hold, the following formulae are satisfiable in the models for \({\mathcal{L}}^{ORG}\):
$$ DO(a: inform(b,\phi)) \wedge \neg [a: inform(b,\phi)] \phi $$
(18)
$$ \begin{aligned} &DO(a: monitor(b,\alpha)) \wedge \neg [a: monitor(b,\alpha)] (K_a DONE(b: \alpha) \nonumber \\ & \vee (K_a \neg DONE(b: \alpha) \wedge K_a O(a: \alpha))) . \end{aligned} $$
(19)
Satisfiability of such formulae can be seen as a sign of faulty design of the organization, where the organizational structure is not tuned on the organizational activities needed for managing the collective agency.
There is however another face of the coin. Given a desired plan, a suitable organizational structure can be designed or, vice versa, given an organizational structure, appropriate plans can be designed to meet the objectives of the organization. Proper plans can be chosen on the basis of the available structural links and enactment configuration. The delegation activity can play an essential role in this sense, improving given plans via attributing tasks to more suitable agents. Again, this cannot successfully happen without appropriate structural links, and Formulae 18 and 19 have a delegation variant:
$$ DO(a: delegate(b,\beta)) \wedge \neg [a: delegate(b,\beta)] O(b:\beta) $$
(20)
which is also satisfiable if no power link is put in place.