1 Introduction
2 Informal preliminaries
2.1 Organizational structure and organizational activities
“Delegation is the primordial organizational act, a precarious venture which requires the continuous elaboration of formal mechanisms of coordination and control”.
“The description of a delegation system [power structure] is incomplete unless the simultaneous signaling system [coordination structure] applied to it is also explicitly described”.
2.2 Notions of responsibility: a sketch
3 A semantic framework for organized action
3.1 Language \({\mathcal{L}}^{ORG}\)
3.2 Models
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\({\mathcal{P}}^{+}(Agents)\) is the non-empty powerset of the finite set of actors Agents, that means the possible groups of actors.
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\({\mathbb{A}}\cup \hbox{skip}\) is the set of actions.
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\({\mathbb{W}}\) is the set of possible states.
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\([\![\;]\!]_{R}\) is a function f s.t. \(f : Evt \times {\mathbb{W}} \longrightarrow {\mathcal{P}}({\mathbb{W}})\), to each event expression-world couple it associates the set of states to which the performance of that event in that world leads. It consists of a composition of the two functions \([\![ \ ]\!]\) and R which will be introduced in Sect. 3.3.
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\(\{ {\mathcal{K}}_{i} \}_{i \in Agents}\) is a family of reflexive symmetric and transitive accessibility relations which are indexed by actors indicating the accessible worlds representing the epistemic alternatives of agent a i .
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run is a structure \(run= \left\langle {\mathbb{W}}_0, \prec \right\rangle\) modeling an actual run of the agent system and such that:
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\({\mathbb{W}}_0\) denotes the set of states of \({\mathbb{W}}\) reached by the run (\({\mathbb{W}}_0 \subseteq {\mathbb{W}}\));
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\( \prec \) is a finite path of length n on \({\mathbb{W}}_0\) (a sequence \(\left\langle w_1,\ldots, w_{n+1} \right\rangle\) of distinct elements of W 0 s.t. \(\forall w_i 1\leq i \leq n\), \(w_{i} \prec w_{i+1}\)) denoting the order in which worlds are reached in the run through actual performances of events. Path \( \prec \) has therefore always a first and a last state. We denote the first state of the run as w start .
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\(\pi\) is a usual truth function \(\pi: Ass \times W \longrightarrow \{ 1, 0 \}\)
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OS is an organizational structure (Definition 3.1).
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\(J = \left\langle J_a , J_r \right\rangle\) where \(J_a: Ag \longrightarrow Agents\), i.e., J a is a function that maps agent names into agents, and \(J_r: AR \longrightarrow Roles\), i.e., J r is a function that maps role names into the corresponding roles.
3.3 Synchronicity sets, steps, synchronicity traces, and worlds
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\(t_{1} \circ t_{2}\) is defined as follows: if \(t_{1} = \left\langle st_{1}, \ldots, st_{n} \right\rangle\) and \(t_{2} = \left\langle st_{1}^\prime, \ldots, st_{m}^\prime \right\rangle\) then, \(t_{1} \circ t_{2} = \left\langle st_{1}, \ldots, st_{n}, st_{1}^\prime, \ldots, st_{m}^\prime \right\rangle\).
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\(t_{1} \doublecap t_{2}\) is defined as follows: \(t_{1} \doublecap t_{2} = \left\{ \begin{array}{ll} t_{1} \ \ \hbox{if} \ \ t_{2} \in start(t_{1}) \\ t_{2} \ \ \hbox{if} \ \ t_{1} \in start(t_{2}) \\ \emptyset \ \ \hbox{otherwise} \end{array} \right.\) where start is a function which associates to a given s-trace all its starting possible s-traces: \(start(t) = \{ t^\prime \ | \ t^\prime = t \; or\; \exists t^{\prime\prime} \neq \emptyset \hbox{ s.t. } t^\prime \circ t^{\prime\prime} = t \}\) .
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If \(t = \left\langle st_{1}, \ldots , st_{n} \right\rangle\) then \(\tilde{t}\) is defined as follows:where \(\tilde{st} = Step - \{ st \}\).7.$$ \tilde{t} = \bigcup_{1 \leq n \leq dur(t)} \left\langle st_{1}, \ldots , \tilde{st_{n}} \right\rangle $$
3.4 Evaluating formulae
3.5 Deontic notions
\(O (X \cup Y : \overline{\gamma}) \rightarrow O (X : \overline{\gamma})\)
| (1) |
\(O (X : \gamma) \rightarrow O (X \cup Y : \gamma)\)
| (2) |
\(O (X : \gamma_{1} + Y: \gamma_{1}) \rightarrow O (X \cup Y : \gamma)\)
| (3) |
\(O (X : \gamma_{1} \ \& \ Y : \gamma_{2}) \rightarrow O (X \cup Y : \gamma_{1} \& \gamma_{2})\)
| (4) |
\(O (X : \gamma_{1} \ ; \ Y : \gamma_{2}) \rightarrow O (X \cup Y : \gamma_{1} ; \gamma_{2}) \)
| (5) |
\(O (X : \alpha_{1} \ \& \ Y : \alpha_{2}) \leftrightarrow O (X: \alpha_{1}) \wedge O(Y: \alpha_{2})\)
| (6) |
\(O (X : \alpha_{1} \ ; \ Y : \alpha_{2}) \leftrightarrow O (X : \alpha_{1}) \wedge [X : \alpha_{1}]O(Y : \alpha_{2})\)
| (7) |
\( O (X : \alpha_{1}) \vee O(Y : \alpha_{2}) \rightarrow O (X : \alpha_{1} + Y : \alpha_{2})\)
| (8) |
4 Organizational actions and organizational structure
4.1 Semantics of organizational actions
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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.
\( \begin{aligned} \;&(Power(r,s) \wedge rea(a,r) \wedge rea(b,s) \wedge K_a O(a: \alpha))\\ &\rightarrow [a: delegate(b,\alpha )] O (b: \alpha) \end{aligned}\)
| (9) |
\((Coord(r,s) \wedge rea(a,r) \wedge rea(b,s) \wedge K_a@_{+1} \phi) \rightarrow [a: inform(b,\phi)] K_b \phi\)
| (10) |
\( \begin{aligned} \;&(Control(r,s) \wedge rea(a,r) \wedge rea(b,s) \wedge K_a O(b:\alpha))\\ &\rightarrow [a: monitor(b, \alpha)] (K_a DONE(b: \alpha) \vee(K_a DONE(b: \overline{\alpha}) \\ &\wedge K_a O(a: \alpha))) \end{aligned}\)
| (11) |
\( K_a ([a: delegate(b,\alpha)] \psi) \rightarrow [a: delegate(b,\alpha)] K_a \psi\)
| (12) |
\( K_a ([a: inform(b,\phi)] \psi) \rightarrow [a: inform(b,\phi)] K_a \psi\)
| (13) |
\( K_a ([a: monitor(b,\alpha)] \psi) \rightarrow [a: monitor(b,\alpha)] K_a \psi\)
| (14) |
\( O(a: delegate(b,\alpha)) \rightarrow K_a O(a: delegate(b,\alpha))\)
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\( O(a: inform(b,\phi)) \rightarrow K_a O(a: inform(b,\phi))\)
| (16) |
\( O(a: monitor(b,\alpha)) \rightarrow K_a O(a: monitor(b,\alpha)).\)
| (17) |
4.2 Organizational actions and knowledge
4.3 Organizational actions, knowledge, and deontics
4.4 Role-based and agent-based plans
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\(Plan(Ag, \tau)\) is obtained from \(Plan(AR, \tau)\) by substitution of the role indexes r i with the agent indexes a i according to Rea,
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\(M, w \models [a_1: \alpha_1 \bullet \cdots \bullet a_n: \alpha_n] \tau\).
4.5 Plans and structure
5 Responsibilities in form
5.1 Causal responsibility
5.2 Blameworthiness
5.3 Task-based responsibility
5.4 Accountability
5.5 Responsibilities and organizational structure
6 Related work
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the notions of objectives and plans (and therefore task division and task allocation) of the organization;
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the organizational actions of delegating, informing and monitoring, that is to say, with the issue of the management of the collective activity;
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the notions of role, organizational structure, role-enacting agent.