2.1 Normative specification

Elements of norms are expressed in first-order predicate language. The element $\alpha$ is either a constant referring to an agent identifier or a variable that, when grounded, refers to an agent identifier. The remainder c elements are expressed through formulae that may include the connectives $\{\neg,\wedge,\vee\}$. Informally, a norm expresses that if, at some point, $c_a$ holds, then the agent $\alpha$ is obliged to see to it that $c_m$ is maintained at least until $c_d$ holds; otherwise, $\alpha$ is obliged to see to it that $c_r$ holds before the timeout $c_t$. For example, the norm

\begin{align*}\langle Ag,driving(Ag),\neg cross_red(Ag,LightID),\neg driving(Ag), fine_\kern0.7ptpaid(Value), time(500)\rangle\end{align*}

expresses that when an agent Ag is driving, he is obliged to not cross the red traffic light LightID until he is not driving; otherwise it has to pay Value before the time 500. Terms starting with uppercase letters are variables that are implicitly universally quantified.

2.2 Normative dynamics

The agents follow norm instances (López y López & Luck, Reference López y López and Luck2003) that are a copy of the specified norms (possibly grounding existing variables). Many instances of the same norm can exist. For example, the norm ‘Buyers are obliged to Pay’ could produce the instances ‘bob is obliged to transfer($100)’ and ‘tom is obliged to deposit($50)’. This section defines how the instances of norms proposed by Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) are represented, and how these instances compose the normative state of an institution.

As shown in Figure 1, a norm instance $n^\prime$ is activated as soon as its activation condition $c_a^\prime$ is satisfied, getting then into AS, creating an obligation for the agent $a^\prime$. The obligation is fulfilled when the deactivation condition $c_d^\prime$ is satisfied, becoming then deactivated (DS). Active norm instances become violated (VS) when the maintenance condition $c_m^\prime$ is no longer satisfied. Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) consider that violations can be repaired before the timeout $c_t^\prime$. If a norm instance is violated, either (i) satisfying its reparation condition $c_r^\prime$ leads it to deactivated state (DS) or (ii) the occurrence of the timeout condition $c_t^\prime$ leads it to the failure state (FS), when misbehaviours can no longer be repaired.

Figure 1 Life cycle of norm instances, s.t. g(p) is the event of some condition p becoming true (based on Panagiotidi et al., Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013)

The predicates active, viol, deactivated, and failed are defined to check a norm instance with respect to the normative state N as follows:

(1)\begin{align} N \models active(n^{\prime})\ \text{iff}\ n^{\prime} \in AS \end{align}

(2)\begin{align} N \models viol(n^\prime)\ \text{iff}\ n^\prime \in VS \end{align}

(3)\begin{align} N \models deactivated(n^\prime)\ \text{iff}\ n^\prime \in\; DS \end{align}

(4)\begin{align} N \models failed(n^\prime)\ \text{iff}\ n^\prime\in\; FS \end{align}

A normative monitor is the element responsible for checking the current normative state with respect to the current world under regulation. In Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013), a normative monitor is defined as a tuple $M_N=\langle\mathcal{N},AS,VS,DS,FS,s\rangle$ where (i) $\mathcal{N}$ is the set of considered norms, (ii) AS, VS, DS, and FS are the sets of active, violated, deactivated, and failed norm instances, and (iii) s indexes the current state of the normative monitor.The transition system for a normative monitor $M_N$ is $TS_{M_N}=\langle\Gamma_{M_N},\triangleright\rangle$, where $\Gamma_{M_N}$ is the set of all possible configurations of the normative monitor and $\triangleright \subseteq {\Gamma _{{M_N}}} \times {\Gamma _{{M_N}}}$ is a transition relation between configurations.The operational semantics of the normative monitor follows the transition rules (5)–(9).

(5)\begin{align} \frac{\langle \alpha,c_a,c_m, c_d,c_r,c_t\rangle\in\mathcal{N}\qquad c_a\theta\qquad \neg c_d\theta}{M_N\triangleright \langle \mathcal{N},AS\cup \langle \alpha^\prime,c_a\theta,c_m\theta, c_d\theta,c_r\theta,c_t\theta\rangle,VS,DS,FS,s_{i+1}\rangle}\end{align}

(6)\begin{align}\frac{n^\prime=\langle \alpha^\prime,c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\qquad n^\prime\in AS\qquad \neg c_m^\prime}{M_N\triangleright \langle \mathcal{N},AS\setminus n^\prime,VS\cup n^\prime,DS,FS,s_{i+1}\rangle}\end{align}

(7)\begin{align}\frac{n^\prime=\langle \alpha^\prime,c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\qquad n^\prime\in AS\qquad c_d^\prime}{M_N\triangleright \langle \mathcal{N},AS\setminus n^\prime,VS,DS\cup n^\prime,FS,s_{i+1}\rangle}\end{align}

(8)\begin{align}\frac{n^\prime=\langle \alpha^\prime,c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\qquad n^\prime\in VS\qquad c_r^\prime}{M_N\triangleright \langle \mathcal{N},AS,VS\setminus n^\prime,DS\cup n^\prime,FS,s_{i+1}\rangle}\end{align}

(9)\begin{align}\frac{n^\prime=\langle \alpha^\prime,c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\qquad n^\prime\in VS\qquad c_t^\prime}{M_N\triangleright \langle \mathcal{N},AS,VS\setminus n^\prime,DS,FS\cup n^\prime,s_{i+1}\rangle}\end{align}

The sets $\mathcal{N}$, AS, VS, DS, and FS are those of the $M_N$. Under a closed world assumption, the conditions $c_a$, $c_d$, $c^\prime_m$, $c^\prime_d$, $c^\prime_r$, and $c^\prime_t$ are evaluated against the state of the world to manage the normative state N as illustrated in Figure 1. For example, by the transition rule (5), if the state of the world satisfies the activation condition but does not satisfy the deactivation condition of a norm—both under a substitution $\theta$—then the monitor adds a norm instance $n^\prime=\langle \alpha^\prime,c_a\theta,c_m\theta, c_d\theta,c_r\theta,c_t\theta\rangle$ into the set AS. Variables in partially grounded conditions are implicitly universally quantified (Brachman & Levesque, Reference Brachman and Levesque2004).

3.1 Constitutive specification

The constitutive specification defines, through constitutive rules, how the elements that may be part of the environment, defined below, are considered from the institutional perspective. This section defines constitutive rules according to the SAI model, starting by the elements which they refer to (the environmental elements) and then presenting the elements that are constituted from the environment (the status functions).Footnote ⁴

Agents $a_\mathcal{X}\in\mathcal{A}_\mathcal{X}$ are represented by constants (e.g. bob). Events $e_\mathcal{X}\in\mathcal{E}_\mathcal{X}$ are pairs (e, a) s.t. e is a first-order logic predicate identifying the event with its arguments and a is (i) either a constant identifying the agent that has triggered the event or (ii) $\varepsilon$ if the event is produced by the environment itself (e.g. a clock tick). Properties $s_\mathcal{X}\in\mathcal{S_X}$ are represented by first-order logic predicates. It is important to remark that the set $\mathcal{X}$ is just a representation of the elements that potentially take part in the environment.Footnote ⁵ For example, when a SAI specification contains an event $e_\mathcal{X}\in\mathcal{E_X}$, it does not mean that $e_\mathcal{X}$ has happened in the environment. Rather, it means that the designer of the institution assumes that $e_\mathcal{X}$ may happen.

Agent-status functions are represented by constants. Event- and state-status functions are represented by first-order logic predicates. The assignment of status functions of $\mathcal{F}$ to the environment elements of $\mathcal{X}$ is specified through constitutive rules.

A constitutive rule where $x=\varepsilon$ specifies a freestanding assignment of the status function y, that is, an assignment where there is no concrete environmental element carrying y (Searle, Reference Searle2009; de Brito et al., Reference de Brito, Hübner and Boissier2018). In the case of $t=\varepsilon \wedge m=\top$, the constitutive rule is simply read as x $\texttt{count-as}$ y since y is assigned to x in any circumstance. The element x may be a variable $var\in Var$ that is substituted by an element belonging either to $\mathcal{F}$ or to $\mathcal{X}$ when the constitutive rule is interpreted. When x actually counts as y (i.e. when the conditions t and m declared in the constitutive rule are true), we say that there is a status function assignment (SFA) of the status function y to the element x. The establishment of a SFA of y to some x is the constitution of y.

According to Definition 6, the constitution of status functions is conditioned by m. The following BNF grammar defines all the well-formed formulae for m (hereafter m-formulae)Footnote ⁷

\begin{align} m ::=\ & s_\mathcal{X}\ |\ e_\mathcal{X}\ |\ s_\mathcal{F}\ |\ e_\mathcal{F} \ | \neg m \ |\ m \vee m |\ m \wedge m\ | \ \bot\ |\ \top \ | \nonumber\\ & (a_\mathcal{X}|Var)\ \texttt{"is"}(\ a_\mathcal{F}|Var)\ |\ (e_\mathcal{X}|Var)\ \texttt{"is"}\ (e_\mathcal{F}|Var)\ |\ (s_\mathcal{X}|Var)\ \texttt{"is"}\ (s_\mathcal{F}|Var) \nonumber\end{align}

The semantics of the m-formulae is given in Section 3.2 as it depends on the environmental and constitutive states that are also defined in that section.

3.2 Constitutive dynamics

Status functions are dynamically assigned to the actual environmental elements by the interpretation of constitutive rules. This section introduces the elements that are relevant to coupling this dynamics with the normative one.

Agents $a_X\in A_X$ are represented by constants referring to their identifiers. States $s_X\in S_X$ are represented by atomic formulae. Events $e_X\in E_X$ are represented by pairs (e, a) where e is the event, represented by an atomic formulae, triggered by the agent a. Events can be triggered by actions of the agents (e.g. the utterance of a bid in an auction) but can also be produced by the environment itself (e.g. a clock tick). In this case, events are represented by pairs ${(e,\varepsilon)}$.

Elements of F are SFAs, that is, relations between environmental elements and status functions. Elements of $A_F$ are pairs $\langle a_X,a_\mathcal{F}\rangle$ meaning that the agent $a_X$ has the status function $a_\mathcal{F}$. Elements of $E_F$ are triples $\langle e_X, e_\mathcal{F},a_X\rangle$ meaning that the event-status function $e_\mathcal{F}$ is assigned to the event $e_X$ produced by the agent $a_X$.Footnote ⁸ Elements of $S_F$ are pairs $\langle s_X,s_\mathcal{F}\rangle$ meaning that the state $s_X$ carries the status function $s_\mathcal{F}$.The process of interpretation of constitutive rules that builds the constitutive state is detailed in de Brito et al. (Reference de Brito, Hübner and Boissier2018). Briefly, if the actual environment holds the elements t and m of a constitutive rule $\langle x,y,t,m\rangle$, then the environmental element x constitutes the status function y, producing an SFA. If the environment is no longer holding the condition m that maintains an existing SFA, then such SFA is dropped from the constitutive state (as illustrated in Figure 3).

Figure 3 Example of constitutive state changes along time. At some moment, while a state m holds, an event t happens (cf. the first arrow in the bottom of the picture). By the constitutive rule $\texttt{x count-as y when t while m}$ ($\langle x,y,t,m\rangle \in \mathcal{C}$), such conditions lead the environmental element x to count as y in the constitutive state (cf. the dashed line). Then, the event $t^\prime$ occurs while the state $m^\prime$ holds. By the constitutive rule $\texttt{y count-as y'}$ $\texttt{when t' while m'}$, the element that counts as y (i.e. x) counts also as $y^\prime$ (cf. the dotted horizontal line). When the state m ceases to hold, the conditions to x count as y also cease to hold, and as x is no longer counting as y, it also ceases to count $y^\prime$. For example, if $x=bob,y=bidder,\text{ and }y^\prime=auction_\kern0.7ptparticipant$, s.t. $bob\in\mathcal{A_X}$ and $\{bidder,auction_\kern0.7ptparticipant\}\subseteq\mathcal{A_F}$, then bob counts as (i) bidder from t and (ii) $auction_\kern0.7ptparticipant$ from $t^\prime$

Both the environmental and constitutive states are used to evaluate the m-formulae defined in Section 3.1. For a model $M=\langle F,X,\mathcal{F}\rangle$, the semantics of m-formulae is defined as follows:

(10)\begin{align} M \models m \text{ iff } \exists\theta:& (m\theta\in E_X\vee m\theta\in S_X)\vee\\[3pt] & (\exists (e,a)\in E_X: \langle e,m\theta,a\rangle\in E_F)\ \vee\nonumber\\[3pt] & (\exists s_X\in S_X: \langle s_X,m\theta\rangle\in S_F)\nonumber\end{align}

(11)\begin{align} M \models x\ {\bf is }\ y \text{ iff }\exists\theta: & (x\theta\in A_X\wedge y\theta\in\mathcal{A_F}\wedge\langle x\theta,y\theta\rangle\in A_F)\vee \\[3pt] & (x\theta\in E_X\wedge x=(e,a)\wedge y\theta\in\mathcal{E_F}\wedge\langle e\theta,y\theta,a\theta \rangle\in E_F)\vee\nonumber\\[3pt] & (x\theta\in S_X\wedge y\theta\in\mathcal{S_F}\wedge\langle x\theta,y\theta\rangle\in S_F)\nonumber\end{align}

From the expression (10), an m-formula m is satisfied by M (i) if it represents either an event actually occurring or a state actually holding in the environment; (ii) if it represents an event-status function assigned to some environmental elements; or (iii) if it represents a state-status function assigned to some environmental state. From the expression (11), an m-formula is satisfied by M if it has the form $x {\ \text{is}\ } y$ and if, under a substitution $\theta$, either (i) $x\theta$ is the identifier of an agent that carries the agent-status function $y\theta$ or (ii) $x\theta$ identifies an event that actually carries the event-status function $y\theta$ or (iii) $x\theta$ identifies a state that actually carries the state-status function $y\theta$. As usual, $M\models \top$ and $M\not\models\bot$.

Figure 4 Example of constitutive specification

3.3 Example of SAI constitutive specification

Based on the described constitutive model, a language to specify the constitution of status functions is proposed in de Brito et al. (Reference de Brito, Hübner and Boissier2018). In that language, the symbols $\texttt{"not"}$, $\texttt{"|"}$, $\rm{"\&"}$, $\texttt{"false"}$, and $\texttt{"true"}$ correspond respectively to $\neg$, $\vee$, $\wedge$, $\bot$, and $\top$ in the m-formulae. Figure 4 shows the constitutive specification for a use case where agents collaborate to manage crisis such as flooding and car crashes in zones that may be either insecure (only professional people intervene) or secure (admits unprofessional intervention) (de Brito et al., Reference de Brito, Thévin, Garbay, Boissier, Hübner, Demazeau, Decker, Pérez and de la Prieta2015c). Norms regulate the collaboration (e.g. firefighters are obliged to evacuate insecure zones). The environment is composed of geographic information systems (GIS) and of tangible tables (Kubicki et al., Reference Kubicki, Lepreux and Kolski2012), where the agents put objects equipped with RFID tags on to signal their intended actions. The relevant information provided by the GIS are $\texttt{security-phase(Zone,Phase)}$, when a $\texttt{Zone}$ is in a specific $\texttt{Phase}$ of the crisis management, and $\texttt{nb_inhabit(Zone,X)}$ when a $\texttt{Zone}$ has $\texttt{X}$ inhabitants. The event $\texttt{put_tangible(Object,X,Y,Agent)}$ is triggered when an $\texttt{Agent}$ puts an $\texttt{Object}$ on the coordinates ($\texttt{X}$, $\texttt{Y}$) of a tangible table. The event $\texttt{send_message(Content,Zone,Agent)}$ is triggered when an $\texttt{Agent}$ sends a message with some $\texttt{Content}$ with respect to a $\texttt{Zone}$.The actions of the agents upon the tables, as well as the information from the GIS,do not have per se any meaning in the crisis scenario and, thus, they cannot ground, by themselves, the checking of the norm compliance. For instance, an agent putting an object on a specific point of the table does not mean, by itself, a command for the evacuation of a zone. Such action becomes meaningful in the crisis scenario through the interpretation of constitutive rules. For example, agents are recognized as mayor and firefighter according to the table where they are acting in (constitutive rules 1 and 2) and putting a launch_object on the coordinates (2,2) of a table signals the evacuation of the downtown (constitutive rule 3).

4.1 Representing norms through status functions

Evaluating norms with respect to the constitutive and to the normative state itself requires to define expressions that can be used to evaluate those states. Conditions over the whole constitutive state are expressed through sf-formulae $m_\mathcal{F}\in M_\mathcal{F}$. The sf-formulae are particular cases of the m-formulae whose atomic formulae are either event/state-status functions or expressions of the type $x\ \mathbf{ is }\ y$. The syntax of sf-formulae is given by the grammar (12) and their semantics follows the expressions (10) and (11).

(12)\begin{align}m_\mathcal{F} ::=&\ s_\mathcal{F}\ |\ e_\mathcal{F} \ | \neg m_\mathcal{F} \ |\ m_\mathcal{F} \vee m_\mathcal{F} |\ m_\mathcal{F} \wedge m_\mathcal{F}\ | \\ %\ a_\mathcal{X}\ \texttt{"is"}\ a_\mathcal{F}\ |\ e_\mathcal{X}\ \texttt{"is"}\ e_\mathcal{F}\ |\ s_\mathcal{X}\ \texttt{"is"}\ s_\mathcal{F} &(a_\mathcal{X}|Var)\ \texttt{"is"}(\ a_\mathcal{F}|Var)\ |\ (e_\mathcal{X}|Var)\ \texttt{"is"}\ (e_\mathcal{F}|Var)\ |\ (s_\mathcal{X}|Var)\ \texttt{"is"}\ (s_\mathcal{F}|Var)\nonumber\end{align}

The sf-formulae can be combined with the predicates active, viol, deactivated, and failed, evaluated according to the expressions (1)–(4) in the n-formulae ********$m_N\in M_N$, that follow the grammar (13).

(13)\begin{align} m_N::=m_\mathcal{F}\arrowvert active(n^\prime)\arrowvert viol(n^\prime)\arrowvert deactivated(n^\prime)\arrowvert\, failed(n^\prime)\arrowvert m_N\wedge m_N\arrowvert m_N\vee m_N\arrowvert\bot\arrowvert\top \end{align}

To link the representation of norms presented in Section 2 to the constitutive state presented in Section 3, we need to introduce status functions in the norms. For a norm $n\in\mathcal{N}$, where $n=\langle \alpha,c_a,c_m, c_d,c_r,c_t\rangle$, we define that $\alpha\in\mathcal{A_F}$, $c_a\in M_N$, $c_m\in M_N$, $c_d\in\mathcal{E_F}\cup\mathcal{S_F}$, $c_r\in\mathcal{E_F}\cup\mathcal{S_F}$, and $c_t\in M_N$. The reasons for linking the elements of the norm with these particular types are:

1. • A norm is directed to agents carrying an agent-status function ($\alpha \in\mathcal{A_F}$) and not to concrete agents.

2. • Deactivation and repairing conditions are expressed with event- and state-status functions. From the institutional perspective, agents must behave as prescribed by the norms and then produce, in the environment, events and states that can be interpreted as the corresponding event- and state-status functions.

3. • Activation, maintenance, and timeout refer to the whole constitutive and normative states and, thus, are referred in norms through particular types of formulae.

Figure 5 shows the norms as conceived in Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) using the status functions defined in the constitutive specification shown in Figure 4 to specify that (i) the mayor is obliged to evacuate secure zones and (ii) firefighters are obliged to evacuate insecure zones.

Figure 5 Norms using status functions

4.2 Linking representations of norm instances to the constitutive state

While norms refer to agent-status functions (i.e. $\alpha\in\mathcal{A_F}$), their instances prescribe the behaviour of the concrete agents acting in the environment. The obligation of an agent $a_X$ to follow a norm instance $n^\prime$ is conditioned by its carrying of the status function $\alpha$ as prescribed in the norm n. As detailed later in the expressions (15)–(18), to check this condition norm instances must record both the agent to whom the norm is directed and the status function carried by that agent when the instance was created. Thus, the representation of norm instance of Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) is extended to $n^\prime=\langle (a_X,\alpha),c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle$, where $a_X\in A_X$ points to the concrete agent targeted by the norm instance and $\alpha\in\mathcal{A_F}$ is the status function carried by that agent when the instance was created. The remainder elements are those from Definition 2. The elements $c_m^\prime, c_d^\prime,c_r^\prime$ may be partially grounded while the remainder ones are fully grounded.

5.1 Norm activation, deactivation, violation, and failure

5.1.1 Activation

Given a normative specification $\mathcal{N}$, a constitutive state F, and a normative state N, the set of norm instances to be created is given by the function activated defined below:

(14)\begin{align} activated(\mathcal{N},F,N) & = \{n^\prime \arrowvert \exists\,\theta\exists\langle \alpha,c_a,c_m, c_d,c_r,c_t\rangle\in\mathcal{N}:\nonumber \\[3pt] & \qquad\quad F\cup N\models c_a\theta \wedge (a_X {\bf{is}} \alpha\theta)\} \end{align}

\begin{align*} \text{s.t. }\ \ n^\prime=\langle (a_X,\alpha\theta),c_a\theta,c_m\theta, c_d\theta,c_r\theta,c_t\theta\rangle\end{align*}

The creation of norm instances is conditioned by the constitutive and normative states satisfying the activation condition $c_a$ for some substitution $\theta$ (i.e. $F\cup N\models c_a\theta$). The evaluation of $c_a$ with respect to N follows the expressions (1)–(4). Its evaluation with respect to F follows the expressions (10) and (11). By the function activated, a norm directed to an agent-status function $\alpha$ produces an instance for every concrete agent $a_X$ carrying $\alpha$. For example, considering the constitutive specification in Figure 4, if both agents bob and tom carry the status function of firefighter (i.e. $\{\langle bob,firefighter\rangle,\langle tom, firefighter\rangle\}\subseteq A_F$) and the downtown is in emergency phase of crisis, being thus insecure (i.e. $\langle security_\kern0.7ptphase(downtown, emergency), insecure(downtown) \rangle\in S_F $), then (i) $F\models insecure(downtown)$, (ii) $F\models bob {\bf{is}} firefighter$, and (iii) $F\models tom {\bf{is}} firefighter$. Thus, as illustrated in Figure 6, the following instances of the norm 2 of Figure 5 are created:

Figure 6 Example of norm activation. The normative specification contains a norm stating that when the downtown is insecure, firefighters are obliged to evacuate it (cf. Figure 5). The environment is in a state that counts as the downtown being insecure in the constitutive state. The holding of such condition in the constitutive state produces an instance of that norm for each agent counting as firefighter

\begin{align*} \langle (bob,firefighter),insecure(downtown),insecure(downtown),evacuate(downtown),\bot,\neg insecure(downtown)\rangle\\[2pt] \langle (tom,firefighter),insecure(downtown),insecure(downtown),evacuate(downtown),\bot,\neg insecure(downtown)\rangle\end{align*}

5.1.2 Deactivation

In the considered normative model, obligations are fulfilled when norm instances are deactivated. Deactivations are considered separately according to the nature of the deactivation condition (event or state). The functions $ f\text{-}deactivated^e$ and $f\text{-}deactivated^s$ deal respectively with deactivations of active instances conditioned by events and by states.

(15)\begin{align} f\hbox{-}deactivated^e(F,N) = \{\langle n^\prime\arrowvert &\exists (e_X,a_X)\in E_X: n^\prime \in AS \wedge c_d^\prime\in \mathcal{E}_\mathcal{F} \wedge \nonumber \\[4pt] & F\models ((e_X,a_X) {\bf{is}} c_d^\prime \vee \neg (a_X\,{\bf{is}}\,\alpha))\wedge F\cup N\models c_m^\prime\} \end{align}

(16)\begin{align} f\hbox{-}deactivated^s(F,N) = \{\langle n^\prime\arrowvert & n^\prime \in AS \wedge \nonumber c_d^\prime\in \mathcal{S_F} \wedge\\[4pt] & F\models (c_d^\prime \vee \neg (a_X\,{\bf{is}}\,\alpha)) \wedge F\cup N\models c_m^\prime\} \end{align}

\begin{align*} \text{s.t. }\ \ n^\prime=\langle (a_X,\alpha),c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\end{align*}

The function $f\hbox{-}deactivated^e$ captures the notion of events as being considered at the individual agent level. The obligation of an agent $a_X$ with respect to the occurrence in the environment of an event that counts as the event-status function $c_d^\prime$ is only fulfilled when $c_d^\prime$ is assigned to an event $e_X$ really produced by the agent $a_X$. This is expressed by the element $ F\models ((e_X,a_X){\bf{is}} c_d^\prime)$, evaluated according to the Expression (11). For instance, if two agents are obliged to produce an event that counts as the event-status function evacuate(downtown), then every obliged agent must produce such an event (Figure 7).

By the function $ f\hbox{-}deactivated^s$, an agent fulfils an obligation to achieve a state when it sees to it that such state holds, no matter by whom it has been produced. This achievement is detected when there is an assignment to the state-status function $c_d^\prime$, evaluated according to the Expression (10). For example, if two agents are obliged to see to them that the environment is in a state that counts as the downtown being evacuated, then the obligations of all the agents are fulfilled when such a state is produced, no matter by whom (Figure 8).

Figure 7 Example of norm deactivation—event-status function obligation. Both the agents bob and tom are obliged to produce, in the environment, an event that counts as the evacuation of the downtown. The agent bob produces such an event, fulfilling, thus, his obligation. The obligation of the agent tom remains active until he does the same

Figure 8 Example of norm deactivation—state-status function obligation. Both the agents bob and tom are obliged to see to them that the environment is in a state that counts as the downtown being evacuated. The agent bob produces such state. Then, both bob and tom have their obligation fulfilled

The functions $f\hbox{-}deactivated^e$ and $f\hbox{-}deactivated^s$ capture the idea of norm instances directed to the concrete agents but conditioned by the agent-SFAs. If an instance is assigned to the agent $a_X$ because it carries the agent-status function $\alpha$, then it is deactivated if $a_X$ ceases to carry $\alpha$ (Figure 9).

Figure 9 Example of norm deactivation—loss of agent-status function. The agent bob loses the status function of firefighter and, then, their standing obligations related to that status function are deactivated

The element $F\cup N\models c_m^\prime$ in expressions (15) and (16) captures the assumption of Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) that an active norm instance is deactivated if the deactivation condition starts to hold while the maintenance condition is still holding. While active norm instances are deactivated when the deactivation condition $c_d^\prime$ is satisfied, violated instances are deactivated by the satisfaction of the repair condition $c_r^\prime$. Deactivations by reparation of violated instances are also considered at the individual agent level when they are conditioned by events (function $r\hbox{-}deactivated^e$). Reparations conditioned by states are achieved when the agents see to them that such state holds (function $r\hbox{-}deactivated^s$). Different of deactivations of active instances, the maintenance condition is not considered in the reparations of violated ones. An instance, to be repaired, must be in the violated state, reached when the maintenance condition $c_m^\prime$ ceased to hold in the past. If it is the case that $c_m^\prime$ holds while the reparation condition of a violated instance is reached, such condition has started to hold again while the instance was violated, having thus no influence on such instance.

(17)\begin{align} r\hbox{-}deactivated^e(F,N) = \{ n^\prime\arrowvert &\exists (e_X,a_X)\in E_X: n^\prime\in VS \wedge c_r^\prime\in \mathcal{E}_\mathcal{F} \wedge \nonumber \\[4pt] & F\models ((e_X,a_X){\bf{is}} c_r^\prime \vee \neg (a_X\,{\bf is }\,\alpha))\} \end{align}

(18)\begin{align} r\hbox{-}deactivated^s(F,N) = \{n^\prime\arrowvert & n^\prime \in VS \wedge \nonumber c_r^\prime\in \mathcal{S_F} \wedge\\[4pt] & F\models (c_r^\prime \vee \neg (a_X\,{\bf is }\,\alpha)) \}\end{align}

\begin{align*} \text{s.t. }\ \ n^\prime=\langle (a_X,\alpha),c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\end{align*}

5.1.3 Violation

Active norm instances are considered violated when the constitutive and normative states do not satisfy the maintenance condition (function violated below).

(19)\begin{align} violated(F,N) = \{ n^\prime\arrowvert & n^\prime\in AS \wedge F\cup N\not\models c_m^\prime\}\end{align}

\begin{align*} \text{s.t. }\ \ n^\prime=\langle (a_X,\alpha),c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\end{align*}

Following the considered normative model, the Expression (19) captures the assumption that a norm instance becomes violated as soon as the maintenance condition ceases to hold, even if the deactivation or any other condition starts to hold at the same time.

5.1.4 Failure

An instance is failed if (i) it is violated and (ii) the current constitutive and normative states satisfy the timeout condition (function failed below).

(20)\begin{align} failed(F,N) = \{n^\prime\arrowvert n^\prime \in VS \wedge F\cup N\models\ c_t\prime\}\end{align}

\begin{align*} \text{s.t. }\ \ n^\prime=\langle (a_X,\alpha),c_a^\prime,c_m^\prime, c_d^\prime,c_r^\prime,c_t^\prime\rangle\end{align*}

5.2 Monitoring norms based on the constitutive state

The original operational semantics for norm monitoring proposed by Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) considers that the life cycle of the norm instances evolves based on the satisfaction of the conditions $c_a$, $c_d$, $c_m$, $c_r$, and $c_t$. Basing the regulation on the constitutive state requires also to consider other conditions captured by the functions (14)–(20). For this reason, to base the operational semantics of the normative monitor proposed by Panagiotidi et al. (Reference Panagiotidi, Álvarez-Napagao, Vázquez-Salceda, Balke, Dignum, van Riemsdijk and Chopra2013) on the SAI constitutive state, we redefine below the transition rules presented in Section 2.

(21)\begin{align}\frac{n^\prime\in activated(\mathcal{N},F,N)\ \ \ \ \ n^\prime \notin f\hbox{-}deactivated^e(F,activated(\mathcal{N},F,N))\cup f\hbox{-}deactivated^s(F,activated(\mathcal{N},F,N))}{M_N\triangleright \langle \mathcal{N},AS\cup n^\prime,VS,DS,FS,s_{i+1}\rangle}\end{align}

(22)\begin{align}\frac{n^\prime\in AS\ \ \ \ \ \ \ \ n^\prime\in violated(F,N)}{M_N\triangleright \langle \mathcal{N},AS\setminus n^\prime,VS\cup n^\prime,DS,FS,s_{i+1}\rangle}\end{align}

(23)\begin{align}\frac{n^\prime\in AS\ \ \ \ \ \ \ \ n^\prime\in\; f\hbox{-}deactivated^e(F,N)\cup f\hbox{-}deactivated^s(F,N)}{M_N\triangleright \langle \mathcal{N},AS\setminus n^\prime,VS,DS\cup n^\prime,FS,s_{i+1}\rangle}\end{align}

(24)\begin{align}\frac{n^\prime\in VS\ \ \ \ \ \ \ \ n^\prime\in r\hbox{-}deactivated^e(F,N)\cup r\hbox{-}deactivated^s(F,N)}{M_N\triangleright \langle \mathcal{N},AS,VS\setminus n^\prime,DS\cup n^\prime,FS,s_{i+1}\rangle}\end{align}

(25)\begin{align}\frac{n^\prime\in VS\ \ \ \ \ \ \ \ n^\prime\in\; failed(F,N)}{M_N\triangleright \langle \mathcal{N},AS,VS\setminus n^\prime,DS,FS\cup n^\prime,s_{i+1}\rangle}\end{align}