## 2 Dense-Timed Deontic Interpreted System  
Each interpreted system's formalism consists of a set of agents functioning within a specified environment. We assume the existence of a non-empty, finite set of agents \(\mathbb{A}=\{1,\ldots,n\}\), along with a unique agent \(\mathcal{E}\), representing the environment. The collection of agents in set \(\mathbb{A}\) and the environment agent \(\mathcal{E}\) together constitute a multi-agent system (MAS). To formally model our agents and define the DTDIS, we establish the following notation:  
* \(\mathbb{R}\) denotes the set of non-negative real numbers, while \(\mathbb{R}_{+}\) represents the set of positive real numbers as \(\mathbb{R}_{+}\). * \(\mathcal{X}=\bigcup_{\mathcal{E}\in\mathbb{A}}\mathcal{X}_{\mathcal{E}}\cup \mathcal{X}_{\mathcal{E}}\) is a finite set of non-negative real variables, called _clocks_, such that \(\mathcal{X}_{\mathbf{c}_{1}}\cap\mathcal{X}_{\mathbf{c}_{2}}=\emptyset\), for all \(\mathbf{c}_{1},\mathbf{c}_{2}\in\mathbb{A}\cup\{\mathcal{E}\}\). * \(v:\mathcal{X}\mapsto\mathbb{R}\) is a _clock valuation function_ that is total and assigns to each clock \(x\in\mathcal{X}\) a non-negative real value \(v(x)\). * \(\mathbb{R}^{|\mathcal{X}|}\) consists of all the clock valuations. * For \(Y\subseteq\mathcal{X}\), the valuation \(v^{\prime}=v[Y:=0]\) is defined as follows: \(\forall x\in Y,v^{\prime}(x)=0\) and \(\forall x\in\mathcal{X}\backslash Y,v^{\prime}(x)=v(x)\). * For \(\delta\in\mathbb{R}\), the valuation \(v^{\prime}=v+\delta\) is defined as follows: \(\forall x\in\mathcal{X},v^{\prime}(x)=v(x)+\delta\). * Let \(x\in\mathcal{X},c\in\mathbb{N}\), and \(\sim\in\{\leq,<,=,>,\geq\}\). The set \(\mathcal{C}(\mathcal{X})\) of _clock constraints_ over the set of clocks \(\mathcal{X}\) is defined by the following grammar: \(\alpha:=x\sim c\mid\alpha\wedge\alpha\). * For any clock valuation \(v\in\mathbb{R}^{|\mathcal{X}|}\) and \(\alpha\in\mathcal{C}(\mathcal{X})\), the satisfaction relation \(v\models\alpha\) is defined as follows: \(v\models x\sim c\) iff \(v\models\alpha\wedge\alpha\), and \(v\models\alpha\wedge\alpha\) if \(v\models\alpha\) and \(v\models\alpha\wedge\alpha\). * \(\mathcal{PV}=\bigcup_{\mathcal{E}\in\mathcal{A}}\mathcal{PV}_{\mathcal{E}} \cup\mathcal{PV}_{\mathcal{E}}\) is a set of propositional variables such that \(\mathcal{PV}_{\mathbf{c}_{1}}\cap\mathcal{PV}_{\mathbf{c}_{2}}=\emptyset\), for all \(\mathbf{c}_{1},\mathbf{c}_{2}\in\mathbb{A}\cup\{\mathcal{E}\}\). The _dense-timed deontic interpreted system_ (DTDIS) is a tuple  
\[\mathbb{D}=(\{L_{\mathbf{c}},l_{\mathbf{c}},\Sigma_{\mathbf{c}},\mathcal{X}_{ \mathbf{c}},P_{\mathbf{c}},\mathcal{V}_{\mathbf{c}},\mathcal{I}_{\mathbf{c}} \}_{\mathbf{c}\in\mathbb{A}\cup\{\mathcal{E}\}},\{t_{\mathbf{c}}\}_{\mathbf{c }\in\mathbb{A}},t_{\mathcal{E}}),\text{ where}\]
* \(L_{\mathbf{c}}=\mathcal{G}_{\mathbf{c}}\cup\mathcal{R}_{\mathbf{c}}\) is a non-empty and finite set of _local states_ of agent (environment) \(\mathbf{c}\). \(\mathcal{G}_{\mathbf{c}}\) represents a non-empty set of _ideal (faultless, norm-compliant, green)_ states, while \(\mathcal{R}_{\mathbf{c}}\) denotes a set of _faulty (non-compliant, red)_ states. Every local state associated with an agent encapsulates comprehensive information about the system available to the agent at any given moment. We operate under the assumption that the local states of \(\mathcal{E}\) are considered \"public\". * \(t_{\mathbf{c}}\subseteq L_{\mathbf{c}}\) is a non-empty set of _initial local states_ of agent (environment) \(\mathbf{c}\). * \(\Sigma_{\mathbf{c}}\) is a non-empty and finite set of _local actions_ available to agent (environment) \(\mathbf{c}\), employed to characterize the temporal progression of the system. It is presumed that the special _null_ action \(\epsilon_{\mathbf{c}}\) is included within \(\Sigma_{\mathbf{c}}\), and that all actions are considered \"public\". Each element of the set \(\Sigma=\Sigma_{1}\times\ldots\times\Sigma_{n}\times\Sigma_{\mathcal{E}}\) is referred to as a _joint action_.