## 1 Introduction
### Model Types  
The modeling of multi-agent systems (MASs) has evolved significantly to capture the complexities of agent interactions, including their epistemic knowledge, temporal constraints, deontic properties, and quantitative factors. Various formal models have been developed to enable reasoning about these aspects. These models can be broadly categorized into the following:  
* Classical interpreted systems--designed primarily for reasoning about epistemic and temporal properties of agents. * Deontic interpreted systems--extend classical models by incorporating reasoning about the correct (normative) behavior of agents. * Weighted and quantitative extensions--introduce cost functions and quantitative measures to enhance expressiveness. * Timed interpreted systems--allow reasoning about time-sensitive properties, either in discrete or continuous time. One of the foundational models for reasoning about multi-agent systems is the interpreted system (IS) formalism [3]. This classical model enables reasoning about epistemic (knowledge-based) and temporal properties of agents, making it particularly useful for verifying distributed protocols and knowledge-based decision-making. However, a key limitation of classical IS models is that they do not account for deontic constraints--that is, the difference between an agent's ideal behavior (following predefined rules) and actual behavior (which may involve norm violations). To address this limitation, deontic interpreted systems (DISs) were introduced [4]. These models extend IS by introducing a distinction between ideal (norm-compliant) and actual (possibly violating) behaviors, allowing for the formal analysis of agent compliance with prescribed norms in multi-agent environments. The key properties of DISs include the following:  
* Violation detection--a DIS enables the identification of norm violations, distinguishing between behaviors that conform to or deviate from ideal execution paths. * Normative state transitions--system states evolve depending on whether agents adhere to or violate norms, affecting the overall behavior of the system. A key assumption in IS and DIS models is that agents' actions do not involve associated costs. However, in many real-world scenarios, actions have quantitative impacts, such as energy consumption, computational resources, or penalties for norm violations. To enhance expressiveness, the weighted deontic interpreted system (WDIS) formalism was introduced [5]. WDIS extends DIS by incorporating quantitative reasoning, allowing agents to be evaluated based on costs, rewards, and trade-offs. This enhancement is particularly useful in decision-making under constraints, where agents must optimize their actions while adhering to deontic rules. Despite these advancements, classical IS, DIS, and WDIS models lack explicit representation of time, which is crucial for real-time multi-agent systems (RTMASs). To incorporate time into MAS modeling, the timed interpreted system (TIS) formalism was introduced [6]. TIS extends IS by enabling reasoning about discrete-time properties, making it applicable to real-time decision-making scenarios where time progression is modeled in discrete steps. A further refinement is the dense-timed interpreted system (DTIS) model [7], which enriches previous models by the following:  
* Supporting continuous (dense) time semantics, allowing for more precise reasoning about real-time constraints. * Modeling real-time properties, such as deadlines and synchronization between agents. * Analyzing temporal dependencies, enabling the verification of systems where agent decisions depend on real-time factors. Despite their advantages, DTIS models do not incorporate deontic reasoning, meaning they cannot explicitly analyze whether agents follow prescribed rules. In this paper, we introduce dense-timed deontic interpreted systems (DTDISs), which integrate norm compliance analysis from DISs and dense-time semantics from DTISs into a single framework.