## 2 Elevator Control Problem  
Domain Description: The elevator system operates in a building with several floors. The floors are numbered from 1 to Nf (where Nf is the total number of floors in the building). The elevator can move up or down one floor at a time, and it can stop at any floor. The elevator has a capacity of c passengers, and each passenger has a destination floor. Passengers can only be picked up and dropped off on the floors where the elevator stops. Within this operational domain, two distinct categories of actions can be delineated:  
* Load(p) and Unload(p) actions for each passenger p to permit that person's entrance and exit;
* 1) actions, where np is the number of passengers and Nf is the number of floors. Note that there is no up action from the top floor or down action from the bottom floor. Our elevator system, denoted as A1, encompasses a finite set of states, specifically Nf states, each symbolizing the elevator's present floor. All states within this automaton are designated as accepting states, with the initial state correspondingly set to f0. Within the context of each state or floor, the up and down actions facilitate transitions that align with the positional relationship between the current floor and the destination floor. Notably, these actions are constrained by specific rules, disallowing transitions that are incompatible with the relative positions of the current and prospective floors. Figure 1 provides a visual representation of the resultant automaton. With these elements in place, a classical planner can solve this problem and find a sequence of actions that will transport all passengers to their desired floors while minimizing the cost of the actions taken. Single-agent classical planning task tuple \(\pi\) = {\(A\), \(F\), \(I\), \(G\)} for an elevator system:  
Figure 1: Elevator automaton: generic case. * _A_: The set of possible actions that the elevator can take, which includes moving up, moving down, stopping, loading passengers, and unloading passengers. * Move up: the elevator moves up one floor. * Move down: the elevator moves down one floor. * Stop: the elevator stops at the current floor. * Load: the elevator picks up a passenger at the current floor. * Unload: the elevator drops off a passenger at the current floor. More details of each action in the elevator system and their preconditions and effects can be found in Table 1.  
Constraints: The elevator cannot carry more passengers than its capacity, and passengers can only be dropped off at their desired floor. The operation of an elevator must follow five essential constraints as outlined by [PERSON] in 1970 [19]. These constraints are as follows:  
* An elevator is not allowed to stop at a floor if no passenger is entering or exiting the elevator;
* An elevator cannot bypass a floor where a passenger wishes to disembark;
* A passenger is not allowed to enter an elevator that is moving in the opposite direction to where they need to go;
* Once a passenger is in the elevator, it cannot change direction of travel;
* Car calls always take priority over floor calls. * F: The set of possible states in the elevator system, which includes the location of the elevator, the passengers in the elevator, and the passengers waiting at each floor. A finite-state transition model is a mathematical representation of the behavior of a system. It is composed of a set of states, transitions between those states, and actions associated with each transition. To be more generic, we introduce the following, as shown in Figure 1.  
The input set, denoted as Is = {R\({}_{i}\), 1 \(\leq\) j \(\leq\) Nf}, represents the floors that have been requested; for example, R\({}_{i}\) indicates that the jth floor has been requested. The output set, denoted as Os = {MoveDown\({}_{i}\), MoveUp\({}_{k}\)}, represents the direction and the number of