## 3 Model Validation
### Model  
An illustrative model for the experiment was developed based on the proposed modeling method. The symbolic structure of the model consists of nine procedural rules concerning the primary task, eight procedural rules concerning the secondary task, and the abovementioned four procedural rules concerning task switching, as shown in Figure 5.  
The primary task, i.e., the memory game, consists of ten repetitions of a subtask, and the goal of the subtask, i.e., the subgoal, is to find a pair of matching cards. The nine procedural rules represent the accomplishment of the unity subgoal. Therefore, by starting the memory game, the start-memory-game rule is fired, and a request is made to the imaginal buffer to initialize the subgoal. Each subtask repeats the process of selecting and clicking on a card until it achieves its subgoal, that is, until it finds a pair of matching cards. If the imaginal buffer is initialized with information concerning the subgoal, the choose-random-card rule or the choose-target-card rule is fired. The choose-random-card rule requests the vision module to find a visual-location chunk, including location information concerning an arbitrary card, and requests the vision module to move the visual attention to the location based on the found visual-location chunk. Next, the open-the-card rule requires the visually attended card to be clicked to flip the card, and subsequently, the re-attend-the-card rule requests the card to be visually attended again. Under higher-difficulty conditions, the procedural rules associated with mentally determining the answer to simple calculations or solving equations are additionally fired after the execution of the re-attend-the-card rule. Through this series of rules, the model can identify the content of a card,  
Figure 5: Symbolic structure of the illustrated model. which, in turn, leads to the inspect-the-opened-card rule, which requests the declarative module to retrieve information concerning the card that was just identified. This represents the cognitive process used to determine whether the location information of the other card that matches the card that has just been identified needs to be recalled. Therefore, if the retrieval fails, it means that the content of the card that has just been identified is not in the declarative module; thus, the choose-random-card rule is fired again. Conversely, if the retrieval succeeds, it means that the declarative module includes the location information of the card with the same content as the card that has just been identified; in this case, the choose-target-card rule is fired and a request is made to the vision module to identify the location. If the clicked cards match and are cleared, the check-the-matched-cards rule is fired instead of the inspect-the-opened-card rule. The check-the-matched-cards rule clears the imaginal buffer to represent the completion of an ongoing subtask. Both the continue-memory-game rule and the decision-to-switching rule are executed when the imaginal buffer is empty; however, because the utility value of the decision-to-switching rule is set to exceed the value of the continue-memory-game rule by 0.5, the decision-to-switching rule is usually executed. Additionally, the interrupt-task, choose-random-card, and choose-target-card rules are executed when the visual buffer is empty. However, because the interrupt-task rule includes an additional condition for the level of imbalance, the rule is more likely to be executed when the scaled \(LI_{T}\) value is greater than the sum of unity and the imbalance threshold value. The imbalance threshold value was set to 0.3, as shown in Table 2. In addition, to represent variability, the scaled \(LI_{T}\) value is added to the value obtained from the noise generation function provided by ACT-R. The ACT-R noise function generates a value from a logistic distribution with a mean of 0 and an \(s\) value of 0.1 (see \(S_{\text{imbalance}}\) in Table 2).