## 3 System Architecture
### Fuzzy Inference Module  
The Fuzzy Inference Module is responsible for analyzing the time students spend on exercises and determining how the user interface should adapt based on this behavior. Unlike traditional binary classification approaches, which would set a strict time threshold to classify learners, fuzzy logic allows for gradual transitions between different learner types. This means that instead of rigidly categorizing a student as either \"fast\" or \"slow\", the system assigns a degree of membership to each category, ensuring a more flexible and adaptive response. The classification is based on triangular membership functions, which define the Fast, Moderate, and Slow learner categories. These categories are determined by three key thresholds:  
* Tlow--the maximum time a fast learner is expected to complete an exercise. * Tmid--the ideal time for a moderate learner to complete an exercise. * Tmax--the threshold beyond which a student is classified as slow. Each student is classified into one or more categories using triangular membership functions. The function for each category gradually increases or decreases based on time spent on an exercise. Figure 1: System architecture. Students who complete an exercise in a short amount of time are considered fast learners. However, the classification is not absolute--a student just above T\({}_{\text{low}}\) still retains partial membership in the fast category. \[\mu_{A_{11}}=\left\{ \begin{array}{cc}1,&x_{1}\leq T_{low}\\ \frac{T_{mid}-x_{1}}{T_{mid}-T_{low}},&T_{low}<x_{1}<T_{mid}\\ 0,&x_{1}\geq T_{mid}\end{array} \right\}\]  
where:  
* x\({}_{1}\) is the time spent on an exercise. * Membership value decreases gradually from 1 (certain fast learner) to 0. A student who spends less than T\({}_{\text{low}}\) minutes on an exercise is fully classified as a fast learner (\(\mu_{\text{A11}}(\text{x}_{1})=1\)). A student who takes more than T\({}_{\text{mid}}\) minutes is no longer considered fast (\(\mu_{\text{A11}}(\text{x}_{1})=0\)). Moderate learners fall in a transition zone between fast and slow learners. The function peaks at T\({}_{\text{mid}}\), meaning that students spending exactly this amount of time on an exercise are most strongly classified as moderate learners. \[\mu_{A_{12}}=\left\{ \begin{array}{cc}0,&x_{1}\leq T_{low}\\ \frac{x_{1}-T_{pair}}{T_{mid}-T_{low}},&T_{low}\leq x_{1}\leq T_{mid}\\ \frac{T_{max}-T_{mid}}{T_{max}-T_{mid}},&T_{mid}\leq x_{1}\leq T_{max}\\ 0,&x_{1}\geq T_{max}\end{array} \right\}\]  
The moderate category has a symmetrical shape around T\({}_{\text{mid}}\), meaning that students just below or just above this value still partially belong to this group. This ensures that students who are slightly slower or faster than expected are not abruptly reclassified. Students who spend significantly more time on an exercise belong to the slow learner category. However, membership gradually increases instead of being assigned abruptly. \[\mu_{A_{13}}=\left\{ \begin{array}{cc}0,&x_{1}\leq T_{mid}\\ \frac{x_{1}-T_{pair}}{T_{max}-T_{mid}},&T_{mid}<x_{1}<T_{max}\\ 1,&x_{1}\geq T_{max}\end{array} \right\}\]
* A student spending more than T\({}_{\text{max}}\) minutes on an exercise is fully classified as slow. * A student just above T\({}_{\text{mid}}\) is partially classified as slow, ensuring a smooth transition. The values of T\({}_{\text{low}}\), T\({}_{\text{mid}}\), and T\({}_{\text{max}}\) are set based on educational research and empirical observations. Typical learners exhibit different problem-solving speeds, and these thresholds allow for a reasonable classification of students [46] (Table 1). These values are not static. While they provide an initial guideline, they are later adjusted dynamically using reinforcement learning to better reflect real-world student behavior.