1. Inputs For input we have two a student response at time t and skill at time t + 1. These responses are one-hot encoded and for RNN: $x_t \in {0, 1}^{2N}$ $x_t^k = 1$ (wrong answer) $x_t^{k + N} = 1$ (correct answer)

N - amount of skills, k - skill at time t, $k^`$ is skill at time t+1 that is one-hot encoded to $e_{t+1}$ in which $k^`$ -th element is set to 1 and all other elements are 0.

2. Knowledge State Query $y_{t+1} = KS_t \cdot s_{t+1}$ $p_{t+1} = \sigma (y_{t+1})$ KS is a knowledge state vector and s is a skill vectors. If the skills are independent than they will be orthogonal and they wouldn't have an effect on each other, otherwise increase/decrease in one vector would lead to increase/decrease in another. The KQN approximates parameter of Bernoulli distribution as:

$P (c_{t+1} = 1| e_{1:t+1}, c_{1:t}) = P(c_{t+1} = 1 | x_{1:t}, e_{t+1}) \approx \sigma(y_{t+1})$