The FSRS Scheduler Algorithm The FSRS scheduler algorithm is a Markov chain with several state variables and four different transition events. It is based on a variant of the DSR (Difficulty, Stability, Retrievability) model which is used to predict human memory states.
$DECAY=-0.5,$
$FACTOR = 0.9^{\frac{1}{DECAY}}-1 = \frac{19}{81}$ ,
$G$ : Grade (card rating)
$1$ : again
$2$ : hard
$3$ : good
$4$ : easy
State:
New
Learning
Review
Relearning
Scheduler weights: $(w_0, w_1,... w_{18})$
HARD PENALTY = $w_{15}$
EASY BONUS = $w_{16}$
$S$ : Stability
$D$ : Difficulty, $D \in [0,10]$
$R$ : Retrievability (probability of recall)
$r$ : Request retention
$t$ : Days since last review ("elapsed days")
$I$ : Interval ("scheduled days"). Number of days before the card is due next
Initial difficulty :
$D_0(G) = w_4-e^{w_5\cdot(G-1)}+1$
Initial stability :
$S_0(G) = w_{G-1}$
Next interval :
$I(r,S) = \frac{S}{FACTOR}\cdot\left(r^{\frac{1}{DECAY}}-1\right)$
Mean reversion :
$MR(a,b) = w_7\cdot a + (1-w_7)\cdot b$
Short-term stability :
$S^\prime_s(S,G) = S\cdot e^{w_{17}\cdot (G-3+w_{18})}$
Forgetting curve :
$R(t,S) = \left(1+FACTOR\cdot \frac{t}{S}\right)^{DECAY}$
Note that this function can be used to calculate a card's current retrievability.
Next forget stability :
$S^\prime_f(D,S,R) = w_{11} \cdot D^{-w_{12}}\cdot \Big[(S+1)^{w_{13}} -1 \Big]\cdot e^{w_{14}\cdot (1-R)}$
Next recall stability :
$$S^\prime_r(D,S,R,G) = S\cdot \left[1+e^{w_{8}}\cdot (11-D)\cdot S^{-w_9}\cdot (e^{w_{10}\cdot (1-R)}-1)\cdot \textrm{HARD PENALTY(if $G$=2)}\cdot \textrm{EASY BONUS(if $G$=4)} \right]$$
Next stability :
$$S^\prime(D,S,R,G) =
\begin{cases}
S^\prime_f\big(D,S,R\big) & \text{if } G = \text{1} \\
S^\prime_r\big(D,S,R,G\big) & \text{if } G = 2,3,4
\end{cases}$$
Next difficulty :
$D^\prime(D,G) = MR\Big(D_0(4), D-w_6\cdot (G-3)\Big)$
New card rated Easy:
State: New -> Review
$S=S_0(4)$
$D=D_0(4)$
$I = I(r,S)$
New card rated Again / Hard / Good :
State: New -> Learning
$S = S_0(G)$
$D = D_0(G)$
$I = 0$
Learning / Relearning card rated Again / Hard:
State: Learning / Relearning -> Learning / Relearning (stays the same )
$S = S^\prime_s(S,G)$
$D = D^\prime(D,G)$
$I=0$
Learning / Relearning card rated Good / Easy:
State: Learning / Relearning -> Review
$S = S^\prime_s(S,G)$
$D = D^\prime(D,G)$
$I = I(r,S)$
Review card rated Again:
State: Review -> Relearning
$S = S^\prime\big(D,S,R(t,S),G\big)$
$D = D^\prime(D,1)$
$I=0$
Review card rated Hard / Good / Easy:
State: Review -> Review (stays the same )
$S = S^\prime\big(D,S,R(t,S),G\big)$
$D = D^\prime(D,G)$
$I = I(r,S)$
If you are interested in learning more about the FSRS algorithm, you can learn more here:
