4.6 Algorithms for the FIRST and FOLLOW Functions

FIRST of a single symbol

1. For a terminal symbol $a$, let FIRST($a$) = $\left\{a\right\}$.
2. For a production rule of the form $A\to a\alpha$, add the terminal symbol $a$ to FIRST($A$).
3. For a production rule of the form $A\to ϵ$, add the empty string $ϵ$ to FIRST($A$).
4. For a production rule of the form $A\to X_1\cdots X_k$, add to FIRST($A$)
   • The empty string $ϵ$, if $ϵ$ is in FIRST($X_1$),…, FIRST($X_k$)
   • The terminal symbol $a$, if $ϵ$ is in FIRST($X_1$),…, FIRST($X_{i-1}$) and $a$ is in FIRST($X_i$)

FIRST of a sequence of symbols

1. The empty string $ϵ$ is in FIRST($X_1\cdots X_k$), if $ϵ$ is in FIRST($X_1$),…, FIRST($X_k$)
2. The terminal symbol $a$ is in FIRST($X_1\cdots X_k$), if the empty string $ϵ$ is in FIRST($X_1\cdots X_{i-1}$) and the terminal symbol $a$ is in FIRST($X_i$).

FOLLOW of a symbol

1. For the start symbol $S$, add the end-of-input marker $ into FOLLOW($S$)
2. For a production rule $A\to \alpha B$, add FOLLOW($A$)-$\left\{ϵ\right\}$ to FOLLOW($B$)
3. For a production rule $A\to \alpha B\beta$ where $\beta \ne ϵ$, add FIRST($\beta$)-$\left\{ϵ\right\}$ to FOLLOW($B$)
4. For a production rule $A\to \alpha B\beta$ where $ϵ$ is in FIRST($\beta$), add FOLLOW($A$) to FOLLOW($B$)