First and Follow

• There are two sets we will compute to aid in parsing.
• First(α) is the set of terminals that begin strings derived from α
• Computing FIRST(α)

  
  FIND_FIRSTS
  
      
   foreach α ∈ TERMINALS
      
      FIRST(α) = {α} 
      
   foreach α ∈ NON_TERMINALS
      
      FIRST(α) ← {}
      
   repeat until no set changes
      
      foreach α ∈ NON_TERMINALS
      
         FIND_FIRST(α)
  
  
  
  
  FIND_FIRST(X)
  
        foreach  production X → Y1Y2 ... Yn
            i ← 1
            while  i ≤ n do
                  FIRST(X) ← FIRST(X) ∪ FIRST(Yi)
                  if Yi =>* ε then
                       i ← i + 1
                  else
                       i ← n+1
             if Y1Y2 ... Yn =>* ε then
                FIRST(X) ← FIRST(X) ∪ {ε}
       if X →  ε then
            FIRST(X) ← FIRST(X) ∪ {ε}
  
      

• in short : all terminals are their own first.
• All nonterminals contain the terminals they first derive, plus possibly ε
• Derive first for

  E  → TE'
  E' → +TE' | ε 
  T  → FT'
  T' → *FT' | ε
  F  → (E) | id
  
  
  FIRST(i ) = {i} where i ∈ { (, ), id, +, *}
  FIRST(F) = { (, id}
  FIRST(T) = FIRST(F) = {(,id}
  FIRST(E) = FIRST(T) = {(,id}
  FIRST(T') = {*} ∪ {ε} = { *, ε}
  FIRST(E') = {+} ∪ {ε} = { +, ε}
  	 

• The FOLLOW(A), for a nonterminal A, is the set of terminals a that can appear immediately to the right of A in some from.
• The set of a such that A =>^* αAaβ
• For this discussion $ is the end of line marker.
• Algorithm

  FIND_FOLLOW
  
   foreach α ∈ NON_TERMINALS
      FOLLOW(α) ← {}
   FOLLOW(S) ← {$}
   repeat until no set changes
      foreach α ∈ NON_TERMINALS
         FIND_FOLLOW(α)
  
  
  FIND_FOLLOW(A) 
  
   foreach production A → α B β
         FOLLOW(B) ← FOLLOW(B) ∪ (FIRST(β)-{ε})
         if ε ∈ FIRST(β) then
              FOLLOW(B) ← FOLLOW(B) ∪ FOLLOW(A)
   foreach production A → α B
              FOLLOW(B) ← FOLLOW(B) ∪ FOLLOW(A)
  
  	 

• With the above grammar.

  FOLLOW(E) = {$}
  
  Consider F → (E)
         FOLLOW(E) = FOLLOW(E) ∪ FIRST()) ({)})     = {$, )}
  
  Consider E → TE'
      A → αTβ
         FOLLOW(T) = FOLLOW(T) ∪ (FIRST(E')-{ε}) = {+}
         since ε ∈ FIRST(E')  
         FOLLOW(T) = FOLLOW(T) ∪ FOLLOW(E)  = { +,),$}
  
      A → α E'
         FOLLOW(E') = FOLLOW(E') ∪ FOLLOW(E)  = {$,)}
  E' → +TE'
      E' → α T β
         FOLLOW(T) = FOLLOW(T) ∪ (FIRST(E')-{ε}) = {+}
         since ε ∈ FIRST(E')  
         FOLLOW(T) = FOLLOW(T) ∪ FOLLOW(E)  = { +,),$}
         (NO CHANGE)
  
  T → FT'
      T → αFβ
         FOLLOW(F) = FOLLOW(T) ∪ (FIRST(T')-{ε}) = {*}
         since ε ∈ FIRST(T')  
         FOLLOW(F) = FOLLOW(F) ∪ FOLLOW(F)  = {*, +,),$}
  
      A → α E'
         FOLLOW(T') = FOLLOW(T') ∪ FOLLOW(T)  = {+,),$}
  E' → +TE'
      E' → α T β
         FOLLOW(T) = FOLLOW(T) ∪ (FIRST(E')-{ε}) = {+}
         since ε ∈ FIRST(E')  
         FOLLOW(T) = FOLLOW(T) ∪ FOLLOW(E)  = { +,),$}
         (NO CHANGE)
  
  T → FT'
      T → αFβ