4.7 LL(k) Grammars

• A context free grammar is said to be a strong LL(k) grammar if for every pair of production rules $A\to \alpha$ and $A\to \beta$ the following condition holds.

  ${\text{ FIRST}}_k\left(\alpha {\text{ FOLLOW}}_k\left(A\right)\right)\cap {\text{ FIRST}}_k\left(\beta {\text{ FOLLOW}}_k\left(A\right)\right)=\varnothing$

• The FIRST, FOLLOW, and SELECT algorithms can be generalized to $k$ symbols for use with strong LL(k) grammars.

  1. $A$ $\to$ $aBaa$ $aa,ab$ 2. $$ $\to$ $bCba$ $bb$ 3. $B$ $\to$ $b$ $ba$ 4. $$ $\to$ $ϵ$ $aa$ 5. $C$ $\to$ $b$ $bb$ 6. $$ $\to$ $ϵ$ $ba$

  FIRST FOLLOW A aa, ab, bb $$ B b, $ϵ$ aa C b, $ϵ$ bb

  aa ab ba bb A 1 1 2 B 4 3 C 6 5

• A grammar is said to be LL(k) grammar if each step in a derivation can be determined from the already constructed parse tree and the k lookahead symbols.

  The following grammar is LL(2) but not strong LL(2).

  $$		$$	strong LL(2)	LL(2)
  $A$	$\to$	$aBaa$	$aa,ab$	$aa,ab$
  $$	$\to$	$bBba$	$bb$	$bb$
  $B$	$\to$	$b$	$ba,bb$	$ba\text{ in 1},bb\text{ in 2}$
  $$	$\to$	$ϵ$	$aa,ba$	$aa\text{ in 1},ba\text{ in 2}$