In order to define LL(1), we need two preliminary definitions: FIRST and FOLLOW. FIRST is defined for a sentential form; FOLLOW is defined for a nonterminal.
),
for any sentential form ,
is the set of terminals that occur left-most on any string
derivable from including
if
is derivable from .
For a nonterminal A in a sentential form, say 
A
where
and are some string of terminals and nonterminals,
FOLLOW(A) = FIRST()
A grammar is LL(1) if and only if given any two productions
A 
,
A 
,
(a) FIRST ()
FIRST(
) = 
, and
(b) If one of or derives ,
...
then FIRST()FOLLOW(A) = .
Elimination of left-recursion and left-factoring aid in producing an LL(1)grammar. If the grammar is LL(1), we need only examine the next token to decide which production to use. The magic table contains a production at the entry for Table[Nonterminal, Input Token].
Top-down parsing uses a stack of symbols, initialized to the start symbol.
Example 4 shows this algorithm in
action with the grammar from Example 2
and the input string a := b * c + d ;
Push a $ on the stack.
Initialize the stack to the start symbol.
REPEAT WHILE stack is nonempty:
CASE top of the stack is:
terminal: IF input symbol matches terminal, THEN
advance input and pop stack, ELSE error
nonterminal: Use nonterminal and current input
symbol to find correct production in table.
Pop stack
Push right-hand side of production
from table onto stack, last symbol
first.
END REPEAT
If input is finished, THEN accept, ELSE error
