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The LL(1) Property
If A ? α and A ? β both appear in the grammar, we would like
FIRST(α) ∩ FIRST(β) = ∅
Predictive parsers accept LL(k) grammars. The two LL stand for Left-to-right scan of
input, left-most derivation. The k stands for number of look-ahead tokens of input. The
LL(1) Property allows the parser to make a correct choice with a look-ahead of exactly
one symbol! What about ε-productions? They complicate the definition of LL(1).
If A ? α and A ? β and ε ∈ FIRST(α) , then we need to ensure that FIRST(β) is
disjoint from FOLLOW(α), too.
Definition: FOLLOW(α) is the set of all words in the grammar that can legally appear
after an α.
For a non-terminal X, FOLLOW(X ) is the set of symbols that might follow the
derivation of X. Define FIRST+(α) as FIRST(α) ∪ FOLLOW(α), if ε ∈ FIRST(α),
FIRST(α), otherwise. Then a grammar is LL(1) iff A ? α and A ? β implies
FIRST+(α) ∩ FIRST+(β) = ∅
Given a grammar that has the is LL(1) property, we can write a simple routine to
recognize each lhs. The code is simple and fast. Consider
A ? β1 | β2 | β3 ,
which satisfies the LL(1) property FIRST+(α)∩FIRST+(β) = ∅
/* find an A */
if(token ∈ FIRST(β1))
find a β1 and return true
else if(token ∈ FIRST(β2))
find a β2 and return true
if(token ∈ FIRST(β3))
find a β3 and return true
else error and return false
Grammar with the LL(1) property are called predictive grammars because the parser can
“predict” the correct expansion at each point in the parse. Parsers that capitalize on the
LL(1) property are called predictive parsers. One kind of predictive parser is the
recursive descent parser.
