LL(1)

LL(1) Grammar

• A context-free grammar G = (V

, V

, S, P) whose

parsing table has no multiple entries is said to

be LL(1). In the name LL(1),

– the first

L

stands for scanning the input from

l

eft to

right,

– the second

L

stands for producing a

l

eftmost

derivation,

LL(1) Grammar

• A language is said to be LL(1) if it can be generated by a LL(1) grammar. It can be shown that LL(1) grammars are not ambiguous and

LL(1) Grammar

• A grammar G is LL(1) if and only if whenever A α | β

are two distinct productions of G the following conditions hold

• For no terminal a do both α and β derive string beginning with a.

• At most one of α and β can derive empty string

• If β=>ε then α does not derive any string beginning with a terminal in FOLLOW(A).Like wise α=> εthen β does not derive any string beginning with a terminal in

Table-Driven LL Parsing

To build the parsing table, we need three concepts:Nullability

Nullability

Definition (Nullable)

A nonterminal A is nullable if

Nullability

Clearly, A is nullable if it has a production

A → ε.

But A is also nullable if there are, for example, productions

A. → BC

B. → A | aC | ε

C. → aB | Cb |

Nullability

In other words, A is nullable if there is a production

A → ε,

or there is a production

A → B1B2 . . .

Bn,

Example

Example (Nullability) In the grammar

E → T E′

E′ _{→ + T E |} ε

T → F T

T′ _→

F →

′

′

* F T′ | ε

(E) | id |

num

E′ _{and T}′ _{are nullable.}

Example

Example (Nullability)

Nonterminal Nullable

E No

E′ Yes

T No

T′ Yes

FIRST and FOLLOW

Definition (FIRST)

FIRST(α) is the set of all terminals that may appear as the first symbol in a replacement string of α.

Definition (FOLLOW)

FOLLOW(α) is the set of all terminals that may follow α in a derivation.

Given a grammar G, we may define the functions FIRST and FOLLOW on the strings of symbols of

FIRST

For a grammar symbol X, FIRST(X) is computed as follows.

For every terminal X, FIRST(X) = {X}. For every nonterminal X, if

X → Y1Y2 . . . Yn

is a production, then FIRST(Y1) ⊆ FIRST(X).

Furthermore, if Y1, Y2, . . . , Yk are nullable, then

FIRST

We are concerned with FIRST(X) only for the nonterminals of the grammar.

FIRST(X) for terminals is trivial.

Example

Example (FIRST) Let the grammar be

E → T E′

E → + T E | ε

T → F T′

T′ _→

* F T′ | ε

F → (E) | id |

num

Example

Example (FIRST) Find FIRST(E).

E occurs on the left in only one production

E → T E′

Therefore, FIRST(T) ⊆ FIRST(E).

Furthermore, T is not nullable. Therefore, FIRST(E) = FIRST(T).

Example

Example (FIRST) Find FIRST(T).

T occurs on the left in only one production

T

→ F T′

Therefore,

FIRST(F) ⊆ FIRST(T). Furthermore, F is not nullable.

Therefore,

Example

Example (FIRST) Find FIRST(F).

FIRST(F) = {(, id,

num}.

Therefore,

FIRST(E) = {(, id,

num}.

FIRST(T) = {(, id,

Example

Example (FIRST) Find FIRST(E′_).

FIRST(E′_{) =}

{+}. Find FIRST(T′_).

FIRST(T′_{) =}

Example

Example (FIRST)

Nonterminal Nullable FIRST

E No {(, id, num}

E′ Yes {+}

T No {(, id, num}

T′ Yes {*}

FOLLOW

For a grammar symbol X, FOLLOW(X) is defined as

follows.

If S is the start symbol, then $ ∈

FOLLOW(S). If A → αBβ is a production, then

FIRST(β) ⊆ FOLLOW(B).

If A → αB is a production, or A → αBβ is a production and β is nullable, then

FOLLOW

We are concerned about FOLLOW(X) only for the nonterminals of the grammar.

Example

Example (FOLLOW) Let the grammar be

E → T E′

E → + T E | ε

T → F T′

T′ _→

* F T′ | ε

F → (E) | id |

num

Example

Example (FOLLOW) Find FOLLOW(E).

E is the start symbol, therefore

$ ∈ FOLLOW(E).

E occurs on the right in only one production.

F → (E)

Therefore

Example

Example (FOLLOW) Find FOLLOW(E′_).

E′ _{occurs on the right in two productions.}

E → T E′

E′ _{→ + T E}

′

Therefore,

FOLLOW(E′_{) = FOLLOW(E) =}

Example

Example (FOLLOW) Find FOLLOW(T).

T occurs on the right in two productions.

E → T E′

E′ _{→ + T E}

′

Therefore, FOLLOW(T) ⊇ FIRST(E′_{) = {+}.}

However, E′ _{is nullable, therefore it also contains}

FOLLOW(E) = {$, )} and FOLLOW(E′_{) =}

{$, )}. Therefore,

Example

Example (FOLLOW) Find FOLLOW(T′_).

T′ _{occurs on the right in two productions.}

T → F T′

T′ _→

* F T′ Therefore,

Example

Example (FOLLOW) Find FOLLOW(F).

F occurs on the right in two productions.

T → F T′

T′ _→

* F T ′_.

Therefore,

FOLLOW(F) ⊇ FIRST(T′_{) = {} *}.

However, T′ _{is nullable, therefore it also contains}

FOLLOW(T) = {+, $, )} and FOLLOW(T′_{) = {$, ),} +}.

Therefore,

Example

Example (FOLLOW)

Nonterminal Nullable FIRST FOLLOW

E No {(, id, num} {$, )}

E′ Yes {+} {$, )}

T No {(, id, num} {$, ₎, ₊}

T′ Yes {*} {$, ), +}

F No {(, id, num_{} {}

*, $, ),

Parsing table

• For each production Aα of the grammar doo the following

• For each terminal a in FIRST(A),add A α to M[A, a]

• If ε is in FIRST (α) then for each terminal b in

Parsing table

Non Terminals

Terminals

id num + * ( ) $

E _{ETE’ ETE’ ETE’}

E’ _{E’+TE’ E’ε E’ε}

T _{TFT’ TFT’ TFT’}

T’ _{T’ε T’FT’ T’ε T’ε}