Chapter 3

Chang Chi-Chung2015.6.8

LL(1) Grammar Predictive parsers, that is, recursive-descent parsers

needing no backtracking, can be constructed for a class of grammars called LL(1)

First “L” means the input from left to right. Second “L” means leftmost derivation. “1” for using one input symbol of lookahead at each

step tp make parsing action decisions. No left-recursive. No ambiguous.

LL(1) 文法 明確性文法 (No Ambiguity) 不可以有左遞迴 (No Left Recursion) 不可以有左因子 (No Left Factor)

Top-Down Parsing LL methods and recursive-descent parsing

Left-to-right, Leftmost derivation Creating the nodes of the parse tree in preorder ( depth-

first )

GrammarE T + TT ( E )T - ET id

Leftmost derivationE lm T + T lm id + T lm id + id

E E

T

+

T

idid

E

TT

+

E

T

+

T

id

Top-down Parsing

Give a Grammar G

E → T E’ E’ → + T E’ | εT → F T’T’ → * F T’ | εF → ( E ) | id

E E + T | TT T * F | FF ( E ) | id

Elimination of Left Recursion Productions of the form

A A |

are left recursive Non-left-recursions

A A’ A’ A’ | ε

When one of the productions in a grammar is left recursive then a predictive parser loops forever on certain inputs

Immediate Left-Recursion Elimination Group the Productions as

A A1 | A2 | … | Am | 1 | 2 | … | n

Where no i begins with an A Replace the A-Productions by

A 1 A’ | 2 A’ | … | n A’

A’ 1 A’ | 2 A’ | … | m A’ | ε

Example Left-recursive grammar

A A | | | A

Into a right-recursive production A AR

| AR

AR AR

| AR |

Non-Immediate Left-Recursion The Grammar

S A a | b A A c | S d | ε

The nonterminal S is left recursive, because S A a Sda But S is not immediately left recursive.

Elimination of Left Recursion Eliminating left recursion algorithm

Arrange the nonterminals in some order A1, A2, …, Anfor (each i from 1 to n) {

for (each j from 1 to i-1){ replace each production

Ai Ajwith

Ai 1 | 2 | … | kwhere

Aj 1 | 2 | … | k}

eliminate the immediate left recursion in Ai}

Example A B C | aB C A | A bC A B | C C | a

i = 1 nothing to doi = 2, j = 1 B C A | A b

B C A | B C b | a b(imm) B C A BR | a b BR

BR C b BR |

i = 3, j = 1 C A B | C C | a C B C B | a B | C C | a

i = 3, j = 2 C B C B | a B | C C | aC C A BR C B | a b BR C B | a B | C C | a(imm)C a b BR C B CR | a B CR | a CR

CR A BR C B CR | C CR |

Exercise

The grammarS A a | b A A c | S d | ε

Answer A A c | A a d | b d |

Left Factoring

Left Factoring is a grammar transformation. Predictive Parsing Top-down Parsing

Replace productionsA 1 | 2 | … | n |

withA AR | AR 1 | 2 | … | n

Example

The Grammar stmt if expr then stmt

| if expr then stmt else stmt Replace with

stmt if expr then stmt stmtsstmts else stmt | ε

Exercise

The following grammarS i E t S | i E t S e S | aE b

AnswerS i E t S S’ | aS’ e S | εE b

LL(1)

LL(1) A grammar G is LL(1) if it is not left recursive

and for each collection of productionsA 1 | 2 | … | n

for nonterminal A the following holds: FIRST(i) FIRST(j) = for all i j如果交集不是空集合，會如何？ if i * then

j * for all i j FIRST(j) FOLLOW(A) = for all i j

ExampleGrammar Not LL(1) because:

S S a | a Left recursive

S a S | a FIRST(a S) FIRST(a) ={a}

S a R | R S |

For R: S * and *

S a R aR S |

For R:FIRST(S) FOLLOW(R)

Top-down Parsing

Give a Grammar G

E → T E’ E’ → + T E’ | εT → F T’T’ → * F T’ | εF → ( E ) | id

E E + T | TT T * F | FF ( E ) | id

Example

Give a Grammar GE → T E’ E’ → + T E’ | εT → F T’T’ → * F T’ | εF → ( E ) | id

FIRSTE ( idE’ + T ( idT’ * F ( id

FOLLOW

E $ )E’ $ )T + $ )T’ + $ )F * + $ )

Non-Recursive Predictive Parsing Table-Driven Parsing Given an LL(1) grammar G = <N, T, P, S>

construct a table M[A,a] for A N, a T and use a driver program with a stack

Predictive parsingprogram (driver)

Parsing tableM

a + b $

XYZ$

stackinput

output

Predictive Parsing Table Algorithm

ExampleE T E’E’ + T E’ | T F T ’ T ’ * F T ’ | F ( E ) | id

id + * ( ) $E E T E’ E T E’E’ E’ + T E’ E’ E’ T T F T’ T F T’T’ T’ T’ * F T’ T’ T’ F F id F ( E )

A FIRST() FOLLOW(A)E T E’ ( id $ )

E’ + T E’ + $ )E’ $ )

T F T’ ( id + $ )T’ * F T’ * + $ )

T’ + $ )F ( E ) ( * + $ )F id id * + $ )

Example MATCHED STACK INPUT ACTIONE$ id + id * id$

TE’$ id + id * id$ E T E’

FT’E’$ id + id * id$ T F T ’

id T’E’$ id + id * id$ F id

id T’E’$ + id * id$ match id

id E’$ + id * id$ T ’

id +TE’$ + id * id$ E’ + T E ’

id + TE’$ id * id$ match +

id + FT’E’$ id * id$ T F T’

id + id T’E’$ id * id$ F id

id + id T’E’$ * id$ match id

id + id * FT’E’$ * id$ T’ * F T’

id + id * FT’E’$ id$ match *

id + id * id T’E’$ id$ F id

id + id * id T’E’$ $ Match id

id + id * id E’$ $ T’

id + id * id $ $ E’

Table-Driven Parsing

E T E’E’ + T E’ | T F T ’ T ’ * F T ’ | F ( E ) | id

Exercise Give a Grammar G as below

S i E t S S’S’ e S | εE b

Calculate the FIRST and FOLLOW Create a predictive parsing table

AnswerAmbiguous grammar

S i E t S S’ | aS’ e S | E b

a b e i t $S S a S i E t S S’

S’ S’ S’ e S S’

E E b

A FIRST() FOLLOW(A)S i E t S S’ i e $

S a a e $S’ e S e e $S’ e $E b b t

Error: duplicate table entry