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Elimination of Left Recursion

A A | Can be replace by non-left-recursive productions

A AA A|

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Elimination of Left RecursionE E + T | TT T*F | FF ( E ) | id

E TEE +TE | T FTT *FT | F (E) | id

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TechniqueA A1 | A2 | A3 | . A m | 1| 2| 3 | n

Replace A productions by

A 1 A| 2 A|..| nA

A 1 A| 2 A|..| m A|

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Example

S Aa | bA Ac | Sd |

A Ac | Aad | bd |

S Aa | bA bdA | AA cA | adA |

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TOP-DOWN Parsing

Can be viewed as an attempt to find aleftmost derivation for an input string

Is an attempt to construct a parse tree for theinput string from the root and creating thenodes of the parse tree in preorder.

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ExampleS cAdA ab | a

Input string w = cad

S S S

c A d c A d c A d

a b a(a) (b) (c)

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Left Factoring

Is a grammar transformation Useful for producing a grammar suitable for

predictive parsing.Example:For a production:

A AA 1 | 2

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The following grammar dangling-

else problem

S iEtS | iEtSeS | aE b

Left factored:S iEtSS | a

S eS | E b

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Nonrecursive Predictive Parsinga + b $

X

Y

Z

$

Predictive ParsingProgram

ParsingTable M

Output

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Parser programThree possibilities1. If X = a = $, the parser halts and announces successful

completion of parsing2. If X = a = $, the parser pops X off the stack and advances

the input pointer to the next input symbol.3. If X is a nonterminal, the program consults entry M[X,a] of

the parsing table M. This entry will be either an X-production of the grammar or an error entry. If for example,

M[X,a] = {X UVW}, the parser replaces X on the top of the stack by WVU ( with U on top).

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Nonrecursive predictive parsing

Input : a string w and a parsing table M forgrammar G

Output: if w is in L(G), a leftmostderivation of w; otherwise, an errorindication.

Setup:Stack is initialized to S$input buffer to w$

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ProgramSet ip to point ot the first symbol of w$;Repeat

let X be top stack symbol and a the symbol pointed to by ip;if X is a terminal or $ then

if X = a then

pop X from the stack and advance ipelse error()else /* X is a nonterminal */

if M[X,a] = X Y1 Y2Y3. Yk then beginpop X from the stack;

push Y k Yk-1YK-2. Y1 onto the stack, with Y 1 on topoutput the production X Y1 Y2Y3. Yk

endelse error()

Until X = $ /* stack is empty */

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Non terminal Input Symbol

id + * ( ) $

E E->TE E TE

E E +TE E e E e

T T FT T FT

T T e T *FT T e T e

F F id F (E)

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STACK INPUT OUTPUT

$E id + id * id$

$ET id + id * id$ E- >ET

$ETF id + id * id$ T->FT

$ETid Id + id * id$ F->id

$ET + id * id$

$E + id * id$ T->e

$ET+ + id * id$ E->+TE

$ET id * id$$ETF id * id$ T->FT

$ETid id * id$ F->id

$ET * id$

$ETF* * id$ T->*FT

$ETF id$$ETid id$ F->id

$ET $

$E $ T->e

$ $ E->e

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FIRST and FOLLOW:

Rules to compute FIRST(X);

1. If X is terminal, then FIRST(X) is {X}

2. If X e is a production, then add e to FIRST(X).

3. If X is nonterminal and X Y1 Y2 ..Yk is a production then placea in FIRST(X) if for some i, a is in FIRST(Y i), and e is in all of FIRST(Y 1) . FIRST(Y i-1); that is Y 1. Yi-1 * e.

If e is in FIRST(Y j) for all j -> 1 to k then add e to FIRST(X).

Computing FIRST(X) for any string X 1 X2 X nas follows

add to FIRST(X 1 X2 .. Xn) all the non-e symbols of FIRST(X 1).

also add the non-e symbols of FIRST(X 2) if e is in FIRST(X 1). and so on

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FOLLOW(S)Rules1. Place $ on FOLLOW(S), where S is the start

symbol and $ is the input right endmarker.

2. If there is a production A B, theneverything in FIRST( ) except for e is placed inFOLLOW(B)

3. If there is a production A B, or a

production A B where FIRST( ) containse then everything in FOLLOW(A) is inFOLLOW(B)

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Example

E TEE +TE | e

T FTT *FT | eF ( E ) | id

FIRST(E) = FIRST(T) = FIRST(F) = {(,id}. R 3

FIRST(E) = {+,e} R 2 for e

FIRST(T) = {*,e}

FOLLOW(E) = FOLLOW(E) = {),$}to compute FOLLOW(E)

$ is due to r 1

) is due to rule 2

to compute FOLLOW(E)

r3 to p E TE $ & ) are addred toFOLLOW(E)

To compute FOLLOW(T) & FOLLOW(T)

Since E e, {),$} are also in FOLLOW(T)plus everything otherthen e in FIRST(E)must be placed in FOLLOW(T).

FOLLOW(F) = {+,*,),$)