lec04-topdownparser

8/7/2019 lec04-topdownparser

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Top-Down Parsing

The parse tree is created top to bottom.

Top-down parser Recursive-Descent Parsing

Backtracking is needed (If a choice of a production rule does not work, we backtrack to try other

alternatives.)

It is a general parsing technique, but not widely used.

Not efficient

Predictive Parsing

no backtracking

efficient

needs a special form of grammars (LL(1) grammars).

Recursive Predictive Parsing is a special form of Recursive Descent parsing withoutbacktracking.

Non-Recursive (Table Driven) Predictive Parser is also known as LL(1) parser.


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Recursive-Descent Parsing (uses Backtracking)

Backtracking is needed.

It tries to find the left-most derivation.

Sp aBc

Bp bc | b

S S

input: abc

a B c a B c

b c bfails, backtrack


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Recursive Predictive Parser

a grammar a grammar suitable for predictive

eliminate left parsing (a LL(1) grammar)

left recursion factor no %100 guarantee.

When re-writing a non-terminal in a derivation step, a predictive parsercan uniquely choose a production rule by just looking the currentsymbol in the input string.

Ap E1 | ... | En input: ... a .......

current token


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Recursive Predictive Parser (example)

stmtp if exprthen stmt else stmt|

while exprdo stmt|

begin .stmt_list end

When we are trying to write the non-terminal stmt, if the current tokenis if we have to choose first production rule.

When we are trying to write the non-terminal stmt, we can uniquely

choose the production rule by just looking the current token. We eliminate the left recursion in the grammar, and left factor it. But it

may not be suitable for predictive parsing (not LL(1) grammar).


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Recursive Predictive Parsing

Recursive Descent parsing without backtracking. Each non-terminal corresponds to a procedure.

Ex: Ap aBb (This is only the production rule for A)

proc A {- match the current token with a, and move to the next token;

- call B;

- match the current token with b, and move to the next token;

}


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Recursive Predictive Parsing (cont.)

Ap aBb | bAB

proc A {

case of the current token {

a: - match the current token with a, and move to the next token;- call B;

- match the current token with b, and move to the next token;

b: - match the current token with b, and move to the next token;

- call A;

- call B;

}

}


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Recursive Predictive Parsing (cont.)

When to apply I-productions.

Ap aA | bB | I

If all other productions fail, we should apply an I-production. For

example, if the current token is not a or b, we may apply the

I-production.

Most correct choice: We should apply an I-production for a non-

terminal A when the current token is in the follow set of A (whichterminals can follow A in the sentential forms).


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Recursive Predictive Parsing (Example)

Ap aBe | cBd | C

Bp bB | I

Cp fproc C { match the current token with f,

proc A { and move to the next token; }

case of the current token {

a: - match the current token with a,

and move to the next token; proc B {

- call B; case of the current token {

- match the current token with e, b: - match the current token with b,

and move to the next token; and move to the next token;

c: - match the current token with c, - call B

and move to the next token; e,d: do nothing

- call B; }

- match the current token with d, }

and move to the next token;

f: - call C

}

}

follow set of B

first set of C


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Non-Recursive Predictive Parsing -- LL(1) Parser

Non-Recursive predictive parsing is a table-driven parser.

It is a top-down parser.

It is also known as LL(1) Parser.

input buffer

stack Non-recursive output

Predictive Parser

Parsing Table


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LL(1) Parser

input buffer our string to be parsed. We will assume that its end is marked with a special symbol $.

output a production rule representing a step of the derivation sequence (left-most derivation) of the string in

the input buffer.

stack contains the grammar symbols

at the bottom of the stack, there is a special end marker symbol $.

initially the stack contains only the symbol $ and the starting symbol S. $S initial stack

when the stack is emptied (ie. only $ left in the stack), the parsing is completed.

parsing table

a two-dimensional array M[A,a] each row is a non-terminal symbol

each column is a terminal symbol or the special symbol $

each entry holds a production rule.


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LL(1) Parser Parser Actions

The symbol at the top of the stack (say X) and the current symbol in the input string(say a) determine the parser action.

There are four possible parser actions.

1. If X and a are $ parser halts (successful completion)

2. If X and a are the same terminal symbol (different from $)

parser pops X from the stack, and moves the next symbol in the input buffer.

3. If X is a non-terminal

parser looks at the parsing table entry M[X,a]. If M[X,a] holds a production ruleXpY1Y2...Yk, it pops X from the stack and pushes Yk,Yk-1,...,Y1 into the stack. Theparser also outputs the production rule XpY1Y2...Ykto represent a step of the

derivation.

4. none of the above error all empty entries in the parsing table are errors.

If X is a terminal symbol different from a, this is also an error case.


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LL(1) Parser Example1

Sp aBa LL(1) Parsing

Bp bB | I Table

stack

inpu

t ou

tpu

t$S abba$ S p aBa

$aBa abba$

$aB bba$ B p bB

$aBb bba$

$aB ba$ B p bB

$aBb ba$$aB a$ B p I

$a a$

$ $ accept, successful completion

a b $

S Sp aBa

B Bp I Bp bB


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LL(1) Parser Example1 (cont.)

Outputs: S p aBa B p bB B p bB B p I

Derivation(left-most): SaBaabBaabbBaabba

S

Ba a

B

Bb

b

I

parse tree


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Constructing LL(1) Parsing Tables

Two functions are used in the construction of LL(1) parsing tables:

FIRST FOLLOW

FIRST(E) is a set of the terminal symbols which occur as first symbols in strings derived fromE where E is any string of grammar symbols.

ifE derives to I, then I is also in FIRST(E) .

FOLLOW(A) is the set of the terminals which occur immediately after (follow) the non-terminal A in the strings derived from the starting symbol.

a terminal a is in FOLLOW(A) if S EAaF

$ is in FOLLOW(A) if S EA

Benefits of FIRST() and FOLLOW()

can be used to prove the LL(k) characteristics of the grammar can be used to Aid in the construction of predictive parsing table

Provides selection information for recursive decent parsers

**


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Compute FIRST for Any String X

If X is a terminal symbol FIRST(X)={X} If X is a non-terminal symbol and X p I is a production rule

I is in FIRST(X).

If X is a non-terminal symbol and X p Y1Y2..Yn is a production rule if a terminal a in FIRST(Yi) and I is in all FIRST(Yj) for j=1,...,i-1

then a is in FIRST(X). ifI is in all FIRST(Yj) for j=1,...,n

then I is in FIRST(X).

If X is I FIRST(X)={I}


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FIRST Example

Ep TE

E p +TE | ITp FT

T p *FT | IFp (E) | id

FIRST(F) = {(,id} FIRST(TE) = {(,id}FIRST(T) = {*, I} FIRST(+TE ) = {+}FIRST(T) = {(,id} FIRST(I) = {I}FIRST(E) = {+, I} FIRST(FT) = {(,id}FIRST(E) = {(,id} FIRST(*FT) = {*}

FIRST(I) = {I}FIRST((E)) = {(}FIRST(id) = {id}


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Compute FOLLOW (for non-terminals)

If S is the start symbol $ is in FOLLOW(S)

if A p EBF is a production rule

everything in FIRST(F) is FOLLOW(B) except I

If ( Ap EB is a production rule ) or

( Ap EBF is a production rule and I is in FIRST(F) )

everything in FOLLOW(A) is in FOLLOW(B).

We apply these rules until nothing more can be added to any follow set.


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FOLLOW Example

Ep TE

E p +TE | I

Tp FT

T p *FT | I

Fp (E) | id

FOLLOW(E) = { $, ) }

FOLLOW(E) = { $, ) }

FOLLOW(T) = { +, ), $ }

FOLLOW(T) = { +, ), $ }

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


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Constructing LL(1) Parsing Table -- Algorithm

for each production rule Ap E of a grammar G

for each terminal a in FIRST(E)

add A p E to M[A,a]

IfI in FIRST(E)

for each terminal a in FOLLOW(A) add A p E to M[A,a] IfI in FIRST(E) and $ in FOLLOW(A)

add A p E to M[A,$]

All other undefined entries of the parsing table are error entries.


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Constructing LL(1) Parsing Table -- Example

Ep TE FIRST(TE)={(,id} Ep TE into M[E,(] and M[E,id]

E p +TE FIRST(+TE )={+} E p +TE into M[E,+]

E p I FIRST(I)={I} nonebut since I in FIRST(I)and FOLLOW(E)={$,)} E p I into M[E,$] and M[E,)]

Tp FT FIRST(FT)={(,id} Tp FT into M[T,(] and M[T,id]

T p *FT FIRST(*FT )={*} T p *FT into M[T,*]

T p I FIRST(I)={I} nonebut since I in FIRST(I)

and FOLLOW(T)={$,),+} T p I into M[T,$], M[T,)] and M[T,+]

Fp (E) FIRST((E) )={(} Fp (E) into M[F,(]

Fp id FIRST(id)={id} Fp id into M[F,id]


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Ep TE

Ep +TE | I

Tp FT

Tp *FT | I

Fp (E) | id

id + * ( ) $

E Ep TE Ep TE

E Ep +TE Ep I Ep I

T Tp FT Tp FT

T Tp I Tp *FT Tp I Tp I

F Fp id Fp (E)


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stack input output

$E id+id$ Ep TE

$ET id+id$ Tp FT

$E TF id+id$ Fp id

$ E Tid id+id$

$ E

T

+id$ T

p I$ E +id$ E p +TE

$ E T+ +id$

$ E T id$ Tp FT

$ E T F id$ Fp id

$ E Tid id$

$ E T $ T p I

$ E $ E p I

$ $ accept


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LL(1) Grammars

A grammar whose parsing table has no multiply-defined entries is said

to be LL(1) grammar.

one input symbol used as a look-head symbol do determine parser action

LL(1) left most derivationinput scanned from left to right

The parsing table of a grammar may contain more than one production

rule. In this case, we say that it is not a LL(1) grammar.


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A Grammar which is not LL(1)

Sp i C t S E | a FOLLOW(S) = { $,e }

Ep e S | I FOLLOW(E) = { $,e }

Cp b FOLLOW(C) = { t }

FIRST(iCtSE) = {i}FIRST(a) = {a}

FIRST(eS) = {e}

FIRST(I) = {I}

FIRST(b) = {b}

two production rules for M[E,e]

Problem ambiguity

a b e i t $

S Sp a Sp iCtSE

E Ep e S

Ep I

Ep I

C Cp b


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Q.1 Consider the following Grammar, test whether the grammar is LL(1)

or not, and construct a predictive parsing table for it.Sp AaAb |BbBa FIRST (S) = { a,b} FOLLOW(S) = { $}

Ap I FIRST (A) = {I} FOLLOW (A) = {a,b}

Bp I FIRST (B) = {I} FOLLOW(B) = {a,b}

a b $S SpAaAb SpBbBa

A Ap I Ap I

B Bp I Bp I


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Q.1 Consider the following Grammar, test whether the grammar is LL(1)

or not, and construct a predictive parsing table for it.Sp 1AB | I FOLLOW(S) = { $}

Ap 1AC | 0C FOLLOW(A) = { 1,0 }

Bp 0S FOLLOW(B) = { $}

Cp 1

FIRST(S) = {1, I }

FIRST(A) = {1,0}

FIRST(B) = {0}

FIRST(C) = {1}

1 0 $

S

A

B

C


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A Grammar which is not LL(1) (cont.)

What do we have to do it if the resulting parsing table contains multiply

defined entries? If we didnt eliminate left recursion, eliminate the left recursion in the grammar.

If the grammar is not left factored, we have to left factor the grammar.

If its (new grammars) parsing table still contains multiply defined entries, that grammar is

ambiguous or it is inherently not a LL(1) grammar.

A left recursive grammar cannot be a LL(1) grammar. Ap AE | F

any terminal that appears in FIRST(F) also appears FIRST(AE) because AE FE.

IfF is I, any terminal that appears in FIRST(E) also appears in FIRST(AE) and FOLLOW(A).

A grammar is not left factored, it cannot be a LL(1) grammar Ap EF1 | EF2

any terminal that appears in FIRST(EF1) also appears in FIRST(EF2).

An ambiguous grammar cannot be a LL(1) grammar.


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Properties of LL(1) Grammars

A grammar G is LL(1) if and only if the following conditions hold for

two distinctive production rules A p E and A p F

1. Both E and F cannot derive strings starting with same terminals.

2. At most one ofE and F can derive to I.

3. IfF can derive to I, then E cannot derive to any string starting

with a terminal in FOLLOW(A).


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Error Recovery in Predictive Parsing

An error may occur in the predictive parsing (LL(1) parsing)

if the terminal symbol on the top of stack does not match with

the current input symbol.

if the top of stack is a non-terminal A, the current input symbol is a,

and the parsing table entry M[A,a] is empty. What should the parser do in an error case?

The parser should be able to give an error message (as much as

possible meaningful error message).

It should be recover from that error case, and it should be able

to continue the parsing with the rest of the input.


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Error Recovery Techniques

Panic-Mode Error Recovery Skipping the input symbols until a synchronizing token is found.

Phrase-Level Error Recovery Each empty entry in the parsing table is filled with a pointer to a specific error routine to

take care that error case.

Error-Productions If we have a good idea of the common errors that might be encountered, we can augment

the grammar with productions that generate erroneous constructs.

When an error production is used by the parser, we can generate appropriate errordiagnostics.

Since it is almost impossible to know all the errors that can be made by the programmers,this method is not practical.

Global-Correction Ideally, we we would like a compiler to make as few change as possible in processing

incorrect inputs.

We have to globally analyze the input to find the error.

This is an expensive method, and it is not in practice.


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Panic-Mode Error Recovery in LL(1) Parsing

In panic-mode error recovery, we skip all the input symbols until a

synchronizing token is found.

What is the synchronizing token? All the terminal-symbols in the follow set of a non-terminal can be used as a synchronizing

token set for that non-terminal.

So, a simple panic-mode error recovery for the LL(1) parsing: All the empty entries are marked as synch to indicate that the parser will skip all the input

symbols until a symbol in the follow set of the non-terminal A which on the top of the

stack. Then the parser will pop that non-terminal A from the stack. The parsing continues

from that state.

To handle unmatched terminal symbols, the parser pops that unmatched terminal symbol

from the stack and it issues an error message saying that that unmatched terminal is

inserted.


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Panic-Mode Error Recovery - Example

Sp AbS | e | I

Ap a | cAd

FOLLOW(S)={$}

FOLLOW(A)={b,d}

stack input output stack input output

$S aab$ Sp AbS $S ceadb$ S p AbS

$SbA aab$ Ap a $SbA ceadb$ Ap cAd

$Sba aab$ $SbdAc ceadb$

$Sb ab$ Error: missing b, inserted $SbdA eadb$ Error:unexpected e (illegal A)

$S ab$ Sp AbS (Remove all input tokens until first b or d, pop A)

$SbA ab$ Ap a $Sbd db$

$Sba ab$ $Sb b$

$Sb b$ $S $ S p I

$S $ Sp I $ $ accept

$ $ accept

a b c d e $S SpAbS sync SpAbS sync Sp e Sp I

A Ap a sync Ap cAd sync sync sync


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Phrase-Level Error Recovery

Each empty entry in the parsing table is filled with a pointer to a special

error routine which will take care that error case.

These error routines may: change, insert, or delete input symbols.

issue appropriate error messages

pop items from the stack.

We should be careful when we design these error routines, because we

may put the parser into an infinite loop.

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