PL Lab, DongGuk University Compiler Lecture Note, LL ParsingPage 1 컴파일러 입문 제 7 장 LL 구문 분석

Compiler Lecture Note, LL Parsing Page 1

PL Lab, DongGuk University

컴파일러 입문

제 제 7 7 장장LL LL 구문 분석구문 분석



I. 결정적 구문 분석 II. Recursive-descent 파서III. PredictivePredictive 파서VI. Predictive 파싱 테이블의 구성 V. Strong LL(k)LL(k) 문법과 LL(k) 문법

목 차



I. 결정적 구문 분석▶ Deterministic Top-Down Parsing

::= deterministic selection of production rules to be applied

in top-down syntax analysis.

▶ One pass nobackup

1. Input string is scanned once from left to right.

2. Parsing process is deterministic.

▶ Top-down parsing with nobackup

::= deterministic top-down parsing.

called LL parsing.

"Left to right scanning and Left parse"



▶ How to decide which production is to be applied:

sentential form : 1 2 … i-1Xα

input string : 1 2 … i-1 i i+1 … n

X 1 | 2 ... | k P∈ 일 때 ,

i를 보고 X-production 중에 uniqueunique 하게 결정 .

the condition for no backtracking : FIRST 와 FOLLOW 가 필요 .

(= LL condition)



FIRST ▶ FIRST() ::= the set of terminalsterminals that begin the strings derived from .

if * , then is also in FIRST().

FIRST(A) ::= { a V∈ T {∪ } | A * a, V∈ * }.

▶ Computation of FIRST(X), where X V.∈

1) if X V∈ T, then FIRST(X) = {X}

2) if X V∈ N and X a P, then FIRST(X) = FIRST(X) ∈ {a}

if X P, then FIRST(X) = FIRST(X) ∈ {}

3) if X Y1Y2 …Yk P and Y∈ 1Y2 …Yi-1 * ,

i

then FIRST(X) = FIRST(X) ( FIRST(Yj) - {}). j=1

if Y1Y2 …Yk * , then FIRST(X) = FIRST(X) {}.



ex1) E TE E +TE | T FT T FT | F (E) | id

FIRST(E) = FIRST(T) = FIRST(F) = {(, id}

FIRST(E) = {+, }

FIRST(T) = {, }

ex2) PROGRAM begin d semi X end

X d semi X

X s Y

Y semi s Y |

FIRST(PROGRAM) = {begin}

FIRST(X) = {d,s}

FIRST(Y) = {semi, }

Text p.268



연습문제 7.4 (1) - p.299

• FIRST 를 구하시오 .

(1) S aRTb | bRR

R cRd | T RS | TaT



▶ left-dependency graph

- the vertices are the terminal and nonterminal symbols and the

arcs go from X to Y if and only if X X1...XnY, where

n 0, and each of X1,...,Xn can produce the empty string.

ex) S AB

A aA | B bB |

S

A

B b

a

FIRST(S) = {a, , b} FIRST(A) = {a, } FIRST(B) = {b, }



★ In general, A A1A2...An

if A1 : non-nullable

if A1 : nullable

if A1A2 : nullable

A

A1

A3

A

A

A1

A1

A2

A2



FOLLOW ▶ FOLLOW(A)

::= the set of terminals that can appear immediately to the right

of A in some sentential form. If A can be the rightmost

symbol in some sentential form, then $ is in FOLLOW(A).

$ is the input right marker.

::= {a V∈ T {$} | S ∪ * Aa, , V∈ *}.

▶ Computation of FOLLOW(A)

1) FOLLOW(S) = {$}

2) if A B P and ∈ ,

then FOLLOW(B) = FOLLOW(B) (FIRST(∪ ) - )

3) if A B P or A ∈ B and * ,

then FOLLOW(B) = FOLLOW(B) FOLLOW(∪ A).



ex) E TE'

E' +TE' | T FT'

T' FT' | F (E) | id

Nullable = { E, T }

FIRST(E) = FIRST(T) = FIRST(F) = {(, id}

FIRST(E) = {+, } FIRST(T) = {, }

FOLLOW(E) = {),$} FOLLOW(E') = {),$}

FOLLOW(T) = {+,),$} FOLLOW(T') = {+,),$}

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

Text p.271



연습문제 7.4 (3) - p.299

• FOLLOW 를 구하시오 .

(3) S aAa | A abS | c



▶ LL condition

::= no backup condition

::= the condition for deterministic parsing of top-down method.

input : 12 ... i-1i ...n

derived string : 12...i-1X

X 1 | 2 ... | m

i를 보고 X-production 들 중에서 X 를 확장할 rule 을 결정적으로 선택 .

★ <LL condition> A | P,∈

1. FIRST() FIRST() =

2. if * , FOLLOW(A) FIRST() =



ex) A aBc | Bc | dAa

B bB |

FIRST(A) = {a,b,c,d} FOLLOW(A) = {$,a}

FIRST(B) = {b, } FOLLOW(B) = {c}

1) A aBc | Bc | dAa 에서 ,

FIRST(aBc) FIRST(Bc) FIRST(dAa)

= {a} {b,c} {d} =

2) B bB | 에서 ,

FIRST(bB) FOLLOW(B) = {b} {c} =

1), 2) 에 의해 LL 조건을 만족한다 .



II. Recursive-descent 파서 ▶ Recursive-descent parsing

::= A top-down method that uses a set of recursive procedures to recognize its input with no backtracking.

▶ create a procedure for each nonterminal.

ex) G : S aA | bB A aA | c B bB | d procedure pS; begin if nextsymbol = qa then begin get_nextsymbol; pA end else if nextsymbol = qb then begin get_nextsymbol; pB end else error end;



procedure pA;

begin if nextsymbol = qa then begin get_nextsymbol; pA end

else if nextsymbol = qc then get_nextsymbol

else error

end;

procedure pB; ...

(* main *) begin get_nextsymbol; pS; if next_symbol = '$' then accept else error end.

= aac$

Procedure call sequence ::= leftmost derivation



▶ The main problem in constructing a recursive-descent syntax

analyzer is the choice of productions when a procedure is first

entered. To resolve this problem, we can compute the lookahead

of each production.

▶ LOOKAHEADLOOKAHEAD of a production

Definition: LOOKAHEAD(A)

= FIRST({ | S * A * V∈ T*}).

Meaning : the set of terminals which can be generated by and

if * , then FOLLOW(A) is added to the set.

Computing formula: LOOKAHEAD(A X1X2...Xn)

= FIRST(X1X2...Xn) FOLLOW(A)



ex) S aSA | A c

Nullable Set = {S}

FIRST(S) = {a, } FOLLOW(S) = {$,c}

FIRST(A) = {c} FOLLOW(A) = {$,c}

LOOKAHEAD(S aSA) = FIRST(aSA) FOLLOW(S) = {a}

LOOKAHEAD(S ) = FIRST() FOLLOW(S) = {$,c}

LOOKAHEAD(A c) = FIRST(c) FOLLOW(A) = {c}

LOOKAHEAD 를 구하는 순서 :

Nullable => FIRST => FOLLOW => LOOKAHEAD



▶ Strong LL condition

Definition : A | P, ∈ LOOKAHEAD(A ) LOOKAHEAD(A ) = .

Meaning : for each distinct pair of productions with the same left-hand side, it can select the unique alternate that derives a string beginning with the input symbol.

Definition : the grammar G is said to be strong LL(1) if it satisfies the strong LL condition.

ex) G : S aSA | A c

LOOKAHEAD(S aSA) = {a}

LOOKAHEAD(S ) = FOLLOW(S) = {$, c}

LOOKAHEAD(S aSA) LOOKAHEAD(S ) =

G 는 strong LL(1) 이다 .



▶ Implementation of Recursive-descent parser

If a grammar is strong LL(1), we can construct a parser for sentences of the

grammar using the following scheme.

a V∈ T,

procedure pa; (* get_nextsymbol=scanner *)

begin

if nextsymbol = qa then get_nextsymbol

else error

end;

get_nextsymbol : 스캐너에 해당하는 루틴으로 입력 스트림으로부터 토큰 한 개를 읽어 변수

nextsymbol 에 할당하는 일을 한다 .



A V∈ N,

procedure pA;

var i: integer;

begin

case nextsymbol of

LOOKAHEAD(A X1X2...Xm): for i := 1 to m do pXi;

LOOKAHEAD(A Y1Y2...Yn): for i := 1 to n do pYi;

:

LOOKAHEAD(A Z1Z2...Zr): for i := 1 to r do pZi;

LOOKAHEAD(A ): ;

otherwise: error

end (* case *)

end;

Text p.278



▶ Improving the efficiency and structure of recursive-descent parser

1) Eliminating terminal procedures

::= In practice it is better not to write a procedure for each terminal.

Instead the action of advancing the input marker can always be initiated

by the nonterminal procedures. In this way many redundant tests can

be eliminated.

ex) text p.279 [ 예 9]

2) BNF EBNF : reduce the number of productions and nonterminals.

① repetitive part : { }

② optional part : [ ]

③ alternation : ( | )



ex) [ 예 10] --- text p.281

< IF_st > ::= ' if ' < C > ' then ' < S > [ ' else ' < S > ]

procedure pIF;

begin if nextsymbol = qif then

begin get_nextsymbol; pC;

if nextsymbol = qthen then

begin get_nextsymbol; pS end

else error(10)

end

else error(20);

if nextsymbol = qelse then

begin get_nextsymbol; pS end

end;



ex) [ 예 11] --- text p.281

<id_list> ::= ' id ' { ' , ' ' id ' }

procedure pID_LIST;

begin if nextsymbol = qid then

begin get_nextsymbol;

while (nextsymbol = qcomma) do

begin get_nextsymbol;

if nextsymbol = qid then get_nextsymbol

else error

end

end

end;



[ 연습문제 7.8 (2)] --- Text p.300

< 문제 > 다음 grammar 를 extended BNF 로 바꾸고 그에 따른

recursive-descent parser 를 위한 procedure 를 작성하시오 .

<D> ::= ' label ' <L> | ' integer ' <L>

<L> ::= <id> <R>

<R> ::= ' ; ' | ' , ' <L>

<L> <id> (' , ' <id> )* ' ; '

<D> ::= ( ' label ' | ' integer ' ) <id> {' , ' <id>} ' ; '

*



procedure pD; begin if nextsymbol in [qlabel,qinteger] then begin get_nextsymbol; if nextsymbol = qid then begin get_nextsymbol; while (nextsymbol = qcomma) do begin get_nextsymbol; if nextsymbol = qid then get_nextsymbol else error(3) end end else error(2); if nextsymbol = qsemi then get_nextsymbol else error(4) end else error(1) end;



Programming Assignment #1

Implement a recursive-descent syntax analyzer for the grammar

given in exercise 5.30(text p. 224).

Problem Specifications

- input : SPL program to find a Minimum and a Maximum.

- output : left parse

- methods : (1) write the get_nextsymbol routine.

(2) compute LOOKAHEADs for each production.

(3) create a procedure for each nonterminal.

(4) assemble the procedures with main program.

a set of productions

Computation of LOOKAHEADs

LOOKAHEADs foreach nonterminal



III. Predictive Parsing ▶ Predictive parsing

::= a deterministic parsing method using a stack. The stack contains a sequence of grammar symbols.

▶ Model of a predictive parser

Driver routine

$

$ : input

output

stack

Table



Current input symbol 과 stack top symbol 사이의 관계에 따라 parsing.

The input buffer contains the string to be parsed, followed by $.

Initial configuration : STACK INPUT

$S $

Parsing table(LL) : parsing action 을 결정지어 줌 .

※ M[X,a] = r : stack top symbol 이 X 이고 current symbol 이 a 일 때 ,

r 번 생성 규칙으로 expand.

r

terminals

nonterminals X

a



▶ Parsing Actions

X : stack top symbol, a : current input symbol

1. if X = a = $, then accept.

2. if X = a, then pop X and advance input.

3. if X V∈ N, then if M[X,a] = r (X),

then replace X by

else error.



▶ Predictive parsing algorithm

set ip to point to the first symbol of $; repeat let X be the 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 ip else error(1) else /* X is nonterminal */

if M[X,a] = X Y1Y2...Yk then begin pop X from the stack;

push YkYk-1,...,Y1 onto the stack, with Y1 on top;

output the production X Y1Y2...Yk

end else error(2) until X = $ /* stack is empty */

Text p.284



ex) G : 1. S aSb

2. S bA

3. A aA

4. A b

string : aabbbb

• Parsing Table:

a b

S

A

terminalsnonterminals

1 2

3 4



STACK INPUT ACTIONS OUTPUT

$S aabbbb$ expand 1 1

$bSa aabbbb$ pop a and advance

$bS abbbb$ expand 1 1

$bbSa abbbb$ pop a and advance

$bbS bbbb$ expand 2 2

$bbAb bbbb$ pop b and advance

$bbA bbb$ expand 4 4

$bbb bbb$ pop b and advance

$bb bb$ pop b and advance

$b b$ pop b and advance

$ $ Accept

※ How to construct a predictive parsing table for the grammar.



VI. Predictive 파싱 테이블의 구성 ▶ main idea : If A is a production with a in FIRST(), then the parser will expand A by when the current input symbol is a. And if * , then we should again expand A by when the current input symbol is in FOLLOW(A).

▶ parsing table(LL):

M[X,a] = r : expand X with r-production blank : error

VT a

X

VN



▶ Algorithm : for each production A,

1. a FIRST(∈ ), M[A,a] := <A>

2. if * , then b FOLLOW(A), M[A,b] := <A∈ >.

ex) G: 1. E TE' 2. E' +TE' 3. E' 4. T FT'

5. T' FT' 6. T' 7. F (E) 8. F id

FIRST(E)=FIRST(T)=FIRST(F)={ ( , id }

FIRST(E')={ + , } FIRST(T')={ , }

FOLLOW(E) = FOLLOW(E') = { ) , $ }

FOLLOW(T) = FOLLOW(T') = { + , ) , $ }

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



1

2

4

• Parsing Table:

Terminalsid + * ( ) $

E

E'

T

T'

F 8

1

3 3

6 6

7

6 5

4

Nonterminals



▶ LL(1) Grammar

::= a grammar whose parsing table has no multiply-defined entries.

multiply 정의되면 어느 rule 로 expand 해야 할 지 결정할 수 없기 때

문에 deterministic 하게 parsing 할 수 없다 .

▶ LL(1) condition: A | ,

1. FIRST( ) FIRST() = .

2. if , then FOLLOW(A) FIRST() = .

ex) G : 1. S iCtSS' 2. S a

3. S' eS 4. S' 5. C b

FIRST(S) = {i,a} FOLLOW(S) = {$,e}

FIRST(S') = {e, } FOLLOW(S') = {$,e}

FIRST(C) = {b} FOLLOW(C) = {t}

*



Parsing Table:

M[S',e] := <3,4> 로 중복으로 정의되었음 .

여기서 , stack top 이 S' 이고 input symbol 이 e 일 때 3 번 rule 로

expand 해야 할 지 , 4 번 rule 로 expand 해야 하는지 알 수 없다 .

그러므로 G 는 LL(1) grammar 가 아니다 .

ex) [ 예제 15] --- text p.291

G : S aA | abA : abab

A Ab | a

a b e i t $

S

S'

C

2

5

1

43,4



V. Strong LL(k) and LL(k) Grammars ▶ FIRSTk() = {| * , || = k or and || < k}

▶ G is said to be strong LL(k)strong LL(k), for some fixed integer k > 0, if

whenever there are two leftmost derivations.

1. S * A * x V∈ T*, and

2. S * A * y V∈ T* such that

3. FIRSTk(x) = FIRSTk(y). It follows that 4. = .

▶ Meaning: Suppose we consider any state of the parse in which A is the nonterminal currently being parsed and FIRSTk(x) is the k-lookahead at the current point. Then, if the k-lookahead is same, the two productions A and A are identical. Any other information provided by the closed portion and the open portion of the current state of the parse will be disregarded.



▶ S A, : closed portion, : open portion

▶ Two states of the parse

FIRSTk(x) = FIRSTk(y) ===> = .

*

S

A

x

S

A

y



▶ Def) LL(k) grammar:

1. S A x V∈ T*, and

2. S A y V∈ T* such that

3. FIRSTk(x) = FIRSTk(y). It follows that

4. = .

ex) S aAaa | bAba

A b |

S S

a A a a b A b a b

lookahead 가 ba 일 때 A b, A 중 어느 rule 을 택할 수 있는가 ? 이제 본 symbol 이 a 이면 A b 를 선택하고 , b

이면 A 를 선택한다 . 따라서 SLL(2) 는 아니며 LL(2) 가 된다 .

*

*

*

*



▶ SLL(k) and LL(k)

▶ <theorem> strong LL(1) LL(1)

Proof) () clear!

() Suppose that G is not strong LL(1).

Then, by definition, there are two distinct productions

A and A such that,

S 1A1 11 111 111

S 2A2 22 222 222

and FIRST(11) = FIRST(22).

SLL(k)SLL(k)

LL(k)LL(k)

*

*

*

*

*

*



Now we must prove that G is not LL(1).

1) 1= 2= , G is not LL(1).

Indeed, it is ambiguous.

2) one (or both) of 1 and 2 is not . 1 .

FIRST1(1 1) = FIRST1(1) = FIRST1(2 2).

but then,

S 2A2 22 2 12 212

S 2A2 22 2 22 222

satisfy the property

FIRST1(1 2) = FIRST1(1) = FIRST1(2 2).

Thus, by definition, G is not LL(1).

*

*

*

*

*

*

Documents

PL Lab, DongGuk University Compiler Lecture Note, LL ParsingPage 1 컴파일러 입문 제 7 장 LL 구문 분석