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Bottom-Up Parsing
LR methods (Left-to-right, Rightmost derivation) LR(0), SLR, Canonical LR = LR(1), LALR
Other special cases: Shift-reduce parsing Operator-precedence parsing
Bottom-Up Parsing
E → T → T * F → T * id → F * id → id * id
rightmost derivation
reduction
E → E + T | T T → T * F | F F → ( E ) | id
Handle Pruning Handle
A handle is a substring of grammar symbols in aright-sentential form that matches a right-hand side of a production
Right Sentential Form Handle Reducing Production
id1 * id2 id1 F → id
F * id2 F T → F
T * id2 id2 F → id
T * F T * F E → T * F
E → E + T | T T → T * F | F F → ( E ) | id
Handle Pruning A rightmost derivation in reverse can be obtai
ned by “handle pruning”S = γ0 →rm γ1 → rm γ2 →rm …. →rm γn-1 →rm γn =ω
Handle definition S →*rm αAω →*rm αβω
S
A
α ωβ
A handle A →β in the parse tree for αβω
Example: Handle
Handle
GrammarS a A B eA A b c | bB d
NOT a handle, becausefurther reductions will fail
(result is not a sentential form)
a b b c d ea A b c d ea A A e… ?
a b b c d ea A b c d ea A d ea A B eS
Shift-Reduce Parsing Shift-Reduce Parsing is a form of bottom-up
parsing A stack holds grammar symbols An input buffer holds the rest of the string to
parsed. Shift-Reduce parser action
shift reduce accept error
Shift-Reduce Parsing Shift
Shift the next input symbol onto the top of the stack. Reduce
The right end of the string to be reduced must be at the top of the stack.
Locate the left end of the string within the stack and decide with what nonterminal to replace the string.
Accept Announce successful completion of parsing
Error Discover a syntax error and call recovery routine
Shift-Reduce Parsing
Stack Input Action
$ id1 * id2 $ shift
$id1 * id2 $ reduce by F → id
$F * id2 $ reduce by T → F
$T * id2 $ shift
$T * id2 $ shift
$T * id2 $ reduce by F → id
$T * F $ reduce by T → T * F
$T $ reduce by E → T
$E $ accept
E → E + T | T T → T * F | F F → ( E ) | id
Example: Shift-Reduce ParsingGrammar:S a A B eA A b c | bB d
Shift-reduce correspondsto a rightmost derivation:S rm a A B e rm a A d e rm a A b c d e rm a b b c d e
Reducing a sentence:a b b c d ea A b c d ea A d ea A B eS
S
a b b c d e
A
A
B
a b b c d e
A
A
B
a b b c d e
A
A
a b b c d e
A
These matchproduction’s
right-hand sides
Conflicts During Shift-Reduce Parsing Conflicts Type
shift-reduce reduce-reduce
Shift-reduce and reduce-reduce conflicts are caused by The limitations of the LR parsing method (even
when the grammar is unambiguous) Ambiguity of the grammar
Shift-Reduce Conflict
Stack$$id$E$E+$E+id$E+E$E+E*$E+E*id$E+E*E$E+E$E
Inputid+id*id$
+id*id$+id*id$
id*id$*id$*id$
id$$$$$
Actionshiftreduce E idshiftshiftreduce E idshift (or reduce?)shiftreduce E idreduce E E * Ereduce E E + Eaccept
GrammarE E + EE E * EE ( E )E id
Find handlesto be reduced
How toresolve
conflicts?
Shift-Reduce Conflict
Stack$…$…if E then S
Input…$
else…$
Action…shift or reduce?
Ambiguous grammar:S if E then S | if E then S else S | other
Resolve in favorof shift, so else
matches closest if
Shift else to if E then S or Reduce if E then S
Reduce-Reduce Conflict
Stack$$a
Inputaa$
a$
Actionshiftreduce A a or B a ?
GrammarC A BA aB a
Resolve in favorof reduce A a,
otherwise we’re stuck!動彈不得
LR LR parser are table-driven
Much like the nonrecursive LL parsers. The reasons of the using the LR parsing
An LR-parsering method is the most general nonbacktracking shift-reduce parsing method known, yet it can be implemented as efficiently as other.
LR parsers can be constructed to recognize virtually all programming-language constructs for which context-free grammars can be written.
An LR parser can detect a syntactic error as soon as it is possible to do so on a left-to-right scan of the input.
The class of grammars that can be parsed using LR methods is a proper superset of the class of grammars that can be parsed with predictive or LL methods.
LR(0) An LR parser makes shift-reduce decisions b
y maintaining states to keep track. An item of a grammar G is a production of G
with a dot at some position of the body. Example
A → X Y Z
A → . X Y Z
A → X . Y Z
A → X Y . Z
A → X Y Z .
A → X . Y Z
stack next derivations with input strings
items
Note that production A has one item [A •]
LR(0) Canonical LR(0) Collection
One collection of sets of LR(0) items Provide the basis for constructing a DFA that is us
ed to make parsing decisions. LR(0) automation
The canonical LR(0) collection for a grammar Augmented the grammar
If G is a grammar with start symbol S, then G’ is the augmented grammar for G with new start symbol S’ and new production S’ → S
Closure function Goto function
Function Closure
If I is a set of items for a grammar G, then closure(I) is the set of items constructed from I.
Create closure(I) by the two rules: add every item in I to closure(I) If A→α . Bβ is in closure(I) and B →γ is a production, then
add the item B → . γ to closure(I). Apply this rule untill no more new items can be added to closure(I).
Divide all the sets of items into two classes Kernel items
initial item S’ → . S, and all items whose dots are not at the left end. Nonkernel items
All items with their dots at the left end, except for S’ → . S
Example The grammar G
E’ → E E → E + T | T T → T * F | F F → ( E ) | id
Let I = { E’ → . E } , then closure(I) = {
E’ → . E E → . E + T E → . T T → . T * F T → . F F → . ( E ) F → . id }
Function Goto Function Goto(I, X)
I is a set of items X is a grammar symbol Goto(I, X) is defined to be the closure of the set of
all items [A α X‧β] such that [A α‧ Xβ] is in I. Goto function is used to define the transitions in th
e LR(0) automation for a grammar.
Example
I = {
E’ → E . E → E . + T } Goto (I, +) = {
E → E + . T
T → . T * F
T → . F
F → . (E)
F → . id
}
The grammar G E’ → E E → E + T | T T → T * F | F F → ( E ) | id
Constructing the LR(0) Collection 1. The grammar is augmented with a new start
symbol S’ and production S’S2. Initially, set C = closure({[S’•S]})
(this is the start state of the DFA)3. For each set of items I C and each gramm
ar symbol X (NT) such thatGOTO(I, X) C and goto(I, X) , add the set of items GOTO(I, X) to C
4. Repeat 3 until no more sets can be added to C
Example: The Parse of id * idLine STACK SYMBOLS INPUT ACTION
(1) 0 $ id * id $ shift to 5
(2) 0 5 $ id * id $ reduce by F → id
(3) 0 3 $ F * id $ reduce by T → F
(4) 0 2 $ T * id $ shift to 7
(5) 0 2 7 $ T * id $ shift to 5
(6) 0 2 7 5 $ T * id $ reduce by F → id
(7) 0 2 7 10 $ T * F $ reduce by T → T * F
(8) 0 2 $ T $ reduce by E → T
(9) 0 1 $ E $ accept
Structure of the LR Parsing Table Parsing Table consists of two parts:
A parsing-action function ACTION A goto function GOTO
The Action function, Action[i, a], have one of four forms: Shift j, where j is a state.
The action taken by the parser shifts input a to the stack, but use state j to represent a.
Reduce A→β. The action of the parser reduces β on the top of the stack to h
ead A. Accept
The parser accepts the input and finishes parsing. Error
Structure of the LR Parsing Table(1) The GOTO Function, GOTO[Ii, A], defined on
sets of items. If GOTO[Ii, A] = Ij, then GOTO also maps a state i
and a nonterminal A to state j.
LR-Parser ConfigurationsConfiguration ( = LR parser state):
($s0 s1 s2 … sm, ai ai+1 … an $)
stack input
($ X1 X2 … Xm, ai ai+1 … an $)
If action[sm, ai] = shift s then push s (ai), and advance input (s0 s1 s2 … sm, ai ai+1 … an $) (s0 s1 s2 … sm s, ai+1 … an $)
If action[sm, ai] = reduce A and goto[sm-r, A] = s with r = || then pop r symbols, and push s ( push A )
( (s0 s1 s2 … sm, ai ai+1 … an $) (s0 s1 s2 … sm-r s, ai ai+1 … an $)
If action[sm, ai] = accept then stopIf action[sm, ai] = error then attempt recovery
Example LR Parse Table
Grammar:1. E E + T2. E T3. T T * F4. T F5. F ( E )6. F id
s5 s4
s6 acc
r2 s7 r2 r2
r4 r4 r4 r4
s5 s4
r6 r6 r6 r6
s5 s4
s5 s4
s6 s11
r1 s7 r1 r1
r3 r3 r3 r3
r5 r5 r5 r5
id + * ( ) $
0
1
2
3
4
5
6
7
8
9
10
11
E T F
1 2 3
8 2 3
9 3
10shift & goto 5
reduce byproduction #1
action gotostate
Line STACK SYMBOLS INPUT ACTION
(1) 0 id * id + id $ shift 5
(2) 0 5 id * id + id $ reduce 6 goto 3
(3) 0 3 F * id + id $ reduce 4 goto 2
(4) 0 2 T * id + id $ shift 7
(5) 0 2 7 T * id + id $ shift 5
(6) 0 2 7 5 T * id + id $ reduce 6 goto 10
(7) 0 2 7 10 T * F + id $ reduce 3 goto 2
(8) 0 2 T + id $ reduce 2 goto 1
(9) 0 1 E + id $ shift 6
(10)
0 1 6 E + id $ shift 5
(11)
0 1 6 5 E + id $ reduce 6 goto 3
(12)
0 1 6 3 E + F $ reduce 4 goto 9
(13)
0 1 6 9 E + T $ reduce 1 goto 1
(14)
0 1 $ E $ accept
Grammar0. S E1. E E + T2. E T3. T T * F4. T F5. F ( E )6. F id
Example
SLR Grammars SLR (Simple LR): a simple extension of LR(0)
shift-reduce parsing SLR eliminates some conflicts by populating
the parsing table with reductions A on symbols in FOLLOW(A)
S EE id + EE id
State I0:S •E E •id + EE •id
State I2:E id•+ EE id•
goto(I0,id) goto(I3,+)
FOLLOW(E)={$}thus reduce on $
Shift on +
SLR Parsing Table
Reductions do not fill entire rows Otherwise the same as LR(0)
s2
acc
s3 r3
s2
r2
id + $
0
1
2
3
4
E
1
4
1. S E2. E id + E3. E id
FOLLOW(E)={$}thus reduce on $
Shift on +
Constructing SLR Parsing Tables Augment the grammar with S’ S Construct the set C={I0, I1, …, In} of LR(0) items State i is constructed from Ii
If [A•a] Ii and goto(Ii, a)=Ij then set action[i, a]=shift j
If [A•] Ii then set action[i,a]=reduce A for all a FOLLOW(A) (apply only if AS’)
If [S’S•] is in Ii then set action[i,$]=accept
If goto(Ii, A)=Ij then set goto[i, A]=j set goto table Repeat 3-4 until no more entries added The initial state i is the Ii holding item [S’•S]
Example SLR Grammar and LR(0) ItemsAugmentedgrammar:0. C’ C1. C A B2. A a3. B a
State I0:C’ •C C •A BA •a
State I1:C’ C•
State I2:C A•BB •a
State I3:A a•
State I4:C A B•
State I5:B a•
goto(I0,C)
goto(I0,a)
goto(I0,A)
goto(I2,a)
goto(I2,B)
I0 = closure({[C’ •C]})I1 = goto(I0,C) = closure({[C’ C•]})…
start
final
Example SLR Parsing Table
s3
acc
s5
r2
r1
r3
a $
0
1
2
3
4
5
C A B
1 2
4
State I0:C’ •C C •A BA •a
State I1:C’ C•
State I2:C A•BB •a
State I3:A a•
State I4:C A B•
State I5:B a•
1
2
4
5
3
0start
a
A
CB
a
Grammar:0. C’ C1. C A B2. A a3. B a
SLR and Ambiguity Every SLR grammar is unambiguous, but not every
unambiguous grammar is SLR, maybe LR(1) Consider for example the unambiguous grammar
S L = R | RL * R | id
R L FOLLOW(R) = {=, $}
I0:S’ •S S •L=RS •RL •*R L •idR •L
I1:S’ S•
I2:S L•=RR L•
I3:S R•
I4:L *•R R •LL •*R L •id
I5:L id•
I6:S L=•RR •L L •*R L •id
I7:L *R•
I8:R L•
I9:S L=R•
action[2,=]=s6action[2,=]=r5 no
Has no SLRparsing table
LR(1) Grammars SLR too simple LR(1) parsing uses lookahead to avoid unnec
essary conflicts in parsing table LR(1) item = LR(0) item + lookahead
LR(0) item[A•]
LR(1) item [A•, a]
SLR Versus LR(1) Split the SLR states by adding LR(1) lookahead Unambiguous grammar
S L = R | RL * R | idR L
I2:S L•=RR L•
action[2,=]=s6
Should not reduce, because noright-sentential form begins with R=
split
R L•S L•=R
LR(1) Items An LR(1) item
[A•, a]contains a lookahead terminal a, meaning already on top of the stack, expect to see a
For items of the form[A•, a]
the lookahead a is used to reduce A only if the next input is a
For items of the form [A•, a]
with the lookahead has no effect
The Closure Operation for LR(1) Items Start with closure(I) = I If [A•B, a] closure(I) then
for each production B in the grammar
and each terminal b FIRST(a)
add the item [B•, b] to I
if not already in I Repeat 2 until no new items can be added
The Goto Operation for LR(1) Items For each item [A•X, a] I, add the set o
f items closure({[AX•, a]}) to goto(I,X) if not already there
Repeat step 1 until no more items can be added to goto(I,X)
Example Let I = { (S’ → •S, $) } I0 = closure(I) = { S’ → •S, $ S → • C C, $ C → •c C, c/d C → •d, c/d } goto(I0, S) = closure( {S’ → S •, $ } )
= {S’ → S •, $ } = I1
The grammar G S’ → S S → C C C → c C | d
Exercise
Let I = { (S → C •C, $) }
I2 = closure(I) = ?
I3 = goto(I2, c) = ?
The grammar G S’ → S S → C C C → c C | d
Construction of the sets of LR(1) Items Augment the grammar with a new start symbo
l S’ and production S’S Initially, set C = closure({[S’•S, $]})
(this is the start state of the DFA) For each set of items I C and each grammar
symbol X (NT) such that goto(I, X) C and goto(I, X) , add the set of items goto(I, X) to C
Repeat 3 until no more sets can be added to C
Construction of the Canonical LR(1) Parsing Tables Augment the grammar with S’S Construct the set C={I0,I1,…,In} of LR(1) items State i of the parser is constructed from Ii
If [A•a, b] Ii and goto(Ii,a)=Ij then set action[i,a]=shift j If [A•, a] Ii then set action[i,a]=reduce A (apply only
if AS’) If [S’S•, $] is in Ii then set action[i,$]=accept
If goto(Ii,A)=Ij then set goto[i,A]=j Repeat 3 until no more entries added The initial state i is the Ii holding item [S’•S,$]
Example The grammar G S’ → S S → C C C → c C | dstate
ACTION GOTO
c d $ S C
0 s3 s4 1 2
1 acc
2 s6 s7 5
3 s3 s4 8
4 r3 r3
5 r1
6 s6 s7 9
7 r3
8 r2 r2
9 r2
Example Grammar and LR(1) Items Unambiguous LR(1) grammar:
S L = R | RL * R | idR L
Augment with S’ S LR(1) items (next slide)
[S’ •S, $] goto(I0,S)=I1 [S •L=R, $] goto(I0,L)=I2 [S •R, $] goto(I0,R)=I3 [L •*R, =/$] goto(I0,*)=I4 [L •id, =/$] goto(I0,id)=I5 [R •L, $] goto(I0,L)=I2
[S’ S•, $]
[S L•=R, $] goto(I0,=)=I6 [R L•, $]
[S R•, $]
[L *•R, =/$] goto(I4,R)=I7 [R •L, =/$] goto(I4,L)=I8 [L •*R, =/$] goto(I4,*)=I4 [L •id, =/$] goto(I4,id)=I5
[L id•, =/$]
[S L=•R, $] goto(I6,R)=I9 [R •L, $] goto(I6,L)=I10
[L •*R, $] goto(I6,*)=I11 [L •id, $] goto(I6,id)=I12
[L *R•, =/$]
[R L•, =/$]
[S L=R•, $]
[R L•, $]
[L *•R, $] goto(I11,R)=I13
[R •L, $] goto(I11,L)=I10
[L •*R, $] goto(I11,*)=I11
[L •id, $] goto(I11,id)=I12
[L id•, $]
[L *R•, $]
I0:
I1:
I2:
I3:
I4:
I5:
I6:
I7:
I8:
I9:
I10:
I12:
I11:
I13:
Example LR(1) Parsing Table
s5 s4
acc
s6 r6
r3
s5 s4
r5 r5
s12 s11
r4 r4
r6 r6
r2
r6
s12 s11
r5
r4
id * = $
0
1
2
3
4
5
6
7
8
9
10
11
12
13
S L R
1 2 3
8 7
10 4
10 13
Grammar:1. S’ S2. S L = R3. S R4. L * R5. L id6. R L
LALR(1) Grammars LR(1) parsing tables have many states LALR(1) parsing (Look-Ahead LR) combines LR(1)
states to reduce table size Less powerful than LR(1)
Will not introduce shift-reduce conflicts, because shifts do not use lookaheads
May introduce reduce-reduce conflicts, but seldom do so for grammars of programming languages
SLR and LALR tables for a grammar always have the same number of states, and less than LR(1) tables. Like C, SLR and LALR >100, LR(1) > 1000
Constructing LALR Parsing Tables Two ways
Construction of the LALR parsing table from the sets of LR(1) items. Union the states Requires much space and time
Construction of the LALR parsing table from the sets of LR(0) items Efficient Use in practice.
Example
stateACTION GOTO
c d $ S C
0 s36 s47 1 2
1 acc
2 s36 s47 5
36 s36 s47 89
47 r3 r3 r3
5 r1
89 r2 r2 r2
stateACTION GOTO
c d $ S C
0 s3 s4 1 2
1 acc
2 s6 s7 5
3 s3 s4 8
4 r3 r3
5 r1
6 s6 s7 9
7 r3
8 r2 r2
9 r2LALR(1)
LR(1)
Constructing LALR(1) Parsing Tables Construct sets of LR(1) items
Combine LR(1) sets with sets of items that share the same first part
[L *•R, =][R •L, =] [L •*R, =] [L •id, =]
[L *•R, $][R •L, $][L •*R, $][L •id, $]
I4:
I11:
[L *•R, =/$][R •L, =/$][L •*R, =/$][L •id, =/$]
Shorthandfor two items
in the same set
Example LALR(1) Grammar
Unambiguous LR(1) grammar:S L = R | RL * R | idR L
Augment with S’ S LALR(1) items (next slide)
[S’ •S, $] goto(I0,S)=I1 [S •L=R, $] goto(I0,L)=I2 [S •R, $] goto(I0,R)=I3 [L •*R, =/$] goto(I0,*)=I4 [L •id, =/$] goto(I0,id)=I5 [R •L, $] goto(I0,L)=I2
[S’ S•, $]
[S L•=R, $] goto(I0,=)=I6 [R L•, $]
[S R•, $]
[L *•R, =/$] goto(I4,R)=I7 [R •L, =/$] goto(I4,L)=I9 [L •*R, =/$] goto(I4,*)=I4 [L •id, =/$] goto(I4,id)=I5
[L id•, =/$]
[S L=•R, $] goto(I6,R)=I8 [R •L, $] goto(I6,L)=I9
[L •*R, $] goto(I6,*)=I4 [L •id, $] goto(I6,id)=I5
[L *R•, =/$]
[S L=R•, $]
[R L•, =/$]
I0:
I1:
I2:
I3:
I4:
I5:
I6:
I7:
I8:
I9:
Shorthandfor two items
[R L•, =][R L•, $]
Example LALR(1) Parsing Table
s5 s4
acc
s6 r6
r3
s5 s4
r5 r5
s5 s4
r4 r4
r2
r6 r6
id * = $
0
1
2
3
4
5
6
7
8
9
S L R
1 2 3
9 7
9 8
Grammar:1. S’ S2. S L = R3. S R4. L * R5. L id6. R L
LL, SLR, LR, LALR Summary LL parse tables computed using FIRST/FOLLOW
Nonterminals terminals productions Computed using FIRST/FOLLOW
LR parsing tables computed using closure/goto LR states terminals shift/reduce actions LR states nonterminals goto state transitions
A grammar is LL(1) if its LL(1) parse table has no conflicts SLR if its SLR parse table has no conflicts LR(1) if its LR(1) parse table has no conflicts LALR(1) if its LALR(1) parse table has no conflicts