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Bottom Up Parsing1
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1BOTTOM UP PARSING
2Shift-Reduce Parsers
There are two main categories of shift-reduce parsers
1. Operator-Precedence Parser
simple, but only a small class of grammars.
2. LR-Parsers
covers wide range of grammars.
SLR simple LR parser
LR most general LR parser
LALR intermediate LR parser (lookahead LR parser)
SLR, LR and LALR work same, only their parsing tables are different.SLR
CFG
LR
LALR
3LR Parsers
The most powerful shift-reduce parsing (yet efficient) is:
LR(k) parsing.
left to right right-most k lookheadscanning derivation (k is omitted it is 1)
LR parsing is attractive because: LR parsing is most general non-backtracking shift-reduce parsing, yet it is still efficient.
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 parsers.
LL(1)-Grammars LR(1)-Grammars
An LR-parser can detect a syntactic error as soon as it is possible to do so a left-to-right
scan of the input.
4LL(k) vs. LR(k)
LL(k): must predict which production to use having seen only first k
tokens of RHS
Works only with some grammars
But simple algorithm (can construct by hand)
LR(k): more powerful
Can postpone decision until seen tokens of entire RHS of a
production & k more beyond
5More on LR(k)
Can recognize virtually all programming language constructs (if CFG
can be given)
Most general non-backtracking shift-reduce method known, but can be
implemented efficiently
Class of grammars can be parsed is a superset of grammars parsed by
LL(k)
Can detect syntax errors as soon as possible
6More on LR(k)
Main drawback: too tedious to do by hand for typical
programming lang. grammars We need a parser generator
Many available
Yacc (yet another compiler compiler) or bison for C/C++
environment
CUP (Construction of Useful Parsers) for Java environment;
JavaCC is another example
We write the grammar and the generator produces the parser for that
grammar
7LR Parsers
LR-Parsers
covers wide range of grammars.
SLR simple LR parser
LR most general LR parser
LALR intermediate LR parser (look-head LR parser)
SLR, LR and LALR work same (they used the same algorithm),
only their parsing tables are different.
8LR Parsing Algorithm
Sm
Xm
Sm-1
Xm-1
.
.
S1
X1
S0
a1 ... ai ... an $
Action Table
terminals and $
st four different a actionstes
Goto Table
non-terminal
st each item isa a state numbertes
LR Parsing Algorithm
stack
input
output
9Parse Table For Expression Grammar
Rules:
1. E E + T
2. E T
3. T T * F
4. T F
5. F ( E )
6. F id
Notation:
s5 = shift 5
r2 = reduce by
E T
action goto
State id + * ( ) $ E T F
0 s5 s4 1 2 3
1 s6 acc
2 r2 s7 r2 r2
3 r4 r4 r4 r4
4 s5 s4 8 2 3
5 r6 r6 r6 r6
6 s5 s4 9 3
7 s5 s4 10
8 s6 s11
9 r1 s7 r1 r1
10 r3 r3 r3 r3
11 r5 r5 r5 r5
10
Entries in Transition Table
Entry Meaning
sn Shift into state n (advance input
pointer to next token)
gn Goto state n
rk Reduce by rule (production) k;
corresponding gn gives next state
a Accept
Error (denoted by blank entry)
11
Actions of A LR-Parser
1. shift s -- shifts the next input symbol and the state s onto the stack
( So X1 S1 ... Xm Sm, ai ai+1 ... an $ ) ( So X1 S1 ... Xm Sm ai s, ai+1 ... an $ )
2. reduce A (or rn where n is a production number)
pop 2|| (=r) items from the stack;
then push A and s
Output is the reducing production reduce A
3. Accept Parsing successfully completed
4. Error -- Parser detected an error (an empty entry in the action table)
12
LR Parsing Algorithm
Refer Text:
Compilers Principles Techniques and Tools by Alfred V Aho, Ravi
Sethi, Jeffery D Ulman
Page No. 218-219
13
Actions of A (S)LR-Parser -- Example
stack input action output
0 id*id+id$ shift 5
0id5 *id+id$ reduce by Fid Fid
0F3 *id+id$ reduce by TF TF
0T2 *id+id$ shift 7
0T2*7 id+id$ shift 5
0T2*7id5 +id$ reduce by Fid Fid
0T2*7F10 +id$ reduce by TT*F TT*F
0T2 +id$ reduce by ET ET
0E1 +id$ shift 6
0E1+6 id$ shift 5
0E1+6id5 $ reduce by Fid Fid
0E1+6F3 $ reduce by TF TF
0E1+6T9 $ reduce by EE+T EE+T
0E1 $ accept
14
Key Idea
Deciding when to shift and when to reduce is based on a DFA
applied to the stack
Edges of DFA labeled by symbols that can be on stack
(terminals + non-terminals)
Transition table defines transitions (and characterizes the type
of LR parser)
15
SLR PARSING
The central idea in the SLR method is first to construct from
the grammar a DFA to recognize viable prefixes. We group
items into sets, which become the states of the SLR parser.
Viable prefixes:
The set of prefixes of a right sentential form that can appear on the
stack of a Shift-Reduce parser is called Viable prefixes.
Example :- a, aa, aab, and aabb are viable prefixes of aabbbbd.
One collection of sets of LR(0) items, called the canonical
LR(0) collection, provides the basis for constructing SLR
parsers.
16
How to make the Parse Table?
Use DFA for building parse tables
Each state now summarizes how much we have seen so far
and what we expect to see
Helps us to decide what action we need to take
How to build the DFA, then?
Analyze the grammar and productions
Need a notation to show how much we have seen so far
for a given production: LR(0) item
17
LR(0) Item
An LR(0) item is a production and a position in its RHS marked by a dot
(e.g., A )
The dot tells how much of the RHS we have seen so far. For example,
for a production S XYZ,
S XYZ: we hope to see a string derivable from XYZ
S XYZ: we have just seen a string derivable from X and we hope to see a string derivable from YZ
SXY.Z : we have just seen a string derivable from XY and we hope to see a string derivable from Z
SXYZ. : we have seen a string derivable from XYZ and going to reduce it to S
(X, Y, Z are grammar symbols)
18
Augmented Grammar
If G is a grammar with start symbol S, then G', the
augmented grammar for G, is G with
new start symbol S' and
the production S' S.
The purpose of the augmenting production is to indicate to
the parser when it should stop parsing and accept the input.
That is, acceptance occurs only when the parser is about to
reduce by the production S' S.
19
Constructing Sets of LR(0) Items
1. Create a new nonterminal S' and a new production S' S where S is
the start symbol.
2. Put the item S' S into a start state called state 0.
3. Closure: If A B is in state s, then add B to state s for
every production B in the grammar.
4. Creating a new state from an old state[ goto operation] : goto(I,X) is
closure of set of all items [A X ] such that [A X] is in I ,
where X is a grammar symbol.
5. Repeat steps 3 and 4 until no new states are created. A state is new if it
is not identical to an old state.
20
The Closure Operation (Example)
Grammar:E E + T | TT T * F | FF ( E )F id
{ [E E] }
closure({[E E]}) =
{ [E E][E E + T][E T] }
{ [E E][E E + T][E T][T T * F][T F] }
{ [E E][E E + T][E T][T T * F][T F][F ( E )][F id] }
Add [E]Add [T]
Add [F]
21
State 0
We start by adding item E' E to
state 0.
This item has a " " immediately to the
left of a nonterminal. Whenever this is
the case, we must perform step 3
(closure) of the set construction
algorithm.
We add the items E E + T and E
T to state 0, giving
I0: { E' E
E E + T
E T }
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
22
State 0
Reapplying closure to E T, we must add the
items T T * F and
T F to state 0, giving
I0: { E' E
E E + T
E T
T T * F
T F
}
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
23
State 0
Reapplying closure to T F, we must
add the items F ( E ) and F id
to state 0, giving
I0: { E' E
E E + T
E T
T T * F
T F
F ( E )
F id
}
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
24
Formal Definition of GOTO operation for constructing
LR(0) Items
1. For each item [AX] I, add the set of items
closure({[AX]}) to goto(I,X) if not already
there
2. Repeat step 1 until no more items can be added to
goto(I,X)
25
The Goto Operation (Example 1)
Suppose I = Then goto(I,E)= closure({[E E , E E + T]})= { [E E ]
[E E + T] }
Grammar:E E + T | TT T * F | FF ( E )F id
{ [E E][E E + T][E T][T T * F][T F][F ( E )][F id] }
26
Creating State 1 From State 0 [ goto(I0,E)]
Final version of state 0:
I0: {
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
}
Using step 4, we create new state 1 from items E'
E and E E + T
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I0 E I1
27
State 1
State 1 starts with the items E' E and E E + T. These items are formed from items E' E and E E + T by moving the "" one grammar symbol to the right. In each case, the grammar symbol is E.
Closure does not add any new items, so state 1 ends up with the 2 items:
I1: {
E' E
E E + T
}
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I0 E I1
28
Creating State 2 From State 0 [ goto(I0,T)]
Using step 4, we create state 2 from items E T
and T T * F by moving the "" past the T.
State 2 starts with 2 items,
I2: {
E T
T T * F
}
Closure does not add additional items to state 2.
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I0 T I2
29
Creating State 3 From State 0 [ goto(I0,F)]
Using step 4, we create state 3 from item T
F.
State 3 starts (and ends up) with one item:
I3: {
T F
}
Since the only item in state 3 is a complete
item, there will be no transitions out of state
3.
The figure on the next slide shows the DFA of
viable prefixes to this point.
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I0 F I3
30
DFA After Creation of State 3
31
Creating State 4 From State 0 [ goto(I0,( )]
Using step 4, we create state 4 from item F
( E ).
State 4 begins with one item:
F ( E )
Applying closure to this item, we add the items
E E + T
E T
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
32
State 4
Applying closure to E T, we add items T T * F and T F to state 4, giving
F ( E )
E E + T
E T
T T * F
T F
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
33
State 4
Applying step 3 to T F, we add items F
( E ) and F id to state 4, giving the
final set of items
I4: {
F ( E )
E E + T
E T
T T * F
T F
F ( E )
F id
}
The next slide shows the DFA to this point.
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I0 ( I4
34
DFA After Creation of State 4
35
Creating State 5 From State 0 [ goto(I0,id)]
Finally, from item F id in state 0, we
create state 5, with the single item:
I5: {
F id
}
Since this item is a complete item, we will
not be able to produce new states from state
5.
The next slide shows the DFA to this point.
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I0 id I4
36
DFA After Creation of State 5
37
Creating State 6 From State 1 [ goto(I1,+)]
State 1 consists of 2 items
E' E E E + T
Create state 6 from item E E + T, giving the item E E + T.
Closure results in the set of items
I6: {
E E + T
T T * F
T F
F ( E )
F id
}
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I1 + I6
38
DFA After Creation of State 6
39
Creating State 7 From State 2 [ goto(I2,*)]
State 2 has two items,
E T
T T * F
We create state 7 from T T * F,
giving the initial item T T * F.
Using closure, we end up with
I7: {
T T * F
F ( E )
F id}
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I2 * I7
40
DFA After Creation of State 7
41
Creating State 8 From State 4 [ goto(I4,E)]
We use the items F ( E ) and E
E + T from State 4 to add the
following items to State 8:
I8: {
F ( E )
E E + T
}
No further items can be added to state 8
through closure.
There are other transitions from state 4,
but they do not result in new states.
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
I4 E I8
42
Other Transitions From State 4 [ goto(I4,T),
goto(I4,F), goto(I4,( ), goto(I4,id)]
If we use the items E T and
T T * F from state 4 to start a
new state, we begin with items
E T
T T * F
This set is identical to state 2.
Similarly, the items
T F will produce state 3
F ( E ) will produce state 4
F id will produce state 5
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
43
DFA After Creation of State 8
44
Creating State 9 From State 6 [ goto(I6,T)]
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
We use items E E + T and T T * F
from state six to create state 9:
I9: {
E E + T
T T * F
}
All other transitions from state 6 go to
existing states. The next slide shows the
DFA to this point.
I6 T I9
45
DFA After Creation of State 9
46
Creating State 10 From State 7 [ goto(I7,F)]
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
We use item T T * F from state 7 to create state 10:
I10: {
T T * F
}
All other transitions from state 7 go to existing states. The next slide shows the DFA to this point.
I7 F I10
47
DFA After Creation of State 10
48
Creation of State 11 From State 8 [ goto(I8,))]
E' E
E E + T
E T
T T * F
T F
F ( E )
F id
We use item F ( E ) from state 8 to create state 11:
I11: {
F ( E )
}
All other transitions from state 8 go to existing states.
State 9 has one transition to an existing state (7). No other new states can be added, so we are done.
The next slide shows the final DFA for viable prefixes. I8 ) I11
49
DFA for Viable Prefixes
50
DFA for Viable Prefixes
51
Constructing Parse Table
Construct the DFA (state graph) as in LR(0)
Action Table
If there is a transition from the state i to state j on a terminal a,
ACTION[i, a] = shift j
If there is a reduce item A (for a production #k in state i, for each a FOLLOW(A),
ACTION[i, a] = Reduce k
If an item S S. is in state i,
ACTION[i, $] = Accept
Otherwise, error
GOTO
Write GOTO for nonterminals: for terminals it is already embedded
in the action table
52
Algorithm Construction of SLR Parsing Table1. Construct the canonical collection of sets of LR(0) items for G.
C{I0,...,In}
2. Create the parsing action table as follows
If a is a terminal, A.a in Ii and goto(Ii,a)=Ij then action[i,a] is
shift j.
If A. is in Ii , then action[i,a] is reduce A for all a in FOLLOW(A) where AS.
If SS. is in Ii , then action[i,$] is accept.
If any conflicting actions generated by these rules, the grammar is
not SLR(1).
Create the parsing goto table
for all non-terminals A, if goto(Ii,A)=Ij then goto[i,A]=j
All entries not defined by (2) and (3) are errors.
4. Initial state of the parser contains S.S
53
(SLR) Parsing Tables for Expression Grammar
1) E E+T
2) E T
3) T T*F
4) T F
5) F (E)
6) F id
54
DFA for Viable Prefixes
55
We use the partial DFA at right
to fill in row 0 of the parse table.
By rule 2a,
action[ 0, ( ] = shift 4
action[ 0, id ] = shift 5
By rule 3,
goto[ 0, E ] = 1
goto[ 0, T ] = 2
goto[ 0, F ] = 3
56
state id + * ( ) $ E T F
0 s5 s4 1 2 3
1
2
3
4
5
6
7
8
9
10
11
1) E E+T
2) E T
3) T T*F
4) T F
5) F (E)
6) F id
Action Table Goto Table
57
We use the partial DFA at right
to fill in row 1 of the parse table.
By rule 2a,
action [ 1, + ] = shift 6
By rule 2c
action [ 1, $ ] = accept
58
state id + * ( ) $ E T F
0 s5 s4 1 2 3
1 s6 acc
2
3
4
5
6
7
8
9
10
11
1) E E+T
2) E T
3) T T*F
4) T F
5) F (E)
6) F id
Action Table Goto Table
59
We use the partial DFA at right
to fill in row 5 of the parse table.
By rule 2b, we set
action[ 5, x ] = reduce Fid
for each x Follow(F).
Since Follow(F) = { ), +, *, $)
we have
action[ 5, ) ] = reduce
Fid
action[ 5, +] = reduce
Fid
action[5, *] = reduce
Fid
action[5, $] = reduce
Fid
60
state id + * ( ) $ E T F
0 s5 s4 1 2 3
1 s6 acc
2
3
4
5 r6 r6 r6 r6
6
7
8
9
10
11
1) E E+T
2) E T
3) T T*F
4) T F
5) F (E)
6) F id
Action Table Goto Table
61
Use the DFA to Finish the SLR Table
The complete SLR parse table for the expression grammar is given on the next slide.
62
(SLR) Parsing Tables for Expression Grammar
state id + * ( ) $ E T F
0 s5 s4 1 2 3
1 s6 acc
2 r2 s7 r2 r2
3 r4 r4 r4 r4
4 s5 s4 8 2 3
5 r6 r6 r6 r6
6 s5 s4 9 3
7 s5 s4 10
8 s6 s11
9 r1 s7 r1 r1
10 r3 r3 r3 r3
11 r5 r5 r5 r5
Action Table Goto Table
1) E E+T
2) E T
3) T T*F
4) T F
5) F (E)
6) F id
63
Actions of A (S)LR-Parser -- Example
stack input action output
0 id*id+id$ shift 5
0id5 *id+id$ reduce by Fid Fid
0F3 *id+id$ reduce by TF TF
0T2 *id+id$ shift 7
0T2*7 id+id$ shift 5
0T2*7id5 +id$ reduce by Fid Fid
0T2*7F10 +id$ reduce by TT*F TT*F
0T2 +id$ reduce by ET ET
0E1 +id$ shift 6
0E1+6 id$ shift 5
0E1+6id5 $ reduce by Fid Fid
0E1+6F3 $ reduce by TF TF
0E1+6T9 $ reduce by EE+T EE+T
0E1 $ accept
64
SLR PARSING
The central idea behind SLR method was first to construct
from the grammar a DFA to recognize viable prefixes. We
group items into sets, which become the states of the SLR
parser.
Viable prefixes:
The set of prefixes of a right sentential form that can appear on the
stack of a Shift-Reduce parser is called Viable prefixes.
Example :- a, aa, aab, and aabb are viable prefixes of aabbbbd.
65
Example SLR Grammar and LR(0) Items
Augmentedgrammar:1. C C2. C A B3. A a4. B a
State I0:C CC A BA a
State I1:C C
State I2:C ABB 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
66
Example SLR Parsing Table
s3
acc
s5
r3
r2
r4
a $
0
1
2
3
4
5
C A B
1 2
4
State I0:C CC A BA a
State I1:C C
State I2:C ABB 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:1. C C2. C A B3. A a4. B a
67
shift/reduce and reduce/reduce conflicts
If a state does not know whether it will make a shift operation or
reduction for a terminal, we say that there is a shift/reduce conflict.
If a state does not know whether it will make a reduction operation using the production rule i or j for a terminal, we say that there is a
reduce/reduce conflict.
If the SLR parsing table of a grammar G has a conflict, we say that that
grammar is not SLR grammar.
68
Conflict Example
S L=R I0: S .S I1:S S. I6:S L=.R I9: S L=R.
S R S .L=R R .L
L *R S .R I2:S L.=R L .*R
L id L .*R R L. L .id
R L L .id
R .L I3:S R.
I4:L *.R I7:L *R.
Problem R .L
FOLLOW(R)={=,$} L .*R I8:R L.
= shift 6 L .id
reduce by R L
shift/reduce conflict I5:L id.
69
Conflict Example2
S AaAb I0: S .S
S BbBa S .AaAb
A S .BbBa
B A .
B .
Problem
FOLLOW(A)={a,b}
FOLLOW(B)={a,b}
a reduce by A b reduce by A
reduce by B reduce by B
reduce/reduce conflict reduce/reduce conflict