CS 2130

1
CS 2130

• Lecture 18
• Bottom-Up Parsing
• or
• Shift-Reduce Parsing

2
Bottom Up Parsing

• Assume we have a group of tokens
• Bottom up parsing tries to group tokens into
  things it can reduce
• Like our precedence rules make
• 1 2 3
• Understood to be
• (1 (2 3))
• Same idea but with bottom up parsing we use
  special operators Wirth-Weber

3
Wirth-Weber Operators
4
Operation

• Given
• a b c d e f g h

Keep in mind that the letters represent tokens
thus they can be operators, operands, special
symbols, etc
5
Operation

• Given
• lt a b c d e f g h gt

Assumed
6
Operation

• Given
• lt a b c d e f g h gt
• Parser moves through tokens comparing precedence

7
Operation

• Given
• lt a b gt c d e f g h gt
• A series starting with lt and ending with gt is
  called a handle.
• When a handle is found the parser looks for a
  match with the right hand side of some production

8
Operation

• Given
• lt a b gt c d e f g h gt
• Assume the following is in effect
• r a b
• s d e f
• t r c s

9
Operation

• Given
• lt r c d e f g h gt
• Assume the following is in effect
• r a b
• s d e f
• t r c s

10
Operation

• Given
• lt r c d e f g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

11
Operation

• Given
• lt r c d e f g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

12
Operation

• Given
• lt r c lt d e f g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

13
Operation

• Given
• lt r c lt d e f g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

14
Operation

• Given
• lt r c lt d e f g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

15
Operation

• Given
• lt r c lt d e f gt g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

16
Operation

• Given
• lt r c lt d e f gt g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

17
Operation

• Given
• lt r c s g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

18
Operation

• Given
• lt r c s g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

19
Operation

• Given
• lt r c s gt g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

20
Operation

• Given
• lt r c s gt g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

21
Operation

• Given
• lt t g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

22
Operation

• Given
• lt t g h gt
• We continue
• Assume the following is in effect
• r a b
• s d e f
• t r c s

23
Operation

• Given
• lt t g h gt
• If there is a rule for "t g h" successful
  parse
• If not, syntax error

24
What kind of algorithm?

• Stack based
• Known as semantic stack or shift/reduce algorithm

25
Example

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e z g gt
• lt y m gt
• lt n gt

If this is the start symbol Successful Parse!
26
Stack
a, lt

• lt a b lt c d gt e lt f gt g gt

We process the tokens adding the token and the
precedence operator on its left to the stack.
Encountering a gt indicates a handle has been
found and thus will initiate special proceesing

27
Stack
b, a, lt

• lt a b lt c d gt e lt f gt g gt

28
Stack
c, lt b, a, lt

• lt a b lt c d gt e lt f gt g gt

29
Stack
d, c, lt b, a, lt

• lt a b lt c d gt e lt f gt g gt

30
Stack
d, c, lt b, a, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt

Finding a "gt" between d and e, we trace back...
31
Stack
x, gt b, a, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt

x is not really on stack because of the gt
32
Stack
x, lt y, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt

We trace back to process the ab handle and then
x goes on stack
33
Stack
e, x, lt y, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt

Continuing...
34
Stack
f, lt e, x, lt y, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt
• lt y lt x e z g gt

Converting f to z
35
Stack
g, f, lt e, x, lt y, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt
• lt y lt x e z g gt

36
Stack
, gt g, z, lt e, x, lt y, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt
• lt y lt x e z g gt

signifies end of input
37
Stack
, gt m, y, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt
• lt y lt x e z g gt
• lt y m gt

Reducing
38
Stack
, gt n, lt

• lt a b lt c d gt e lt f gt g gt
• lt a b gt x e lt f gt g gt
• lt y lt x e lt f gt g gt
• lt y lt x e z g gt
• lt y m gt
• lt n gt

If n is our start symbol successful parse!
39
Shift/Reduce Parsing

• x lt y
• x y
• x gt y
• ?

Shift
Reduce
Error No relationship ? Syntax error
Not allowed More than one relationship
Actually lt would be okay, gt or gtlt is called
shift/reduce error
40
Shift/Reduce Parsing

• Reduce-Reduce Error
• When trying to match, parser finds two rules
  which match
• No Right Hand Side Match
• Syntax error

41
Simple Precedence

• Only Possibilities
• x lt y
• x y
• x gt y
• No relationship exists
• (No others allowed)

42
Bottom-Up Parsing

• No issues regarding left-recursive versus
  right-recursive such as those found with Top-down
  parsing
• Note There are grammars that will break a
  bottom-up parser.

43
Precedence

• Consider our simple grammar...
• Between symbols there must be ?
• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

44
Precedence

• Consider our simple grammar...
• Between symbols there must be
• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

45
Precedence

• To determine "end points" we must look at
  multiple rules to see how they interact...

x
c
d

46
Precedence

• To determine "end points" we must look at
  multiple rules to see how they interact...

x
c
d

To determine what goes here...
47
Precedence

• To determine "end points" we must look at
  multiple rules to see how they interact...

x
c
d

We look here.
48
Precedence

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

lttermgt
ltfactorgt
lt ltfactorgt
lt (
ltexprgt )
lt (
ltexprgt )


49
Precedence Table

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

L R
ltexprgt
lttermgt
ltfactorgt

(
num

id
)
ltexprgt
lttermgt
ltfactorgt


(
)
num
id
50
Precedence Table

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

L R
ltexprgt
lttermgt
ltfactorgt

(
num

id
)
ltexprgt
lttermgt
ltfactorgt


(
)
num
id
51
Precedence Table

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

L R
ltexprgt
lttermgt
ltfactorgt

(
num

id
)


ltexprgt
gt

gt
lttermgt
gt
gt
gt
ltfactorgt
lt
lt
lt
lt


lt
lt
lt

lt
lt
lt
(
gt
gt
gt
)
gt
gt
gt
num
gt
gt
gt
id
52
Precedence Table

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

L R
ltexprgt
lttermgt
ltfactorgt

(
num

id
)


ltexprgt
gt

gt
lttermgt
gt
gt
gt
ltfactorgt
lt
lt
lt
lt


lt
lt
lt

lt
lt
lt
(
'(' ltexprgt... '(' lt '(' ltexprgt...
gt
gt
gt
)
gt
gt
gt
num
gt
gt
gt
id
53
Precedence Table

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

L R
ltexprgt
lttermgt
ltfactorgt

(
num

id
)


ltexprgt
gt

gt
lttermgt
gt
gt
gt
ltfactorgt
lt
lt
lt
lt
lt


lt
lt
lt

lt
lt
lt
(
lt
gt
gt
gt
)
gt
gt
gt
num
gt
gt
gt
id
54
Precedence Table

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id

L R
ltexprgt
lttermgt
ltfactorgt

(
num

id
)


ltexprgt
gt

gt
lttermgt
'(' ltexprgt ')'
gt
gt
gt
ltfactorgt
lt
lt
lt
lt
lt


lt
lt
lt

lt
lt
lt
(
lt
gt
gt
gt
)
'(' lt ltexprgt
gt
gt
gt
num
gt
gt
gt
id
55
Resolving Ambiguity

• lttermgt
• lt lttermgt ltfactorgt
• Solve by lookahead
• lttermgt or )
• lt lttermgt
• Ambiguity can be resolved by increasing k in
  LR(k) but that's not the only way
• We could rewrite grammar

56
Original Grammar

• ltexprgt ltexprgt lttermgt lttermgtlttermgt
  lttermgt ltfactorgt ltfactorgtltfactorgt '('
  ltexprgt ')' num id

57
Sources of Ambiguity

• ltexprgt ltexprgt lttermgt lttermgtlttermgt
  lttermgt ltfactorgt ltfactorgtltfactorgt '('
  ltexprgt ')' num id

58
Add 2 New Rules

• ltexprgt ltexprgt lttermgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltexprgt ')' num id
• ltegt ltexprgt
• lttgt lttermgt

59
Modify

• ltexprgt ltexprgt lttgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltegt ')' num id
• ltegt ltexprgt
• lttgt lttermgt

60
Original Grammar
ltexprgt ltexprgt lttermgt lttermgtlttermgt
lttermgt ltfactorgt ltfactorgtltfactorgt '('
ltexprgt ')' num id
Rewritten Grammar

• ltexprgt ltexprgt lttgt lttermgt
• lttermgt lttermgt ltfactorgt ltfactorgt
• ltfactorgt '(' ltegt ')' num id
• ltegt ltexprgt
• lttgt lttermgt

61
Performance

• Size of table is O(n2)
• If we use operator precedence then it only uses
  terminals thus the table size is not affected by
  adding non-terminals

62
Question 1

• a lt b c d gt e
• a ? x e

What happens if there is no relationship here?
63
Question 2

• Where do precedence relationships come from?
• Make a table by hand
• Write a program to make table
• How such a program works or how to write it are
  topics beyond the scope of this course.

64
Example
65

• lt num gt

1 2 3 Tokenized num num num
66

• lt num gt
• lt ltfactorgt gt

1 2 3 Tokenized num num num
67

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt

1 2 3 Tokenized num num num
68

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt

1 2 3 Tokenized num num num
69

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num

1 2 3 Tokenized num num num
70

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt

1 2 3 Tokenized num num num
71

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt

1 2 3 Tokenized num num num
72

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt
• lt ltexprgt lt lttermgt

1 2 3 Tokenized num num num
73

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt
• lt ltexprgt lt lttermgt
• lt ltexprgt lt lttermgt lt num

1 2 3 Tokenized num num num
74

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt
• lt ltexprgt lt lttermgt
• lt ltexprgt lt lttermgt lt num
• lt ltexprgt lt lttermgt lt num gt

1 2 3 Tokenized num num num
75

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt
• lt ltexprgt lt lttermgt
• lt ltexprgt lt lttermgt lt num
• lt ltexprgt lt lttermgt lt num gt
• lt ltexprgt lt lttermgt ltfactorgt gt

1 2 3 Tokenized num num num
76

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt
• lt ltexprgt lt lttermgt
• lt ltexprgt lt lttermgt lt num
• lt ltexprgt lt lttermgt lt num gt
• lt ltexprgt lt lttermgt ltfactorgt gt
• lt ltexprgt lttermgt gt

1 2 3 Tokenized num num num
77

• lt num gt
• lt ltfactorgt gt
• lt lttermgt gt
• lt ltexprgt
• lt ltexprgt lt num
• lt ltexprgt lt num gt
• lt ltexprgt lt ltfactorgt gt
• lt ltexprgt lt lttermgt
• lt ltexprgt lt lttermgt lt num
• lt ltexprgt lt lttermgt lt num gt
• lt ltexprgt lt lttermgt ltfactorgt gt
• lt ltexprgt lttermgt gt
• lt ltexprgt gt

1 2 3 Tokenized num num num
78
Questions?
79
(No Transcript)