Chapter 4. Syntax Analysis (2)

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Chapter 4. Syntax Analysis (2)
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Fig. 4.23. Operator-precedence relations.
id
id gt gt gt
lt gt lt gt
lt gt gt gt
lt lt lt
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Operator-Precedence Relations from Associativity
and Precedence (1/2)

• 1. If operator ?1 has higher precedence than
  operator ?2, make ?1 gt ?2 and ?2 lt ?1 . For
  example, if has higher precedence than , make
  gt and lt . These relations ensure that,
  in an expression of the form EEEE, the central
  EE is the handle that will be reduced first.
• 2. If ?1 and ?2 are operators of equal
  precedence (they may in fact be the same
  operator), then make ?1 gt ?2 and ?2 gt ?1 if the
  operators are left-associative, or make ?1 lt ?2
  and ?2 lt ?1 if they are right-associative. For
  example, if and are left-associative, then
  make gt , gt -, - gt - and - gt . If ? is
  right associative, then make ? lt ?. These
  relations ensure that E-EE will have handle E-E
  selected and E?E?E will have the last E?E
  selected.

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Operator-Precedence Relations from Associativity
and Precedence (2/2)

• 3. Make ? lt id, id gt ?, ? lt (, ( lt ? , ) gt
  ?, ? gt ), ? gt , and lt ? for all operators
  ?. Also, let
• These rules ensure that both id and (E) will be
  reduced to E. Also, serves as both the left and
  right endmarker, causing handles to be found
  between s wherever possible.

( ) lt ( lt id
( lt ( id gt ) gt
( lt id id gt ) ) gt )
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- / ? id ( )
gt gt lt lt lt lt lt gt gt
- gt gt lt lt lt lt lt gt gt
gt gt gt gt lt lt lt gt gt
/ gt gt gt gt lt lt lt gt gt
? gt gt gt gt lt lt lt gt gt
id gt gt gt gt gt gt gt
( lt lt lt lt lt lt lt
) gt gt gt gt gt gt gt
lt lt lt lt lt lt lt

Fig. 4.25. Operator-precedence relations.
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Precedence Functions

• Example 4.29 The Precedence table of Fig. 4.25
  has the following pair of precedence functions,
• For example, lt id and, f() lt g(id). Note
  that f(id) gt g(id) suggests that id gt id but,
  in fact, no precedence relation holds between id
  and id. Other error entries in Fig. 4.25 are
  similarly replaced by one or another precedence
  relation.

- / ? ( ) id
f 2 2 4 4 4 0 6 6 0
g 1 1 3 3 5 5 0 5 0
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Fig. 4.26. Graph representing precedence
functions.
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Fig. 4.28. Operator-precedence matrix with error
entries
id ( )
id e3 e3 gt gt
( lt lt e4
) e3 e3 gt gt
lt lt e2 e1

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Fig. 4.29. Model of an LR parser.
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STATE action action action action action action goto goto 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
Fig. 4.31. Parsing table for expression grammar.
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Fig. 4.32. Moves of LR parser on id id id.
STACK STACK INPUT ACTION
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 0 0 id 5 0 F 5 0 T 2 0 T 2 7 0 T 2 7 id 5 0 T 2 7 F 10 0 T 2 0 E 1 0 E 1 6 0 E 1 6 id 5 0 E 1 6 F 3 0 E 1 6 T 9 0 E 1 id id id id id id id id id id id id id id id id shift reduced by F ? id reduced by T ? F shift shift reduced by F ? id reduced by T ? TF reduced by E ? T shift shift reduced by F ? id reduced by T ? F E ? E T accept
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Fig. 4.35. Canonical LR(0) collection for
grammar (4.19)
I0 E' ? E E ? E T E ? T T ? T F T ? F F ? (E) F ? id I5 F ? id
I0 E' ? E E ? E T E ? T T ? T F T ? F F ? (E) F ? id I6 E ? E T T ? T F T ? F F ? (E) F ? id
I0 E' ? E E ? E T E ? T T ? T F T ? F F ? (E) F ? id I7 T ? T F F ? (E) F ? id
I1 E' ? E E ? E T I7 T ? T F F ? (E) F ? id
I2 E ? T T ? T F I8 F ? ( E ) E ? E T
I2 E ? T T ? T F I9 E ? E T T ? T F
I3 T ? F I9 E ? E T T ? T F
I4 F ? ( E ) E ? E T E ? T T ? T F T ? F F ? (E) F ? id I9 E ? E T T ? T F
I4 F ? ( E ) E ? E T E ? T T ? T F T ? F F ? (E) F ? id I10 T ? T F
I4 F ? ( E ) E ? E T E ? T T ? T F T ? F F ? (E) F ? id I11 F ? ( E )
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Fig. 4.36. Transition diagram of DFA D form
viable prefixes.
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Fig. 4.37. Canonical LR(0) collection for grammar
(4.20).
I0 S' ? S S ? L R S ? R L ? R L ? id R ? L I5 L ? id
I0 S' ? S S ? L R S ? R L ? R L ? id R ? L I6 S ? L R R ? L L ? R L ? id
I1 S' ? S I7 L ? R
I2 S ? L R R ? L I8 R ? L
I2 S ? L R R ? L I9 S ? L R
I3 S ? R I9 S ? L R
I4 L ? R R ? L L ? R L ? id I9 S ? L R
I4 L ? R R ? L L ? R L ? id
I4 L ? R R ? L L ? R L ? id
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Fig. 4.39. The goto graph for grammar (4.21).
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STATE action action action goto goto
STATE 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
Fig. 4.40. Canonical parsing table for grammar
(4.21).
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Constructing LALR Parsing Table

• Example 4.44. Consider the grammar
• S' ? S
• S ? aAd bBd aBe bAe
• A ? c
• B ? c
• which generates the four strings acd, ace,
  bcd, and bce. The reader can check that the
  grammar is LR(1) by constructing the sets of
  items. Upon doing so, we find the set of items
  A ? c , e , B ? c , e valid for
  viable prefix ac and A ? c , e , B ? c
  , d valid for bc. Neither of these sets
  generates a conflict, and their cores are the
  same. However, their union, which is
• A ? c , d / e
• B ? c , d / e
• generates a reduce/reduce conflict, since
  reductions by both A ? c and B ? c are called for
  on inputs d and e.

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STATE action action action goto goto
STATE 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
Fig. 4.41. LALR parsing table for grammar (4.21).
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Efficient Construction of LALR Parsing Tables

• Example 4.46. Let us again consider the augmented
  grammar
• S' ? S
• S ? L R R
• A ? R id
• B ? L
• The kernels of the sets of LR(0) items for this
  grammar are shown in Fig. 4.42.

I0 S' ? S
I1 S' ? S
I2 S ? L R R ? L
I3 S ? R
I4 L ? R
I5 L ? id
I6 S ? L R
I7 L ? R
I8 R ? L
I9 S ? L R
Fig. 4.42. Kernels of the sets of LR(0) items for
grammar (4.20).
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Efficient Construction of LALR Parsing Tables

• Example 4.47. Let us construct the kernels of the
  LALR(1) items for the grammar in the previous
  example. The kernels of the LR(0) items were
  shown in Fig. 4.42. When we apply Algorithm 4.12
  to the kernel of set of items I0, we compute
  closure (S' ? S, ), which is
• S' ? S,
• S ? L R,
• S ? R,
• L ? R, /
• L ? id, /
• R ? L,

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Fig.4.44. Propagation of lookaheads.
FROM FROM TO TO
I0 S' ? S I1 I2 I2 I3 I4 I5 S' ? S S ? L R R ? L S ? R L ? R L ? id
I2 S ? L R I6 S ? L R
I4 L ? R I4 I5 I7 I8 L ? R L ? id L ? R R ? L
I6 S ? L R I4 I5 I8 I9 L ? R L ? R R ? L S ? L R
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Fig. 4.45. Computation of lookaheads.
SET ITEM LOOKAHEADS LOOKAHEADS LOOKAHEADS LOOKAHEADS
SET ITEM INIT PASS1 PASS2 PASS3
I0 S' ? S
I1 S' ? S
I2 S ? L R
I2 R? L
I3 S ? R
I4 L ? R / / /
I5 L ? id / / /
I6 S ? L R
I7 L ? R / /
I8 R ? L / /
I9 S ? L R
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Fig. 4.46. Set of LR(0) items for augmented
grammar (4.22).
I0 E' ? E E ? E E E ? E E E ? ( E ) E ? id I5 E ? E E E ? E E E ? E E E ? ( E ) E ? id
I1 E' ? E E' ? E E E' ? E E I6 E' ? ( E ) E' ? E E E' ? E E
I2 E ? ( E ) E ? E E E ? E E E ? ( E ) E ? id I7 E' ? E E E' ? E E E' ? E E
I2 E ? ( E ) E ? E E E ? E E E ? ( E ) E ? id I8 E' ? E E E' ? E E E' ? E E
I3 E ? id I8 E' ? E E E' ? E E E' ? E E
I4 E ? E E E ? E E E ? E E E ? ( E ) E ? id I9 E' ? ( E )
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STATE action action action action action action goto
STATE id ( ) E
0 s3 s2 1
1 s4 s5 acc
2 s3 s2 6
3 r4 r4 r4 r4
4 s3 s2 8
5 s3 s2 8
6 s4 s5 s9
7 r1 s5 r1 r1
8 r2 r2 r2 r2
9 r3 r3 r3 r3
Fig. 4.47. Parsing table for grammar (4.22).