Parse Trees and Ambiguity (Dr. Torng)

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Parse Trees and Ambiguity(Dr. Torng)

• Parse/Derivation Trees
• Leftmost derivations, rightmost derivations
• Ambiguous Grammars
• Examples
• Arithmetic expressions
• If-then-else Statements
• Inherently ambiguous CFLs

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Context-Free Grammars
Parse Trees Leftmost/rightmost derivations Ambiguo
us grammars
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Parse Tree

• Parse/derivation trees are structured derivations
• The structure graphically illustrates semantic
  information about the string
• Formalization of concept we encountered in
  regular languages unit
• Note, what we saw before were not exactly parse
  trees as we define them now, but they were close

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Example

• S gt SS gt SSS gt (S)SS gt ( )SS gt ( )S gt
  ( )
• The above is a leftmost derivation of the string
  ( ) from the grammar BALG
• Draw the corresponding parse tree
• Draw the corresponding rightmost derivation
• S gt (S) gt (SS) gt (S(S)) gt (S( )) gt ((
  ))
• The above is a rightmost derivation of the string
  (( )) from the grammar BALG
• Draw the corresponding parse tree
• Draw the corresponding leftmost derivation

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Ambiguous Grammars
Examples Arithmetic Expressions If-then-else
statements Inherently ambiguous grammars
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Ambiguous Grammars

• A grammar G is ambiguous if there exists a string
  x in L(G) with two or more distinct parse trees
• (2 or more distinct leftmost/rightmost
  derivations)
• Example
• Grammar AG is ambiguous
• String aaa in L(AG) has 2 rightmost derivations
• S gt SS gt SSS gt SSa gt Saa gt aaa
• S gt SS gt Sa gt SSa gt Saa gt aaa

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2 Simple Examples

• Grammar BALG is ambiguous
• String ( ) in L(BALG) has more than 1 leftmost
  derivation
• S gt (S) gt ( )
• S gt (S) gt (SS) gt(S) gt( )
• Give another leftmost derivation of ( ) from BALG
• Grammar ABG is NOT ambiguous
• Consider any string x in aibi i gt 0
• There is a unique parse tree for x

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Legal Arithmetic Expressions

• Develop a grammar MATHG (V, S, S, P) for the
  language of legal arithmetic expressions
• ? 0, 1, , , -, /, (, )
• Strings in the language include
• 0
• 10
• 1011111100
• 10(11111100)
• Strings not in the language include
• 10
• 11101
• )(

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Grammar MATHG1

• V E, N
• S 0, 1, , , -, /, (, )
• S E
• P
• N ? N0 N1 0 1
• E ? N EE EE E/E E-E (E)

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MATHG1 is ambiguous
E ? N EE EE E/E E-E (E)N ? N0 N1
0 1

• Come up with two distinct leftmost derivations of
  the string 11011
• E gt EE gt NE gt N1E gt 11E gt 11EE
  gt 11NE gt 110E gt 110N gt 110N1 gt
  11011
• E gt EE gt EEE gt NEE gt N1EE gt
  11EE gt 11NE gt 110E gt 110N gt
  110N1 gt11011
• Draw the corresponding parse trees

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Implications

• Two interpretations of string 11011
• 11(011) 11
• (110)11 1001
• What if a line in a program is
• MSU_Tuition 11011
• What is MSU_Tuition?
• Depends on how the expression 11011 is parsed.
• This is not good.
• Ambiguity in grammars is undesirable,
  particularly if the grammar is used to develop a
  compiler for a programming language like C.

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Grammar MATHG2

• V E, T, F, N
• S 0, 1, , , -, /, (, )
• S E
• P
• N ? N0 N1 0 1
• E ? ET E-T T
• T ? T F T / F F
• F ? (E) N

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If-Then-Else Statements

• A grammar ITEG (V, ?, S, P) for the language of
  legal If-Then-Else statements
• V (S, BOOL)
• S Dlt85, Dgt50, grade3.5, grade3.0, if, then,
  else
• S S
• P
• BOOL ? Dlt85 Dgt50
• S ? if BOOL then S else S if BOOL then S
  grade3.5 grade3.0

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ITEG is ambiguous
S ? if BOOL then S grade3.5 grade3.0 if
BOOL then S else S BOOL ? Dlt85 Dgt50

• Come up with two distinct leftmost derivations of
  the string
• if Dlt85 then if Dgt50 then grade3.5 else
  grade3.0
• S gtif BOOL then S else S gt if Dlt85 then S
  else S gt if Dlt85 then if BOOL then S else S gt
  if Dlt85 then if Dgt50 then S else S gt if Dlt85
  then if Dgt50 then grade3.5 else S gt if Dlt85
  then if Dgt50 then grade3.5 else grade3.0
• S gtif BOOL then S gt if Dlt85 then S gt if
  Dlt85 then if BOOL then S else S gt if Dlt85 then
  if Dgt50 then S else S gt if Dlt85 then if Dgt50
  then grade3.5 else S gt if Dlt85 then if Dgt50
  then grade3.5 else grade3.0
• Draw the corresponding parse trees

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Parse Tree Meanings
S
S
if
B
then
S
if
S
B
then
S
else
S
else
if
Dlt85
B
then
S
if
Dlt85
grade3.0
B
then
S
Dgt50
grade3.5
grade3.0
Dgt50
grade3.5
If you receive a 90 on type D points, what is
your grade? By parse tree 1 By parse tree 2
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Implications

• Two interpretations of string
• if Dlt85 then if Dgt50 then grade3.5 else
  grade3.0
• Issue is which if-then does the last ELSE attach
  to?
• This phenomenon is known as the dangling else
• Answer Parse tree 2.
• C language An ELSE belongs to the last preceding
  IF for which there is no ELSE already.
• In this case, there is an unambiguous grammar for
  handling if-thens as well as if-then-elses

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Grammar ITEG2

• V (S, M, U, BOOL)
• S Dlt85, Dgt50, grade3.5, grade3.0, if, then,
  else
• S S
• P
• BOOL ? Dlt85 Dgt50
• S ? M U
• M ? if BOOL then M else M grade3.5 grade3.0
  
• U ? if BOOL then S if BOOL then M else U
• Before ELSE only M appears and M has balance
  IF/ELSE

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Inherently ambiguous CFLs

• A CFL L is inherently ambiguous iff for all CFGs
  G such that L(G) L, G is ambiguous
• Examples so far
• None of the CFLs weve seen so far are
  inherently ambiguous
• While the CFGs weve seen ambiguous, there do
  exist unambiguous CFGs for those CFLs.
• There exist inherently ambiguous CFLs
• Example aibjck ij or jk or ijk
• Note ijk is unnecessary, but I added it here
  for clarity

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Summary

• Parse trees illustrate semantic information
  about strings
• Ambiguous grammars are undesirable
• This means there are multiple parse trees for
  some string
• These strings can be interpreted in multiple ways
• There are some heuristics people use for taking
  an ambiguous grammar and making it unambiguous,
  but this is not the focus of this course
• There are some inherently ambiguous CFLs
• Thus, the above heuristics do not always work

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