Module 29

1
Module 29

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

2
Context-Free Grammars

• Parse Trees
• Leftmost/rightmost derivations
• Ambiguous grammars

3
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

4
Parse Tree Example

• Parse tree for string ( )(( )) and grammar BALG
• BALG (V, S, S, P)
• V S, S (, ), S S
• P S --gt SS (S) l
• One derivation of ( )(( ))
• S gt SS gt (S)S gt ( )S gt ( )(S) gt (
  )((S)) gt ( )(( ))
• Parse tree

5
Comments about Example

• Syntax
• draw a unique arrow from each variable to each
  character that is a direct child of that variable
• A line instead of an arrow is ok
• The derived string can be read in a left to right
  traversal of the leaves
• Semantics
• The tree graphically illustrates the nesting
  structure of the string of parentheses

6
Leftmost/Rightmost Derivations

• There is more than one derivation of the string (
  )(( )).
• S gt SS gt (S)S gt( )S gt ( )(S)
• gt ( )((S)) gt ( )(( ))
• S gt SS gt (S)S gt (S)(S) gt ( )(S)
• gt ( )((S)) gt ( )(( ))
• S gt SS gt S(S) gt S((S)) gt S(( ))
• gt (S)(( )) gt( )(( ))
• Leftmost derivation
• Leftmost variable is always expanded
• Which one of the above is leftmost?
• Rightmost derivation
• Rightmost variable is always expanded
• Which one of the above is rightmost?

7
Comments

• Fix a string and a grammar
• Any derivation corresponds to a unique parse tree
• Any parse tree can correspond to many different
  derivations
• Example
• The one parse tree corresponds to all three
  derivations
• Unique mappings
• For any parse tree, there is a unique
  leftmost/rightmost derivation that it corresponds
  to

• S gt SS gt (S)S gt( )S gt ( )(S)
• gt ( )((S)) gt ( )(( ))
• S gt SS gt (S)S gt (S)(S) gt ( )(S)
• gt ( )((S)) gt ( )(( ))
• S gt SS gt S(S) gt S((S)) gt S(( ))
• gt (S)(( )) gt( )(( ))

8
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

9
Ambiguous Grammars

• Examples
• Arithmetic Expressions
• If-then-else statements
• Inherently ambiguous grammars

10
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

11
2 Simple Examples

• Grammar BALG is ambiguous
• String ( ) in L(BALG) has gt1 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

12
Legal Arithmetic Expressions

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

13
Grammar MATHG1

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

14
MATHG1 is ambiguous
E --gt N EE EE E/E E-E (E)N --gt 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

15
Corresponding Parse Trees

• 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

E
E
16
Parse Tree Meanings
Note how the parse trees captures the semantic
meaning of string 11011. More specifically,
what number does the first parse tree
represent? What number does the second parse
tree represent?
17
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.
• In this case, there is an unambiguous grammar for
  the language of arithmetic expressions

18
If-Then-Else Statements

• A grammar ITEG (V, S, 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
• S --gt if BOOL then S else S if BOOL then S
  grade3.5 grade3.0
• BOOL --gt Dlt85 Dgt50

19
ITEG is ambiguous
S --gt if BOOL then S grade3.5 grade3.0 if
BOOL then S else S BOOL --gt 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

20
Corresponding Parse Trees

• 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

S
S
21
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
22
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 Typically, else binds to NEAREST if-then
• In this case, there is an unambiguous grammar for
  handling if-thens as well as if-then-elses

23
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.
• Later result
• There exist inherently ambiguous CFLs
• Example aibjck ij or jk or ijk
• Note ijk is unnecessary, but I added it here
  for clarity

24
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