1. Module 29 Parse/Derivation Trees Ambiguous Grammars
Leftmost derivations, rightmost derivations
Ambiguous Grammars
Examples
Arithmetic expressions
If-then-else Statements
Inherently ambiguous CFL’s

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, Σ, S, P)
V = {S}, Σ = {(, )}, S = S
P = S → SS | (S) | l
One derivation of ( )(( ))
S ==> SS ==> (S)S ==> ( )S ==> ( )(S) ==> ( )((S)) ==> ( )(( ))
Parse tree l S ( )

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 l S ( )

6. Leftmost/Rightmost Derivations
There is more than one derivation of the string ( )(( )).
S ==> SS ==> (S)S ==>( )S ==> ( )(S)
==> ( )((S)) ==> ( )(( ))
S ==> SS ==> (S)S ==> (S)(S) ==> ( )(S)
S ==> SS ==> S(S) ==> S((S)) ==> S(( ))
==> (S)(( )) ==>( )(( ))
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? l S ( )

7. Comments Fix a string and a grammar Example Unique mappings( )
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 ==> SS ==> (S)S ==>( )S ==> ( )(S)
==> ( )((S)) ==> ( )(( ))
S ==> SS ==> (S)S ==> (S)(S) ==> ( )(S)
S ==> SS ==> S(S) ==> S((S)) ==> S(( ))
==> (S)(( )) ==>( )(( ))

8. Example *
S ==> SS ==> SSS ==> (S)SS ==> ( )SS ==> ( )S ==> ( )
The above is a leftmost derivation of the string ( ) from the grammar BALG
Draw the corresponding parse tree
Draw the corresponding rightmost derivation
S ==> (S) ==> (SS) ==> (S(S)) ==> (S( )) ==> (( ))
The above is a rightmost derivation of the string (( )) from the grammar BALG
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 ==> SS ==> SSS ==> SSa ==> Saa ==> aaa
S ==> SS ==> Sa ==> SSa ==> Saa ==> aaa

11. 2 Simple Examples Grammar BALG is ambiguous
String ( ) in L(BALG) has more than 1 leftmost derivation
S ==> (S) ==> ( )
S ==> (S) ==> (SS) ==>(S) ==>( )
Give another leftmost derivation of ( ) from BALG
Grammar ABG is NOT ambiguous
Consider any string x in {aibi | i > 0}
There is a unique parse tree for x

12. Legal Arithmetic Expressions
Develop a grammar MATHG = (V, Σ, S, P) for the language of legal arithmetic expressions
S = {0, 1, +, *, -, /, (, )}
Strings in the language include 10
10*
10*( )
Strings not in the language include
10+ )(

13. Grammar MATHG1 V = {E, N} Σ = {0, 1, +, *, -, /, (, )} S = E P:
E → N | E+E | E*E | E/E | E-E | (E)
N → N0 | N1 | 0 | 1

14. MATHG1 is ambiguous
E → N | E+E | E*E | E/E | E-E | (E) N → N0 | N1 | 0 | 1
Come up with two distinct leftmost derivations of the string 11+0*11
E ==> E+E ==> N+E ==> N1+E ==> 11+E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==> 11+0*11
E ==> E*E ==> E+E*E ==> N+E*E ==> N1+E*E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==>11+0*11
Draw the corresponding parse trees

15. Corresponding Parse Trees
E ==> E+E ==> N+E ==> N1+E ==> 11+E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==> 11+0*11
E ==> E*E ==> E+E*E ==> N+E*E ==> N1+E*E ==> 11+E*E ==> 11+N*E ==> 11+0*E ==> 11+0*N ==> 11+0*N1 ==>11+0*11 E E * E E + + E N 1 N 1 E * N 1 N N N 1

16. Parse Tree Meanings E + * N 1
Note how the parse trees captures the semantic meaning of string 11+0*11.
More specifically, what number does the first parse tree represent?
What number does the second parse tree represent?

17. Implications Two interpretations of string 11+0*11
11+(0*11) = 11
(11+0)*11 = 1001
What if a line in a program is
MSU_Tuition = 11+0*11;
What is MSU_Tuition?
Depends on how the expression 11+0*11 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)
Σ = {D<85, D>50, grade=3.5, grade=3.0, if, then, else}
S = S P:
S → if BOOL then S else S | if BOOL then S |grade=3.5 | grade=3.0
BOOL → D<85 | D>50

19. ITEG is ambiguous
S → if BOOL then S |grade=3.5 | grade=3.0 | if BOOL then S else S BOOL → D<85 | D>50
Come up with two distinct leftmost derivations of the string
if D<85 then if D>50 then grade=3.5 else grade=3.0
S ==>if BOOL then S else S ==> if D<85 then S else S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0
S ==>if BOOL then S ==> if D<85 then S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0
Draw the corresponding parse trees

20. Corresponding Parse Trees
S ==>if BOOL then S else S ==> if D<85 then S else S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0
S ==>if BOOL then S ==> if D<85 then S ==> if D<85 then if BOOL then S else S ==> if D<85 then if D>50 then S else S ==> if D<85 then if D>50 then grade=3.5 else S ==> if D<85 then if D>50 then grade=3.5 else grade=3.0 S S if B
then S
else if B
then S
D<85 if B
then S
grade=3.0
D<85
else S if B
then
D>50
grade=3.5
D>50
grade=3.5
grade=3.0

21. Parse Tree Meanings S S if B
then S if B
then S
else S
D<85 if B
then S
else S
D<85 if B
then S
grade=3.0
D>50
grade=3.5
grade=3.0
D>50
grade=3.5
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 D<85 then if D>50 then grade=3.5 else grade=3.0
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-then’s as well as if-then-else’s

23. Inherently ambiguous CFL’s
A CFL L is inherently ambiguous iff for all CFG’s G such that L(G) = L, G is ambiguous
Examples so far
None of the CFL’s we’ve seen so far are inherently ambiguous
While the CFG’s we’ve seen ambiguous, there do exist unambiguous CFG’s for those CFL’s.
There exist inherently ambiguous CFL’s
Example: {aibjck | i=j or j=k or i=j=k}
Note i=j=k 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 CFL’s
Thus, the above heuristics do not always work