1. Left Recursion
Lecture 7
Fri, Feb 4, 2005

2. A Problem with Recursive Descent Parsers
Suppose the grammar were
S  A B | C D
A  B A | C A
B  C A | A D
C  B A | A D | a
D  A C | B D | b
How could a top-down parser decide which production for S to use?

3. Another Problem with Recursive Descent Parsers
Suppose the grammar were
S  S S | a
How could the parser decide how many times to use the production S  S S before using the production S  a?

4. Futile Attempt #1
void S() {
if (token == a)
match(a);
else
S(); }

5. Futile Attempt #2
void S() {
if (token != EOF)
S(); }

6. Left Recursion
The method of recursive descent does not work if the grammar is left recursive.
A grammar is left recursive if there is a derivation
A + A
for some nonterminal A and string .
In particular, a production is left recursive if it is of the form
A  A.

7. Left Recursion
Applying the method of recursive descent would lead to the function
void A() {
A();
// Process  }
which leads to infinite recursion.

8. Left Recursion
Recall that in the earlier example, we added the production
S'  S S' | ,
not the production
S'  S' S | 
Why?
Are they equivalent as far as the language of the grammar is concerned?

9. Eliminating Left Recursion
Left recursion in a production may be removed by transforming the grammar in the following way.
Replace
A  A | 
with
A  A'
A'  A' | .

10. Eliminating Left Recursion
Under the original productions, a derivation of  is
A  A  A  A  .
Under the new productions, a derivation of  is
A  A'  A'  A'
 A'  .

11. Example: Eliminating Left Recursion
Consider the left recursive grammar
E  E + T | T
T  T * F | F
F  (E) | id
Apply the transformation to E:
E  T E'
E'  + T E' | .

12. Example: Eliminating Left Recursion
Then apply the transformation to T:
T  F T'
T'  * F T' | .

13. Example: Eliminating Left Recursion
Now the grammar is
E  T E'
E'  + T E' | .
T  F T'
T'  * F T' | .
F  (E) | id

14. Example: Eliminating Left Recursion
The function for E' would be
void Eprime() {
if (token == PLUS)
match(PLUS);
T();
Eprime(); }
return;

15. Advantages of Left Recursion
A left recursive grammar is often more intuitive than the transformed grammar.
A left recursive grammar will match expressions earlier, leading to shallow recursion.
Consider parsing a + b + c + d + e.
Bottom-up parsing takes advantage of the benefits of left recursion.

16. Example Consider the simple grammar E  E + E | num Convert it to
E  num E'
E'  + E E' | 
Example: SimpleParser

17. Exercise The grammar R  R  R | RR | R* | (R) | a | b
generates all regular expressions on the alphabet {a, b}.
Rewrite the grammar to reflect the precedence rules.
Eliminate left recursion.