1. Context-Free Languages
Simplifications of
Context-Free Languages

2. A Substitution Rule
equivalent
grammar
Substitute

3. A Substitution Rule
Substitute
equivalent
grammar

4. In general:
Substitute
equivalent
grammar

5. Nullable Variables
Nullable Variable:

6. Removing Nullable Variables
example Grammar:
Nullable variable

7. Final Grammar
Substitute

8. Unit-Productions
Unit Production:
(a single variable in both sides)

9. Removing Unit Productions
Observation:
Is removed immediately

10. example Grammar:

11. Substitute

12. Remove

13. Substitute

14. Remove repeated productions
Final grammar

15. Useless Productions Useless Production
Some derivations never terminate...

16. Another grammar:
Useless Production
Not reachable from S

17. then variable is useful
In general:
contains only
terminals if
then variable is useful
otherwise, variable is useless

18. A production is useless
if any of its variables is useless
Productions
useless
Variables
useless

19. Removing Useless Productions
example Grammar:

20. First: find all variables that can produce strings with only terminals
Round 1:
Round 2:

21. Keep only the variables that produce terminal symbols:
(other variables are useless)
Remove useless productions

22. Second: Find all variables reachable from Use a Dependency Graph not

23. Keep only the variables reachable from S
(other variables are useless)
Final Grammar
Remove useless productions

24. Removing All
Step 1: Remove Nullable Variables
Step 2: Remove Unit-Productions
Step 3: Remove Useless Variables

25. Normal Forms for Context-free Grammars

26. Chomsky Normal Form each productions has form: or variable variable
terminal

27. examples:
Chomsky
Normal Form
Not Chomsky
Normal Form

28. Conversion to Chomsky Normal Form
example:
Not Chomsky
Normal Form

29. Introduce variables for terminals:

30. Introduce intermediate variable:

31. Introduce intermediate variable:

32. Final grammar in Chomsky Normal Form:
Initial grammar

33. In general:
From any context-free grammar
(which doesn’t produce )
not in Chomsky Normal Form
we can obtain:
An equivalent grammar
in Chomsky Normal Form

34. The Procedure
First remove:
Nullable variables
Unit productions

35. Then, for every symbol :
Add production
In productions: replace with
New variable:

36. Replace any production
with
New intermediate variables:

37. Theorem: For any context-free grammar (which doesn’t produce )
there is an equivalent grammar
in Chomsky Normal Form

38. Observations Chomsky normal forms are good
for parsing and proving theorems
It is very easy to find the Chomsky normal
form for any context-free grammar

39. exercise Find CNF for this grammar: S -> 0A0 | 1B1 | BB A -> C
B -> S | A
C -> S | epsilon
(exercise 7.1.3, Hopcroft, Motwani, Ullman)

40. Greibach Normal Form
All productions have form:
symbol
variables

41. examples:
Greibach
Normal Form
Not Greibach
Normal Form

42. Conversion to Greibach Normal Form:

43. Theorem: For any context-free grammar (which doesn’t produce )
there is an equivalent grammar
in Greibach Normal Form

44. Observations Greibach normal forms are very good for parsing
For many context-free grammars,
it's hard to find the Greibach normal form.

45. Why? Try to find Greibach normal form for grammar in CNF exercise ...