1. Courtesy Costas Busch - RPI
Grammars
Courtesy Costas Busch - RPI

2. Courtesy Costas Busch - RPI
Grammars
Grammars express languages
Example: the English language
Courtesy Costas Busch - RPI

3. Courtesy Costas Busch - RPI

4. Courtesy Costas Busch - RPI
A derivation of “the dog walks”:
Courtesy Costas Busch - RPI

5. Courtesy Costas Busch - RPI
A derivation of “a cat runs”:
Courtesy Costas Busch - RPI

6. Courtesy Costas Busch - RPI
Language of the grammar:
L = { “a cat runs”,
“a cat walks”,
“the cat runs”,
“the cat walks”,
“a dog runs”,
“a dog walks”,
“the dog runs”,
“the dog walks” }
Courtesy Costas Busch - RPI

7. Courtesy Costas Busch - RPI
Notation
Production Rules
Variable
Terminal
Courtesy Costas Busch - RPI

8. Courtesy Costas Busch - RPI
Another Example
Grammar:
Derivation of sentence :
Courtesy Costas Busch - RPI

9. Courtesy Costas Busch - RPI
Grammar:
Derivation of sentence :
Courtesy Costas Busch - RPI

10. Courtesy Costas Busch - RPI
Other derivations:
Courtesy Costas Busch - RPI

11. Courtesy Costas Busch - RPI
Language of the grammar
Courtesy Costas Busch - RPI

12. Courtesy Costas Busch - RPI
More Notation
Grammar
Set of variables
Set of terminal symbols
Start variable
Set of Production rules
Courtesy Costas Busch - RPI

13. Courtesy Costas Busch - RPI
Example
Grammar :
Courtesy Costas Busch - RPI

14. Courtesy Costas Busch - RPI
More Notation
Sentential Form:
A sentence that contains
variables and terminals
Example:
Sentential Forms
sentence
Courtesy Costas Busch - RPI

15. Courtesy Costas Busch - RPI
We write:
Instead of:
Courtesy Costas Busch - RPI

16. Courtesy Costas Busch - RPI
In general we write:
If:
Courtesy Costas Busch - RPI

17. Courtesy Costas Busch - RPI
By default:
Courtesy Costas Busch - RPI

18. Courtesy Costas Busch - RPI
Example
Grammar
Derivations
Courtesy Costas Busch - RPI

19. Courtesy Costas Busch - RPI
Example
Grammar
Derivations
Courtesy Costas Busch - RPI

20. Another Grammar Example
Derivations:
Courtesy Costas Busch - RPI

21. Courtesy Costas Busch - RPI
More Derivations
Courtesy Costas Busch - RPI

22. Courtesy Costas Busch - RPI
Language of a Grammar
For a grammar
with start variable :
String of terminals
Courtesy Costas Busch - RPI

23. Courtesy Costas Busch - RPI
Example
For grammar :
Since:
Courtesy Costas Busch - RPI

24. Courtesy Costas Busch - RPI
A Convenient Notation
Courtesy Costas Busch - RPI

25. Courtesy Costas Busch - RPI
Linear Grammars
Courtesy Costas Busch - RPI

26. Courtesy Costas Busch - RPI
Linear Grammars
Grammars with
at most one variable at the right side
of a production
Examples:
Courtesy Costas Busch - RPI

27. Courtesy Costas Busch - RPI
A Non-Linear Grammar
Grammar :
Number of in string
Courtesy Costas Busch - RPI

28. Another Linear Grammar
Courtesy Costas Busch - RPI

29. Right-Linear Grammars
All productions have form:
Example: or
string of
terminals
Courtesy Costas Busch - RPI

30. Courtesy Costas Busch - RPI
Left-Linear Grammars
All productions have form:
Example: or
string of
terminals
Courtesy Costas Busch - RPI

31. Courtesy Costas Busch - RPI
Regular Grammars
Courtesy Costas Busch - RPI

32. Courtesy Costas Busch - RPI
Regular Grammars
A regular grammar is any
right-linear or left-linear grammar
Examples:
Courtesy Costas Busch - RPI

33. Courtesy Costas Busch - RPI
Observation
Regular grammars generate regular languages
Examples:
Courtesy Costas Busch - RPI

34. Regular Grammars Generate Regular Languages
Courtesy Costas Busch - RPI

35. Courtesy Costas Busch - RPI
Theorem
Languages
Generated by
Regular Grammars
Regular
Languages
Courtesy Costas Busch - RPI

36. Courtesy Costas Busch - RPI
Theorem - Part 1
Languages
Generated by
Regular Grammars
Regular
Languages
Any regular grammar generates
a regular language
Courtesy Costas Busch - RPI

37. Courtesy Costas Busch - RPI
Theorem - Part 2
Languages
Generated by
Regular Grammars
Regular
Languages
Any regular language is generated
by a regular grammar
Courtesy Costas Busch - RPI

38. Courtesy Costas Busch - RPI
Proof – Part 1
Languages
Generated by
Regular Grammars
Regular
Languages
The language generated by
any regular grammar is regular
Courtesy Costas Busch - RPI

39. The case of Right-Linear Grammars
Let be a right-linear grammar
We will prove: is regular
Proof idea: We will construct NFA
with
Courtesy Costas Busch - RPI

40. Courtesy Costas Busch - RPI
Grammar is right-linear
Example:
Courtesy Costas Busch - RPI

41. Courtesy Costas Busch - RPI
Construct NFA such that
every state is a grammar variable:
special
final state
Courtesy Costas Busch - RPI

42. Courtesy Costas Busch - RPI
Add edges for each production:
Courtesy Costas Busch - RPI

43. Courtesy Costas Busch - RPI

44. Courtesy Costas Busch - RPI

45. Courtesy Costas Busch - RPI

46. Courtesy Costas Busch - RPI

47. Courtesy Costas Busch - RPI

48. Courtesy Costas Busch - RPI
NFA
Grammar
Courtesy Costas Busch - RPI

49. Courtesy Costas Busch - RPI
In General
A right-linear grammar
has variables:
and productions: or
Courtesy Costas Busch - RPI

50. Courtesy Costas Busch - RPI
We construct the NFA such that:
each variable corresponds to a node:
special
final state
Courtesy Costas Busch - RPI

51. Courtesy Costas Busch - RPI
For each production:
we add transitions and intermediate nodes
………
Courtesy Costas Busch - RPI

52. Courtesy Costas Busch - RPI
For each production:
we add transitions and intermediate nodes
………
Courtesy Costas Busch - RPI

53. Courtesy Costas Busch - RPI
Resulting NFA looks like this:
It holds that:
Courtesy Costas Busch - RPI

54. The case of Left-Linear Grammars
Let be a left-linear grammar
We will prove: is regular
Proof idea:
We will construct a right-linear
grammar with
Courtesy Costas Busch - RPI

55. Courtesy Costas Busch - RPI
Since is left-linear grammar
the productions look like:
Courtesy Costas Busch - RPI

56. Courtesy Costas Busch - RPI
Construct right-linear grammar
Left
linear
Right
linear
Courtesy Costas Busch - RPI

57. Courtesy Costas Busch - RPI
Construct right-linear grammar
Left
linear
Right
linear
Courtesy Costas Busch - RPI

58. Courtesy Costas Busch - RPI
It is easy to see that:
Since is right-linear, we have:
Regular
Language
Regular
Language
Regular
Language
Courtesy Costas Busch - RPI

59. Courtesy Costas Busch - RPI
Proof - Part 2
Languages
Generated by
Regular Grammars
Regular
Languages
Any regular language is generated
by some regular grammar
Courtesy Costas Busch - RPI

60. Courtesy Costas Busch - RPI
Any regular language is generated
by some regular grammar
Proof idea:
Let be the NFA with
Construct from a regular grammar
such that
Courtesy Costas Busch - RPI

61. Courtesy Costas Busch - RPI
Since is regular
there is an NFA such that
Example:
Courtesy Costas Busch - RPI

62. Courtesy Costas Busch - RPI
Convert to a right-linear grammar
Courtesy Costas Busch - RPI

63. Courtesy Costas Busch - RPI

64. Courtesy Costas Busch - RPI

65. Courtesy Costas Busch - RPI

66. Courtesy Costas Busch - RPI
In General
For any transition:
Add production:
variable
terminal
variable
Courtesy Costas Busch - RPI

67. Courtesy Costas Busch - RPI
For any final state:
Add production:
Courtesy Costas Busch - RPI

68. Courtesy Costas Busch - RPI
Since is right-linear grammar
is also a regular grammar
with
Courtesy Costas Busch - RPI