The document discusses Chomsky normal form and Greibach normal form, which are simplified forms of context-free grammars. It defines Chomsky normal form as having rules that are either of the form A->BC or A->a, where A, B, C are nonterminals and a is a terminal. It states that any context-free language can be generated by a context-free grammar in Chomsky normal form by applying a series of transformations. An example is provided to demonstrate the conversion of a grammar into Chomsky normal form.

1. Chomsky and Greibach Normal
Forms
Chomsky and Greibach Normal Forms – p.1/24

2. Simplifying a CFG
It is often convenient to simplify CFG
Chomsky and Greibach Normal Forms – p.2/24

3. Simplifying a CFG
It is often convenient to simplify CFG
One of the simplest and most useful simplified forms of
CFG is called the Chomsky normal form
Chomsky and Greibach Normal Forms – p.2/24

4. Simplifying a CFG
It is often convenient to simplify CFG
One of the simplest and most useful simplified forms of
CFG is called the Chomsky normal form
Another normal form usually used in algebraic
specifications is Greibach normal form
Chomsky and Greibach Normal Forms – p.2/24

5. Simplifying a CFG
It is often convenient to simplify CFG
One of the simplest and most useful simplified forms of
CFG is called the Chomsky normal form
Another normal form usually used in algebraic
specifications is Greibach normal form
Note the difference between grammar cleaning and simplification
Chomsky and Greibach Normal Forms – p.2/24

6. Note
Normal forms are useful when more advanced topics in
computation theory are approached, as we shall see further
Chomsky and Greibach Normal Forms – p.3/24

7. Definition
A context-free grammar is in Chomsky normal form if
every rule is of the form:
where
is a terminal,
are nonterminals, and
may not be the start variable (the axiom)
Chomsky and Greibach Normal Forms – p.4/24

8. Note
The rule
, where is the start variable, is not ex-
cluded from a CFG in Chomsky normal form.
Chomsky and Greibach Normal Forms – p.5/24

9. Theorem 2.9
Any context-free language is generated by a context-free
grammar in Chomsky normal form.
Chomsky and Greibach Normal Forms – p.6/24

10. Theorem 2.9
Any context-free language is generated by a context-free
grammar in Chomsky normal form.
Proof idea:
Chomsky and Greibach Normal Forms – p.6/24

11. Theorem 2.9
Any context-free language is generated by a context-free
grammar in Chomsky normal form.
Proof idea:
Show that any CFG can be converted into a CFG
in
Chomsky normal form
Chomsky and Greibach Normal Forms – p.6/24

12. Theorem 2.9
Any context-free language is generated by a context-free
grammar in Chomsky normal form.
Proof idea:
Show that any CFG can be converted into a CFG
in
Chomsky normal form
Conversion procedure has several stages where the rules that
violate Chomsky normal form conditions are replaced with
equivalent rules that satisfy these conditions
Chomsky and Greibach Normal Forms – p.6/24

13. Theorem 2.9
Any context-free language is generated by a context-free
grammar in Chomsky normal form.
Proof idea:
Show that any CFG can be converted into a CFG
in
Chomsky normal form
Conversion procedure has several stages where the rules that
violate Chomsky normal form conditions are replaced with
equivalent rules that satisfy these conditions
Order of transformations: (1) add a new start variable, (2)
eliminate all -rules, (3) eliminate unit-rules, (4) convert other rules
Chomsky and Greibach Normal Forms – p.6/24

14. Theorem 2.9
Any context-free language is generated by a context-free
grammar in Chomsky normal form.
Proof idea:
Show that any CFG can be converted into a CFG
in
Chomsky normal form
Conversion procedure has several stages where the rules that
violate Chomsky normal form conditions are replaced with
equivalent rules that satisfy these conditions
Order of transformations: (1) add a new start variable, (2)
eliminate all -rules, (3) eliminate unit-rules, (4) convert other rules
Check that the obtained CFG
defines the same language
Chomsky and Greibach Normal Forms – p.6/24

15. Proof
Let
be the original CFG.
Chomsky and Greibach Normal Forms – p.7/24

16. Proof
Let
be the original CFG.
Step 1: add a new start symbol
to
, and the rule
to
Chomsky and Greibach Normal Forms – p.7/24

17. Proof
Let
be the original CFG.
Step 1: add a new start symbol
to
, and the rule
to
Note: this change guarantees that the start symbol of
does not occur
on the
of any rule
Chomsky and Greibach Normal Forms – p.7/24

18. Step 2: eliminate -rules
Repeat
1. Eliminate the rule
from
where
is not the start symbol
2. For each occurrence of
on the
of a rule, add a new rule to
with that occurrence of
deleted
Example: replace
by
;
replace
by
3. Replace the rule
, (if it is present) by
unless the
rule
has been previously eliminated
until all
rules are eliminated
Chomsky and Greibach Normal Forms – p.8/24

19. Step 3: remove unit rules
Repeat
1. Remove a unit rule
2. For each rule
, add the rule
to
, unless
was a unit rule previously removed
until all unit rules are eliminated
Note:
is a string of variables and terminals
Chomsky and Greibach Normal Forms – p.9/24

20. Convert all remaining rules
Repeat
1. Replace a rule
,
, where each
,
,
is a variable or a terminal, by:
,
,
,

21. where
,
,
,

22. are new variables
2. If
replace any terminal
with a new variable
and add the
rule
until no rules of the form
with
remain
Chomsky and Greibach Normal Forms – p.10/24

23. Convert all remaining rules
Repeat
1. Replace a rule
,
, where each
,
,
is a variable or a terminal, by:
,
,
,

24. where
,
,
,

25. are new variables
2. If
replace any terminal
with a new variable
and add the
rule
until no rules of the form
with
remain
Chomsky and Greibach Normal Forms – p.10/24

26. Convert all remaining rules
Repeat
1. Replace a rule
,
, where each
,
,
is a variable or a terminal, by:
,
,
,

27. where
,
,
,

28. are new variables
2. If
replace any terminal
with a new variable
and add the
rule
until no rules of the form
with
remain
Chomsky and Greibach Normal Forms – p.10/24

29. Example CFG conversion
Consider the grammar
whose rules are:
Notation: symbols removed are green and those added are red.
After first step of transformation we get:
Chomsky and Greibach Normal Forms – p.11/24

30. Example CFG conversion
Consider the grammar
whose rules are:
Notation: symbols removed are green and those added are red.
After first step of transformation we get:
Chomsky and Greibach Normal Forms – p.11/24

31. Example CFG conversion
Consider the grammar
whose rules are:
Notation: symbols removed are green and those added are red.
After first step of transformation we get:
Chomsky and Greibach Normal Forms – p.11/24

32. Removing rules
Removing
:
Removing
:

33. Chomsky and Greibach Normal Forms – p.12/24

34. Removing rules
Removing
:
Removing
:

35. Chomsky and Greibach Normal Forms – p.12/24

36. Removing rules
Removing
:
Removing
:

37. Chomsky and Greibach Normal Forms – p.12/24

38. Removing unit rule
Removing
:
Removing
:

39. Chomsky and Greibach Normal Forms – p.13/24

40. Removing unit rule
Removing
:
Removing
:

41. Chomsky and Greibach Normal Forms – p.13/24

42. Removing unit rule
Removing
:
Removing
:

43. Chomsky and Greibach Normal Forms – p.13/24

44. More unit rules
Removing
:
Removing
:
Chomsky and Greibach Normal Forms – p.14/24

45. More unit rules
Removing
:
Removing
:
Chomsky and Greibach Normal Forms – p.14/24

46. More unit rules
Removing
:
Removing
:
Chomsky and Greibach Normal Forms – p.14/24

47. Converting remaining rules
Chomsky and Greibach Normal Forms – p.15/24

48. Converting remaining rules
Chomsky and Greibach Normal Forms – p.15/24

49. Converting remaining rules
Chomsky and Greibach Normal Forms – p.15/24

50. Note
The conversion procedure produces several
variables
along with several rules
.
Since all these represent the same rule, we
may simplify the result using a single variable
and a single rule
Chomsky and Greibach Normal Forms – p.16/24

51. Note
The conversion procedure produces several
variables
along with several rules
.
Since all these represent the same rule, we
may simplify the result using a single variable
and a single rule
Chomsky and Greibach Normal Forms – p.16/24

52. Note
The conversion procedure produces several
variables
along with several rules
.
Since all these represent the same rule, we
may simplify the result using a single variable
and a single rule
Chomsky and Greibach Normal Forms – p.16/24

53. Greibach Normal Form
A context-free grammar
is in
Greibach normal form if each rule
has the
property:
,
,
and
.
Note: Greibach normal form provides a justifica-
tion of operator prefix-notation usually employed
in algebra.
Chomsky and Greibach Normal Forms – p.17/24

54. Greibach Normal Form
A context-free grammar
is in
Greibach normal form if each rule
has the
property:
,
,
and
.
Note: Greibach normal form provides a justifica-
tion of operator prefix-notation usually employed
in algebra.
Chomsky and Greibach Normal Forms – p.17/24

55. Greibach Normal Form
A context-free grammar
is in
Greibach normal form if each rule
has the
property:
,
,
and
.
Note: Greibach normal form provides a justifica-
tion of operator prefix-notation usually employed
in algebra.
Chomsky and Greibach Normal Forms – p.17/24

56. Greibach Theorem
Every CFL where
can be generated by a
CFG in Greibach normal form.
Proof idea: Let
be a CFG generating
. Assume that
is in Chomsky normal form
Let

57. be an ordering of nonterminals.
Construct the Greibach normal form from Chomsky normal form
Chomsky and Greibach Normal Forms – p.18/24

58. Greibach Theorem
Every CFL where
can be generated by a
CFG in Greibach normal form.
Proof idea: Let
be a CFG generating
. Assume that
is in Chomsky normal form
Let

59. be an ordering of nonterminals.
Construct the Greibach normal form from Chomsky normal form
Chomsky and Greibach Normal Forms – p.18/24

60. Greibach Theorem
Every CFL where
can be generated by a
CFG in Greibach normal form.
Proof idea: Let
be a CFG generating
. Assume that
is in Chomsky normal form
Let

61. be an ordering of nonterminals.
Construct the Greibach normal form from Chomsky normal form
Chomsky and Greibach Normal Forms – p.18/24

62. Construction
1. Modify the rules in
so that if
then
2. Starting with
and proceeding to

63. this is done as follows:
(a) Assume that productions have been modified so that for
,
only if
(b) If
is a production with
, generate a new set of
productions substituting for the
the rhs of each
production
(c) Repeating (b) at most
times we obtain rules of the form
,
(d) Replace rules
by removing left-recursive rules
Chomsky and Greibach Normal Forms – p.19/24

64. Construction
1. Modify the rules in
so that if
then
2. Starting with
and proceeding to

65. this is done as follows:
(a) Assume that productions have been modified so that for
,
only if
(b) If
is a production with
, generate a new set of
productions substituting for the
the rhs of each
production
(c) Repeating (b) at most
times we obtain rules of the form
,
(d) Replace rules
by removing left-recursive rules
Chomsky and Greibach Normal Forms – p.19/24

66. Construction
1. Modify the rules in
so that if
then
2. Starting with
and proceeding to

67. this is done as follows:
(a) Assume that productions have been modified so that for
,
only if
(b) If
is a production with
, generate a new set of
productions substituting for the
the rhs of each
production
(c) Repeating (b) at most
times we obtain rules of the form
,
(d) Replace rules
by removing left-recursive rules
Chomsky and Greibach Normal Forms – p.19/24

68. Removing left-recursion
Left-recursion can be eliminated by the following
scheme:
If
are all
left recursive rules, and
are all remaining
-rules then chose a new
nonterminal, say
Add the new
-rules
,
Replace the
-rules by
,
This construction preserve the language .
Chomsky and Greibach Normal Forms – p.20/24

69. Removing left-recursion
Left-recursion can be eliminated by the following
scheme:
If
are all
left recursive rules, and
are all remaining
-rules then chose a new
nonterminal, say
Add the new
-rules
,
Replace the
-rules by
,
This construction preserve the language .
Chomsky and Greibach Normal Forms – p.20/24

70. Removing left-recursion
Left-recursion can be eliminated by the following
scheme:
If
are all
left recursive rules, and
are all remaining
-rules then chose a new
nonterminal, say
Add the new
-rules
,
Replace the
-rules by
,
This construction preserve the language .
Chomsky and Greibach Normal Forms – p.20/24

71. 21-1

72. More on Greibach NF
See Introduction to Automata Theory, Languages,
and Computation, J.E, Hopcroft and J.D Ullman,
Addison-Wesley 1979, p. 94–96
Chomsky and Greibach Normal Forms – p.22/24

73. Example
Convert the CFG
where
into Greibach normal form.
Chomsky and Greibach Normal Forms – p.23/24

74. Example
Convert the CFG
where
into Greibach normal form.
Chomsky and Greibach Normal Forms – p.23/24

75. Example
Convert the CFG
where
into Greibach normal form.
Chomsky and Greibach Normal Forms – p.23/24

76. Solution
1. Step 1: ordering the rules: (Only
rules violate ordering
conditions, hence only
rules need to be changed).
Following the procedure we replace
rules by:
2. Eliminating left-recursion we get:
,
3. All
rules start with a terminal. We use them to replace
. This introduces the rules
4. Use
production to make them start with a terminal
Chomsky and Greibach Normal Forms – p.24/24

77. Solution
1. Step 1: ordering the rules: (Only
rules violate ordering
conditions, hence only
rules need to be changed).
Following the procedure we replace
rules by:
2. Eliminating left-recursion we get:
,
3. All
rules start with a terminal. We use them to replace
. This introduces the rules
4. Use
production to make them start with a terminal
Chomsky and Greibach Normal Forms – p.24/24

78. Solution
1. Step 1: ordering the rules: (Only
rules violate ordering
conditions, hence only
rules need to be changed).
Following the procedure we replace
rules by:
2. Eliminating left-recursion we get:
,
3. All
rules start with a terminal. We use them to replace
. This introduces the rules
4. Use
production to make them start with a terminal
Chomsky and Greibach Normal Forms – p.24/24