# Theory of Automata

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(CS402)
Theory of Automata
Lecture N0. 35
Introduction to Computer Theory
Chapter 13
Summary
Examples of building TG's corresponding to the Regular Grammar, Null productions with examples, Nullable
productions with examples, Unit production with example, Chomsky Normal Form (Definition)
Example
Consider the following CFG
S Æ aA|bB
A Æ aS|a
B Æ bS|b
A
then the corresponding TG will be
a
a
a
+
S-
b
b
b
B
+
The corresponding RE may be (aa+bb) .
Example
Consider the following CFG
S Æ aaS|bbS|abX|baX|L
X Æ aaX|bbX|abS|baS,
aa,bb
aa,bb
then the corresponding TG will be
ab,ba
L
X
+
S-
ab,ba
The corresponding language is EVEN-EVEN.
Null Production
Definition
The production of the form nonterminal Æ L is said to be null production.
Example: Consider the CFG, S Æ aA|bB|L, A Æ aa|L, B Æ aS
Here S Æ L and A Æ L are null productions.
Note
If a CFG has a null production, then it is possible to construct another CFG without null production accepting
the same language with the exception of the word L i.e. if the language contains the word L then the new
language cannot have the word L.
Following is a method to construct a CFG without null production for a given CFG
Method
Delete all the Null productions and add new productions e.g.
consider the productions of a certain CFG  X Æ aNbNa, N Æ L, delete the production N Æ L and using the
production X Æ aNbNa, add the new productions X Æ aNba, X Æ abNa and X Æ aba
Thus the new CFG will contain the productions X Æ aNba|abNa|aba|aNbNa
Note: It is to be noted that X Æ aNbNa will still be included in the new CFG.
Nullable Production
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Definition
A production is called nullable production if it is of the form N Æ L
or
there is a derivation that starts at N and leads to L i.e. N1 Æ N2, N2 Æ N3, N3 Æ N4, ..., Nn ÆL, where N, N1, N2,
..., Nn are non terminals.
Example
Consider the following CFG
S Æ AA|bB, A Æ aa|B, B Æ aS | L
Here S Æ AA and A Æ B are nullable productions, while B Æ L is null a production.
Following is an example describing the method to convert the given CFG containing null productions and
nullable productions into the one without null productions
Example
Consider the following CFG
S Æ XaY|YY|aX|ZYX
X Æ Za|bZ|ZZ|Yb
Y Æ Ya|XY|L
Z Æ aX|YYY
It is to be noted that in the given CFG, the productions S Æ YY, X Æ ZZ, Z Æ YYY are Nullable productions,
while Y Æ L is Null production.
Here the method of removing null productions, as discussed earlier, will be used along with replacing
nonterminals corresponding to nullable productions like nonterminals for null productions are replaced.
Thus the required CFG will be
S ÆXaY|Xa|aY|a|YY|Y|aX|ZYX|YX|ZX|ZY
X Æ Za|a|bZ|b|ZZ|Z|Yb
Y Æ Ya|a|XY|X|Y
Z Æ aX|a|YYY|YY|Y
Example
Consider the following CFG
S Æ XY, X Æ Zb, Y Æ bW
Z Æ AB, W Æ Z, A Æ aA|bA|L
B ÆBa|Bb|L.
Here A Æ L and B Æ L are null productions, while Z Æ AB, W Æ Z are nullable productions. The new CFG
after, applying the method, will be
S Æ XY
X Æ Zb|b
Y Æ bW|b
Z Æ AB|A|B
WÆZ
A Æ aA|a|bA|b
B ÆBa|a|Bb|b
Note
While adding new productions all Nullable productions should be handled with care. All Nullable productions
will be used to add new productions, but only the Null production will be deleted.
Unit production
The productions of the form nonterminal Æ one nonterminal, is called the unit production.
Following is an example showing how to eliminate the unit productions from a given CFG.
Example
Consider the following CFG
S Æ A|bb
A Æ B|b
B Æ S|a
Separate the unit productions from the nonunit productions as shown below
105 Theory of Automata
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unit prods.
nonunit prods.
SÆA
S Æ bb
AÆB
AÆb
BÆS
BÆa
S Æ A gives S Æ b
(using A Æ b)
S Æ A Æ B gives S Æ a
(using B Æ a)
A Æ B gives A Æ a
(using B Æ a)
A Æ B Æ S gives A Æ bb
(using S Æ bb)
B Æ S gives B Æ bb
(using S Æ bb)
B Æ S Æ A gives B Æ b
(using A Æ b)
Thus the new CFG will be
S Æ a|b|bb, A Æ a|b|bb, B Æ a|b|bb.
Which generates the finite language {a,b,bb}.
Chomsky Normal Form
If a CFG has only productions of the form
nonterminal Æ string of two nonterminals
or
nonterminal Æ one terminal
then the CFG is said to be in Chomsky Normal Form (CNF).
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