Theory of Computation- Chomsky Normal Form & Leftmost Derivation - 1CNF - 2CNF - 3CNF - BCNF and Examples

1. CHOMSKY NORMAL FORM
(CNF)
&
LEFTMOST DERIVATION (LMD)
M A D E B Y :
D I M P Y C H U G H ( 1 8 3 3 )
D R I S H T I B H A L L A ( 1 8 3 8 )

2. DEFINITION
If a Context Free Grammar or CFG has only productions of the form
Nonterminal -> string of exactly two non-terminals
or
Nonterminal -> one terminal
then, it is said to be in Chomsky Normal Form, CNF.
Example: S → AS | a
A → SA | b

3. STEPSFOR CONVERSION
• The conversion to Chomsky Normal Form has four main steps:
1. Get rid of all λ-Productions .
2. Eliminate Non-terminal Unit Productions.
A → B , B → b
A → b
3. Elimination of more than one terminals by a non- terminal in a production.
4. Replace every production that is too long by shorter productions.
A → BCD
A → BE
E → CD

4. THEOREM-1
STATEMENT- If L is a language generated by some CFG, then there is another CFG that
generates all the non- λ words of L, all of whose productions are of one of two basic forms:
Nonterminal-> string of only Non-terminals
or
Nonterminal -> one terminal
PROOF- Let us start with the CFG-
S -> X1 | X2aX2 | aSb | b
Xl -> X2X2 | b
X2 -> aX2 | aaX1
Now, in order to convert it into the above form , we introduce new productions :
A->a and B->b

5. After conversion we have,
S -> X1 X1 -> X2X2
S -> X2AX2 X1 -> B
S -> ASB X2 -> AX2
S -> B X2 -> AAX1
Thus, we have converted the original CFG into the desired form.

6. STATEMENT- For any context- free language L, the non- λ words of L can be generated by a
grammar in which all the productions are in CNF.
PROOF - Let our initial CFG is
S -> XI X2 X3 X8
S -> X3 X5
S -> b
Now, in order to get the given CFG in CNF form we’ll add certain new non-terminals , i.e.
S -> XI X2 X3 X8
S -> XI R1
R1 -> X2 X3 X8
R1 -> X2 R2
R2 -> X3 X8
THEOREM-2

7. Consider another CFG
X -> X1aX2bX3
Apply Theorem 1 and add productions A->a and B->b to get string of solid non-terminals.
Now,
X -> X1AX2BX3
X -> X1R1
R1 -> AX2BX3
R1 -> AR2
R2 -> X2BX3
R2 -> X2R3
R3 -> BX3
In this way, we have converted the long productions with long strings of non-terminals into sequences
of productions with exactly 2 non-terminals on RHS. Thus, the new grammar in CNF generates the
same language as the old grammar.

8. Ques16 i). The following CFG has unit productions. Convert the following CGF into CNF.
S -> X
X -> Y
Y-> Z
Z ->aa
Sol16). S -> X -> Y => S -> Y
S -> Y -> Z => S -> Z
S -> Z -> aa =>S ->aa
Since S->aa is not in CNF , we introduce a new production A->a to
transform it to CNF.
Thus, the new grammar in CNF form is :
A -> a
S -> AA
QUESTIONS

9. Ques14 v). Convert the following CGF into CNF.
S -> ABABAB
A -> a|λ
B -> b|λ
Sol13). Removing λ productions,
A -> a ---------1
B -> b ---------2
S -> ABABAB
Using Theorem 2,
S -> AR1 ---------3
R1 -> BABAB
R1 -> BR2 ---------4
R2 -> ABAB
R2 -> AR3 ---------5
R3 -> BAB
R3 -> BR4 ---------6
R4 -> AB ---------7
Thus, the new grammar in CNF form comprises of (1,2,3,4,5,6,7).

10. Ques14 iv). Convert the following CGF into CNF.
E -> E + E
E -> E*E
E -> (E)
E -> 7
The terminals here are +, *, (, ), 7.
Sol12). E -> 7 --------------1
Using Theorem 1, we add new productions that will help in converting CGF to CNF.
F -> + --------------2
G -> * --------------3
H-> ( --------------4
I -> ) --------------5
Now , we have
E -> EFE
E -> ER1 --------------6
R1 -> FE --------------7
E -> EGE
E -> ER2 --------------8
R2 -> GE --------------9
E -> HEI
E -> HR3 -------------10
R3 -> EI -------------11
Thus, the new grammar in CNF form comprises of (1,2,3,4,5,6,7,8,9,10,11).

11. LEFT MOST DERIVATIONS
• The leftmost non-terminal in a working string is the first non-terminal that we encounter
when we scan the string from left to right.
For example : In the string abNbaXYa , the leftmost nonterminal is N.
• If a word w is generated by a CFG by a certain derivation and at each step in the derivation, a
rule of production is applied to the leftmost non-terminal in the working string, then this
derivation is called a leftmost derivation.

12. Ques17 i)Find the leftmost derivation for the word “abba” in the grammar:
S -> AA
A -> aB
B -> bB | λ
Sol18)(i). The following is a leftmost derivation:
S -> AA
S -> aBA
S -> abBA
S -> abbBA
S -> abbλA
S -> abbaB
S -> abbaλ
S -> abba
QUESTIONS

13. Ques17 ii) Find the leftmost derivation for the word “abbabaabbbabbab” in the CFG:
S->SSS|aXb
X->ba|bba|abb
Sol18)(ii). The following is a leftmost derivation:
S -> SSS
S -> aXbSS
S -> abbabSS
S -> abbabaXbS
S -> abbabaabbbS
S -> abbabaabbbaXb
S -> abbabaabbbabbab

14. ADVANTAGES..
• CNF Grammar can be used to determine whether a string is a member of the
language associated with it or not .
• It can be used to find the no. of production steps required to generate a string. The
key advantage is that in Chomsky Normal Form, every derivation of a string of n
letters has exactly 2 n − 1 steps.
Example : S -> AX|AB
A -> a
X -> SB
B -> b
The string to be generated is “ aabb” having n(no. of letters in a string ) =4.
The following is a leftmost derivation:
1) S -> AX
2) S -> aX
3) S -> aSB
4) S -> aABB
5) S -> aaBB
6) S -> aabB
7) S -> aabb
Here, A string of n(n=4) letters has exactly 2n-1(2*4-1=7) steps .

15. THANK YOU !!!