Question

Automata Theory

1. Give a context-free grammar producing the following language over

Σ = {0, 1}:

{w : every odd position of w is 1 and w = w^R}

(HINT: All strings in the language will be of odd length).

Answer 1

TO UNDERSTAND THE QUESTION :
Lets see for some of the strings which are  accepted or not, by language over

Σ = {0, 1}: {w : every odd position of w is 1 and w = wR}
(YES : Accepted )
(NO: Not accepted)

YES | w: 1 (w = wR), (1st position is odd positiion, every odd position of w is 1)
NO | w: 10 (all the strings of the language will be of odd length)
YES | w : 101 (even position can be 0 or 1)
YES | w: 111 ( even position can be 0 or 1)

and so on...

So, for all strings w in language given,

1. ALL strings should be of odd length.
2. ALL strings start with 1 (because starting position of any string 'w' is one which is odd position).
3. ALL strings end with 1 (because 'w' starts with 1 and string has to be palindrome (w=wR) ).
4. EVEN position of string can have only 1 or only 0.

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So Context Free Grammar (CFG) for such a language is ,

S --> RPR | R
P --> TST | RSR | T | R
T --> 0
R --> 1

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Where,

• {S,P,T,R} belongs to NON TERMINALS / VARIABLES
• {S} is Start Symbol
• {1,0} belongs to terminals.

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