Question

Give context-free grammars that generate the following languages. In all parts, the alphabet \Sigma is {0,1}.

a. w
b.w

Answer 1

Context-free grammars for generating languages.

Step-by-step explanation:

a. To generate the language w starts and ends with the same symbol, we can define a context-free grammar as follows: S -> 0S0 | 1S1 | 0 | 1. This grammar has the start symbol S and four production rules.

The rule S -> 0S0 generates strings that start and end with '0', S -> 1S1 generates strings that start and end with '1', and the rules S -> 0 and S -> 1 generate strings consisting of a single '0' and '1' respectively.

b. To generate the language w, we can define a context-free grammar as follows: S -> 0S0 | 1S1 | 0 | 1 | epsilon. The rule S -> epsilon allows for the possibility of an empty string.

The other rules generate palindromic strings by recursively adding a '0' or '1' before and after the non-terminal symbol S.