Question

Give context-free grammars generating each of the following languages over Σ={0,1} :

a) {0ⁿ1ᵐ0ⁿ :m,n≥1}

Answer 1

To generate a context-free grammar for the language {0ⁿ1ᵐ0ⁿ}, use rules such as S → 0S0 | AB, A → 0A | ε, and B → 1B | ε. An example of the language 0²1³0² would be produced by following these steps: S → 0S0 → 00S00 → 001S100 → 0011S1100 → 00111B11100 → 001110B111100 → 0011100.

Step-by-step explanation:

In order to generate a context-free grammar for the language {0ⁿ1ᵐ0ⁿ : m,n ≥ 1}, we need to create rules that specify the structure of the language. One approach is to split the language into three parts: the beginning, middle, and end.

1. Beginning: This part consists of one or more 0's.
2. Middle: This part consists of one or more 1's.
3. End: This part consists of the same number of 0's as the beginning.

To generate the context-free grammar, we can use the following rules:

• S → 0S0 | AB
• A → 0A | ε
• B → 1B | ε

For example, if n=2 and m=3, the language would be '0011100', which can be generated using the grammar rules by following these steps:

1. S → 0S0 → 00S00 → 001S100 → 0011S1100 → 00111B11100 → 001110B111100 → 0011100