Find regular grammars for the following languages on {a, b}:(a) L = {w : na(w) is even, nb(w) ≥ 4}

To find a regular grammar for the language L = {w : na(w) is even, nb(w) ≥ 4}, we need to define a set of production rules that generate all strings in the language and only those strings.

Here is one possible regular grammar:

1. S -> aA | bB | bS | ε
2. A -> aS | bB | bA
3. B -> aA | bC | bB
4. C -> aA | bD | bC
5. D -> aA | bD

This grammar works as follows:

• The start symbol S can produce an 'a' and transition to state A, or produce a 'b' and transition to state B or stay in state S, or produce an empty string (ε). This allows for strings with an even number of 'a's and at least 4 'b's.
• State A can produce an 'a' and transition back to state S, or produce a 'b' and transition to state B or stay in state A. This allows for strings with an even number of 'a's and at least 4 'b's.
• State B can produce an 'a' and transition to state A, or produce a 'b' and transition to state C or stay in state B. This allows for strings with an even number of 'a's and at least 4 'b's.
• State C can produce an 'a' and transition to state A, or produce a 'b' and transition to state D or stay in state C. This allows for strings with an even number of 'a's and at least 4 'b's.
• State D can produce an 'a' and transition to state A, or produce a 'b' and stay in state D. This allows for strings with an even number of 'a's and at least 4 'b's.

This grammar generates all strings in the language L and only those strings.