Context-free grammar (CFG)

 To create a context-free grammar (CFG) for the language represented by the regular expression 

$a(a+b)^*$, you need to define a set of production rules that generate all strings of the form where:

• The string starts with an 'a'.
• Followed by zero or more occurrences of 'a' or 'b'.

Here’s a CFG for this language:

1. Variables (Non-terminals):

   • $S$ (start symbol)
   • $A$ (an auxiliary non-terminal)

2. Terminals:

   • $a$
   • $b$

3. Production rules:

   • $S \rightarrow aA$
   • $A \rightarrow aA \mid bA \mid \epsilon$

4. Start symbol:

   • $S$

Explanation:

• The production rule $S \rightarrow aA$ ensures that every string generated by the grammar starts with an 'a'.
• The non-terminal $A$ is used to generate the rest of the string. It can produce any combination of 'a' and 'b', or nothing ($\epsilon$), representing the zero or more occurrences of 'a' and 'b'.

In more detail:

• $A \rightarrow aA$ allows the generation of strings with additional 'a's.
• $A \rightarrow bA$ allows the generation of strings with additional 'b's.
• $A \rightarrow \epsilon$ allows the generation of the empty string after the initial 'a'.

This CFG correctly generates all strings where the first character is 'a' and the remainder of the string consists of any combination of 'a's and 'b's.