S → X2A2 | ABa
A → AC
C → A
B → X1B1 | bb
B1 → bB
A2 → a
Step-by-step explanation:
To convert the grammar G to Chomsky normal form (CNF), we need to eliminate productions with more than two non-terminals on the right-hand side and handle productions with terminals.
1. Convert A-rules to CNF:
- The production A → AA can be replaced by introducing a new non-terminal. Let's say we introduce a new non-terminal C. So, A → AC can be added, and then C → A can be added to handle the remaining production C → A.
- Now the A-rules are in CNF: A → AC and C → A.
2. Convert B-rules to CNF:
- The production B → AbB can be split into two productions: B → X1B1 and B1 → bB, where X1 is a new non-terminal. This breaks down the production into two non-terminals, each with a single terminal or non-terminal.
- The production B → bb already satisfies CNF as it has two terminals.
3. Convert S-rules to CNF:
- The production S → aA can be rewritten as S → X2A2, where X2 is a new non-terminal, and A2 → a.
Now the grammar G in CNF is:
S → X2A2 | ABa
A → AC
C → A
B → X1B1 | bb
B1 → bB
A2 → a
The grammar G is now in Chomsky normal form, where all productions have either two non-terminals or a single terminal.