To convert the given context-free grammar into Greibach Normal Form (GNF), we need to follow a set of rules. GNF is a specific form of a context-free grammar where each production has the form A → aα, where 'a' is a terminal symbol, and α is a possibly empty string of variables (non-terminals).
The given grammar is:
```
S → AB
A → BS | b
B → SA | a
```
Let's convert this grammar into GNF:
**Step 1:** Eliminate ε-productions (productions that derive the empty string ε).
There are no ε-productions in the given grammar, so no changes are needed for this step.
**Step 2:** Eliminate unit productions (productions of the form A → B).
In the given grammar, we have unit productions:
```
A → BS
B → SA
```
We can replace these unit productions with their respective right-hand sides, recursively if necessary. After this step, we get:
```
A → bS | b
B → aS | a
```
**Step 3:** Remove left-recursion.
In the given grammar, we have indirect left-recursion through A and B. To remove left-recursion, we'll introduce new non-terminals and rewrite the productions.
The modified grammar becomes:
```
S → AB
A → bSA' | b
A' → bSA' | b
B → aSB' | a
B' → aSB' | a
```
**Step 4:** Remove productions with more than one non-terminal on the right-hand side.
The modified grammar has productions with more than one non-terminal on the right-hand side:
```
A' → bSA' | b
B' → aSB' | a
```
We can convert these productions into GNF by introducing new non-terminals. Here's the final GNF:
```
S → AB
A → bX | b
X → SA'
A' → bX | b
B → aY | a
Y → SB'
B' → aY | a
```
Now, every production in this grammar has the form A → aα, where 'a' is a terminal symbol, and α is a possibly empty string of variables (non-terminals), which is the characteristic of Greibach Normal Form (GNF).
