Final answer:
The question involves eliminating all ε-productions from the given CFG by identifying nullable variables and modifying the production rules so that the grammar generates the same language without ε-productions.
Step-by-step explanation:
Eliminating ε-productions in CFG
The question asks how to eliminate all ε-productions from the given context-free grammar (CFG) G. To eliminate ε-productions, we identify nullable variables (variables that derive the empty string directly or indirectly) and modify the production rules accordingly. In the given grammar, variable B has an ε-production. We remove the ε-production and then modify other productions that include variable B to ensure that the grammar remains consistent and generates the same language.
In this CFG, we can proceed as follows:
Remove the production B → ε
For every occurrence of B in other productions, we create new productions: one with the variable B, and one without it. This handles the cases where B could have derived ε.
For example, in production S → AB, we would create a new production S → A to account for the possibility of B deriving ε.
By applying this process, we would have a modified CFG without the ε-productions, while retaining the language that the original grammar generated.