Final answer:
The provided grammar is ambiguous since a string like '00' can be derived in multiple ways. A refactored unambiguous grammar provides unique derivation paths for each string, preserving the language generation while eliminating ambiguity.
Step-by-step explanation:
The grammar provided is ambiguous because there can be more than one leftmost derivation for a certain string. For example, using the given grammar rules, the string 00 can be derived in two different ways. It can be derived from S → 0A → 00AA → 00 and it can also be derived from S → 0A → 00A → 00, which makes the grammar ambiguous as there isn't a unique way to derive the string. An example of an unambiguous grammar that generates the same language would be to refactor the production rules to ensure a unique parsing tree for each string.
Example of Unambiguous Grammar:
- S → 0X | 1Y
- X → 0XX | 1Z
- Z → 0X | 1S
- Y → 1YY | 0Z
This refactoring maintains the language generation capability while ensuring that each string derived has a unique derivation path. Notice that we introduce new nonterminals X, Y, and Z to disambiguate choices between appending strings created by A and B. The choice point is now at X and Y, and Z ensures that the correct sequence is followed when a substring of the original A or B starts with a different numeric prefix.