Final answer:
The CFG for the language of strings that are not of the form ww over the alphabet {a, b}, ensures asymmetry to avoid generating ww strings. It includes rules that pair each letter with both possible letters in the middle, including rules for the empty string and single letters.
Step-by-step explanation:
The question is asking to provide a Context-Free Grammar (CFG) for the language of strings over the alphabet \{a, b\} that are not of the form ww, where w is any string over that alphabet. Designing such a CFG is challenging because typical CFGs describe what is in a language, not what is excluded from it. Therefore, one approach can be to create rules that generate strings that are guaranteed not to be of the form ww by ensuring some asymmetry in the strings produced.
Here is one possible CFG for this language:
- S -> aSa | bSb | aSb | bSa | a | b | ε
This CFG produces strings where an a in the middle of the string is matched with either an a or a b on the opposite side, and similarly for b. This prevents the production of strings of the form ww. The inclusion of a,b, and ε (the empty string) ensures that all singleton strings and the empty string, which cannot be of the form ww, are included in the language.