178k views
2 votes
suppose for a language and a string , where (the pumping length), there exists one decomposition of for which . what can we then conclude about ? select all that apply. group of answer choices is not context free. is context free. is regular. is not regular. none of the other choices is a valid conclusion about .

User Rusev
by
7.7k points

1 Answer

1 vote

The correct answer is 1 and 2:

L is not context-free.

L is not regular.

How to explain

If we have a language L and a string w where |w| >= p, and there exists a decomposition w = uvxyz such that |uv| <= p, |vy| > 0, and for all i >= 0, uvixy^iz is not in L, then we can conclude:

1. L is not context-free:

The pumping lemma for context-free languages states that for any string in a context-free language longer than the pumping length, we can "pump" a substring within the string and still get a valid string in the language. The existence of a decomposition where pumping results in a string not in the language contradicts this property.

2. L is not regular:

Regular languages are a subset of context-free languages, and any property true for context-free languages also applies to regular languages. Therefore, if L is not context-free, it cannot be regular.

3. None of the other choices is a valid conclusion about L:

Since we have already established that L is not context-free and not regular, the remaining options (L is regular or L is context-free) are not valid conclusions based on the given information.

Therefore, the correct answer is 1 and 2:

L is not context-free.

L is not regular.

User Brazil
by
7.7k points