Context Free Languages, Pumping Lemma

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In summary, the question is whether a subset of a language being not context-free implies that the language itself is not context-free. The example given is to show that L, a language with an equal number of a's, b's, and c's, is not context-free by showing that the subset T, containing strings of the form a^n b^n c^n, is not context-free. The final question asks whether the language of all strings is context-free.
  • #1
dopeyranger
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Hello fellow mathematicians/computer-scientists!

I have a question:

If a subset of a language is not context free, does that mean the language itself is not context-free?

For example, I want to show that the following is not context free, using the pumping lemma:

L = {[itex]\omega[/itex] [itex]\in[/itex] {a,b,c}* | [itex]\omega[/itex] has an equal # of a's, b's, and c's}

And since T ={[itex]a^{n}b^{n}c^{n}[/itex] | n [itex]\geq[/itex] 0} [itex]\subset[/itex] L

If I show that T is not context free, does that show that L is not context free?
 
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  • #2
The language of all strings is context-free (even regular), right? ...
 

1. What are context-free languages?

Context-free languages are a type of formal language in computer science that can be generated by a context-free grammar. They are used to describe patterns in strings of characters, such as the syntax of a programming language.

2. What is the Pumping Lemma?

The Pumping Lemma is a theorem used to prove that a language is not context-free. It states that for any context-free language, there exists a pumping length, where all strings of that length or longer can be divided into smaller parts that can be repeated an infinite number of times while still remaining in the language.

3. How is the Pumping Lemma used?

The Pumping Lemma is used as a tool to prove that a language is not context-free. If it is possible to divide a string into smaller parts and pump them to create a string that is not in the language, then the language is not context-free.

4. What is the importance of the Pumping Lemma?

The Pumping Lemma is important in understanding the limitations of context-free languages and in proving the non-context-freeness of certain languages. It is also used in the development and analysis of programming languages and compilers.

5. Are there any exceptions to the Pumping Lemma?

Yes, there are some languages that satisfy the conditions of the Pumping Lemma but are still context-free. These are known as non-regular languages and require more advanced techniques for their analysis.

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