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Prerita Agarwal

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23 Jul, 2024 @ 01:30 PM

Introduction

In the world of language theory, we explore how computers understand and process words and sentences. In this world, we come across a powerful tool called the Pumping Lemma for Context Free Languages(CFL). This tool is like a magic key that helps us unlock the secrets of languages and grammar in computer science.

In this article, we will discuss about Pumping Lemma for Context Free Language(CFL). We will also discuss the theorem and steps to apply the pumping lemma.

There are two Pumping Lemmas

Regular languages

Context-free languages

Pumping Lemma for Regular Languages?

The Pumping Lemma for Regular Languages is a fundamental theorem in formal language theory that helps demonstrate certain properties of regular languages.

Pumping lemma for regular languages provides a way to show that a language is not regular, which is a significant insight when analyzing the complexity of languages in computer science.

For any regular language L, there exists a pumping length p (a positive integer) such that any string s in L with a length of at least p can be divided into three parts, xyz, where:

For any non-negative integer i, the string xy^iz is also in L

The length of the string |xy| is at most p

The string y is not empty (|y| > 0)

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Pumping Lemma for Context Free Languages

Context-free grammar generates context-free languages (CFLs). The set of all context-free languages is the same as the set of languages accepted by the pushdown Automata. The set of regular languages is a subset of context-free languages.

Pumping Lemma for CFL states that for any Context-Free Language L, it is possible to find two substrings that can be 'pumped' any number of times and still be in the same language. We break its strings into five parts for any language L and pump the second and fourth substrings. If any string does not satisfy its conditions, then the language is not CFL.

If L is a context-free language, there is a constant 'n' that depends exclusively on L, such that if w? L and |w| >= n, w can be divided into five pieces, w = uvxyz, meeting the following requirements.

|vxy| >=n

|vy| # ε

For all k>=0, the string uv^{k}xy^{y}z∈L

Steps to apply pumping lemma

Assume that L is context-free.

The pumping length will be n.

All strings longer than the length(n) can be pumped-> |w|>=n.

Now we have to find a string 'w' in L such that |w|>=n.

We will divide string 'w' into uvxyz.

Now show that uv^{k}xy^{k}z ∉L for some constant k.

Then, we have to consider the ways that w can be divided into uvxyz.

Show that none of these can satisfy all the 3 pumping conditions simultaneously.

A string 'w' cannot be pumped (contradiction).

Example

Find out whether L={x^{n}yn^{z}n|n>=1} is context-free or not

Let L be context-free.

'L' must be satisfying the pumping length, say n.

Now we can take a string such that s=x^{n}y^{n}z^{n}

We divide s into 5 strings uvxyz.

Let n=4 so, s=x^{4}y^{4}z^{4}

Case 1

v and y each contain only one type of symbol.

{we are considering only v and y because v and y has power uv2xy2z}

X xx x yyyyz z zz

=uv^{k}xy^{k}z when k=2

=uv^{2}xy^{2}z

=xxxxxxyyyyzzzzz

=x^{6}y^{4}z^{5}

(Number of x # number of y #number of z)

Therefore, The resultant string is not satisfying the condition

x^{6}y^{4}z^{5} ∉ L

If one case fails, there is no need to check another condition.

Case 2

Either v or y has more than one kind of symbols

Xx xx yy y y zzzz

=uv^{k}xy^{k}z (k=2)

=uv^{2}xy^{2}z

=xxxxyyxxyyyyyzzzz

=x^{4}y^{2}x^{2}y^{5}z^{2}

This string is not following the pattern of our string x^{n}y^{n}z^{n}

Pumping Lemma is also used to prove whether the language is regular or not.

The term Pumping Lemma is made up of two words: Pumping: The word pumping refers to repeatedly generating many input strings by pushing a symbol in an input string. Lemma: The word Lemma refers to the intermediate theorem in a proof.

What is context-free language?

Context-free grammar generates context-free languages (CFLs). The set of all context-free languages is identical to the set of languages accepted by pushdown automata, and the set of regular languages is a subset of context-free languages.

What is the difference between language and context-free language?

A language encompasses any set of strings composed of specific symbols without strict rules. In contrast, a context-free language is a specific category governed by context-free grammars. These grammars generate strings adhering to well-defined rules, and context-free languages are recognized by pushdown automata.

Why is it called context free language?

The term "context-free" comes from the rules governing these languages, which do not consider the context or positioning of symbols. In simpler terms, context-free languages are named so because they're generated and recognized solely based on structured rules, regardless of the symbols' surrounding context.

Conclusion

In this blog, we have talked about the following things:

What do you mean by pumping lemma?

How is the pumping lemma used for context-free languages?

Along with some examples, prove whether the language is context-free or not.