APPLICATIONS OF PUMPING LEMMA Pumping Lemma is to be applied to show that certain languages are not regular. It should never be used to show a language is regular.  If L is regular, it satisfies Pumping Lemma.  If L does not satisfy Pumping Lemma, it is non- regular.

Method to prove that a language L is not regular

1. At first, we have to assume that L is regular.
2. So, the pumping lemma should hold for L.
3. Use the pumping lemma to obtain a contradiction − Select w such that |w| ≥ c. Select y such that |y| ≥ 1. Select x such that |xy| ≤ c. Assign the remaining string to z.

The significance of the pumping lemma is that its contrapositive gives us a way to prove that certain languages are not regular. It is a fundamental law of logic that if a theorem is true then its contrapositive is automatically true as well.

1. The Pumping Lemma is used for proving that a language is not regular. Here is the Pumping Lemma.
2. Let L = {0k1k : k ∈ N}. We prove that L is not regular.
3. Let L = {(10)p1q : p, q ∈ N, p ≥ q}. We prove that L is not regular.
4. There are 3 cases to consider: (a) v starts with 0 and ends with 0.

Which of the following is not an application of Pumping Lemma? Explanation: None of the mentioned are regular language and are an application to the technique Pumping Lemma. Each one of the mentioned can be proved non regular using the steps in Pumping lemma.

3. Which of the following is an application of Finite Automaton? Explanation: There are many applications of finite automata, mainly in the field of Compiler Design and Parsers and Search Engines.

3. Which of the following relates to Chomsky hierarchy? Explanation: All the regular languages are the subset to context free languages and thus can be accepted using push down automata. 5.

To prove a language is regular: construct a DFA, NFA or RE that recognizes it. To prove a language is not regular: show that recognizing it requires keeping track of infinite state (hard to be completely convincing in most cases) or use the pumping lemma to get a contradiction.

(Kleene’s Theorem) A language is regular if and only if it can be obtained from finite languages by applying the three operations union, concatenation, repetition a finite number of times. And it is an infinite language. Thus, by Kleene’s Theorem it cannot be a regular language.

Palindromes are not regular: It is easy to prove it using pumping lemma. Let us assume a language L which is palindrome so L can be expressed as x. xR where xR is reverse of x. If x is a regular language then xR will be too and hence the palindrome will be regular.

Well, the alphabet /Sigma is finite, and therefore regular, and the star operation preserves regularity (by the definition of regular languages).

The Kleene closure is defined to only have finite strings. There are an infinite number of such strings, just as there are an infinite number of integers. A string from the Kleene closure can not contain every character from an infinte alphabet.

Σ* (Sigma Star) Σ* is the language that consists of all possible strings over the symbol set Σ .

Given Σ, then the Kleene Star Closure of the alphabet Σ, denoted by Σ*, is the collection of all strings defined over Σ, including Λ. Plus Operation is same as Kleene Star Closure except that it does not generate Λ (null string), automatically. You can use other symbol for alphabet but we are mostly use sigma symbol.

The special empty string denoted ϵ can be a string over any alphabet. The length of a string is a number of symbols in it; the length of ϵ is 0. • A set of all strings over a given alphabet Σ is denoted Σ∗ (“sigma star”).

TOC can be used to identify the constraint (process of significant impact) while statistical tools of Six Sigma can be used to quantitatively measure and analysis process performance. In other words TOC will enable an organization to identify where to judiciously launch a Six Sigma based improvement project.

Now, by definition, Σ∗ is the set of all finite strings that can be written using the characters of Σ. This always includes the empty string ϵ and, as long as Σ≠∅, it also contains strings of all finite lengths.

Can a string be of infinite length? In an infinite language, there is no limit to the length of a string, but the length of each string is finite.

Epsilon in math, represented by the Greek letter “E,” is a positive infinitesimal quantity. ∑ is the Greek capital sigma symbol. Used commonly in algebraic functions, you may also notice it in Excel – the AutoSum button has a sigma as its icon.

The empty set is a language which has no strings. The set { } is a language which has one string, namely . For any alphabet , the set of all strings over (including the empty string) is denoted by . Thus a language over alphabet is a subset of .

Empty Set: The empty set (or null set) is a set that has no members. Note: {∅} does not symbolize the empty set; it represents a set that contains an empty set as an element and hence has a cardinality of one. Equal Sets. Two sets are equal, if they have exactly the same elements.

The empty string is the identity element of the concatenation operation. The set of all strings forms a free monoid with respect to ⋅ and ε. εR = ε. Reversal of the empty string produces the empty string.

The set containing one empty string has one element. The empty set has zero elements. The one with one element is “bigger” (its cardinality is larger).

The Java programming language distinguishes between null and empty strings. An empty string is a string instance of zero length, whereas a null string has no value at all. An empty string is represented as “” . It is a character sequence of zero characters.

null is an empty variable – there’s literally nothing there.