Finite Automata - Prof. Meghalay, Assignments of Theory of Automata

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Unit I: Finite Automata

Finite Automata: (05 hrs)

● Automaton as a model of computation,

● Symbols,

● Alphabets,

● Strings,

● Languages,

● Deterministic Finite Automata (DFA),

● Nondeterministic finite Automata (NFA),

● State Minimization algorithm,

● NFA with epsilon transition,

● pumping lemma

Symbols

Symbols are called as a characters. it is the smallest building block,

which can be any alphabet, letter, or picture.

or

Symbol is an atomic unit, such as a digit, character, lowercase letter, etc. For example, a,b,c,……………z 0,1,2,………….. +,-,*,%,………special characters($,#,@).

Alphabet

The set of characters is called as an alphabet.

An alphabet is a finite, non-empty set of symbols.

It is denoted by Σ.

For Example,

Σ ={0,1} set of binary alphabets.

Σ ={a,b,c,……..,z} set of all lower case letters.

Σ ={A,B,C,………Z} set of all upper case letters.

Σ ={+,&,%,……….} set of all special characters.

String For example, If Σ={a,b,c,d} is set of alphabet then string w=aabbcabcd and |w|= If, w = 0110 y = baa x = aabcaa z = 111. Then, Concatenation − wz = 0110111 Special string −s= { 𝟄 } Length − |w| = 4 , |s| = 0, |x| = 6, |wz| = Reversal − yR^ = aab

Find Number of Strings (of length 2) that can be

generated over the alphabet Σ= {a, b}:

a a a b b a b b Length of String |w| = 2 Number of Strings = 4

For alphabet Σ= {a, b} with length n, number of

strings can be generated = 2 n .

Language **A language is a set of strings from some alphabet (finite or infinite). For example, If alphabet Σ={a,b}then

1. Language over these alphabets is L={a,b,aa,ab,ba,bb,aaa,bbb,abb,aab,baa,bba,aba,bab,aaaa,bb bb,aabb,bbaa……………..} (infinite)
2. Language over these alphabets is having length 3 is L={aaa,bbb,abb,aab,baa,bba,aba,bab} Finite**

Language Some special sets of strings are as follows − {} The empty set/language, containing no string, also called as ɸ Phi. { 𝟄 } A language containing one string, the empty string. Σ*^ All strings of symbols from Σ including 𝟄 Σ

= Σ* - { 𝟄 } or Σ*=Σ

+{ 𝟄 } For example, Σ = {0, 1} Σ* = { 𝟄 , 0, 1, 00, 01, 10, 11, 000, 001,...} Σ+ = {0, 1, 00, 01, 10, 11, 000, 001,.}

Set,Subsets,Superset A set is a collection of objects or elements, grouped in the curly braces. Ex. A= {a,b,c,d} , B={ 1,2,3,4,5} Subsets are a part of one of the mathematical concepts called Sets. Ex. A={2,4,6,8,10,12,14,16,18,20} B={2,4,6} B ⊆A If a set A is a collection of even number and set B consists of {2,4,6}, then B is said to be a subset of A, denoted by B ⊆A and A is the superset of B.

All Subsets of a Set The subsets of any set consists of all possible sets including its elements and the null set. Let us understand with the help of an example. Example: Find all the subsets of set A = {1,2,3,4} Solution: Given, A = {1,2,3,4} Subsets = {} {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3},{2,4}, {3,4}, {1,2,3}, {2,3,4}, {1,3,4}, {1,2,4} {1,2,3,4}

Proper Subsets Set A is considered to be a proper subset of Set B if Set B contains at least one element that is not present in Set A. Example: If set A has elements as {12, 24} and set B has elements as {12, 24, 36}, then set A is the proper subset of B because 36 is not present in the set A. Proper Subset Symbol A proper subset is denoted by ⊂ and is read as ‘is a proper subset of’. Using this symbol, we can express a proper subset for set A and set B as; A ⊂B

Example 1: Given A = {1, 2, 4} and B = {1, 2, 3, 4, 5}, what is the relationship between these sets? We say that A is a subset of B , since every element of A is also in B. This is denoted by: A Venn diagram for the relationship between these sets is shown to the right. A ⊂B

Proper Subset Subset Example : Conclude whether X is a subset of Y. X = {All writing material in a stationary workshop}, Y = {Pencils} Set X involves pens, sketch pens, markers, pencils, notepads, etc. Whereas set Y only carries pencils. So we cannot state that all X’s elements are present in Y, which is a requirement for X to be a subset of Y. In this particular case, we can state that Y is a subset of X, but X is not a subset of Y. Subset Example : Determine whether A is a subset of B. A = {Toyota}, B = {All brands of cars} Set B covers all brands of cars; Maruti Suzuki, Hyundai, Toyota Mahindra, Tata Motors, Mercedes Benz, etc. Moreover, A is a set of Toyota. Then we can say that all elements of A are incorporated into B. Hence, A is a subset of B.

How many subsets and proper subsets does a set have? If a set has “n” elements, then the number of subset of the given set is 2n and the number of proper subsets of the given subset is given by 2n-1. Consider an example, If set A has the elements, A = {a, b}, then the proper subset of the given subset are { }, {a}, and {b}. Here, the number of elements in the set is 2. We know that the formula to calculate the number of proper subsets is 2n – 1. = 2 2 – 1 = 4 – 1 = 3 Thus, the number of proper subset for the given set is 3 ({ }, {a}, {b}).