#TheoryOfComputationpdf

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NFA To DFA Conversion In Theory Of Computation

NFA To DFA Conversion In Theory Of Computation Construct The DFA For The Following NFA

    DFA Using DFA Transition Table If All States Are Final The Minimal DFA Will Be

My Initial State Will Be My Final State And This Is The DFA.   Example-2

Conversion Of NFA To DFA

Find The Minimal No Of States In NFA

Solution:- Transition Table For Given Diagram

This Is One Of The Different Questions. To Solve This Question We Are Going To Use Short Cut. If You Are Expert In Constructing DFA, You Can Solve This Question Using Short Cut.   The Question Given Is "Find The Minimal No.Of States In NFA" Means We Have To Find The Minimum No Of States In Given NFA.   There Will Be No Uniqueness In NFA. If It Is Not Unique, How Can We Find Minimal States?  

Remember

If The Question Is Asked To Find Minimal No Of States, The Diagram Should Be Unique And Should Be Minimal(Optimized) We Know DFA Means Minimized, We Can Construct Unique DFA, We Can Construct A Unique DFA For A Language. It Means The No Of States In DFA wILL Be Unique.   "Every DFA Is Not NFA" We Know But Every NFA Is Not DFA   Simple We Have To Draw DFA For Given NFA We Already Drawn Many DFA From NFA's In Previous Topics.   We All Know That Procedure.   But Here To Solve This Problem Using Short Cut. If We Can Predict The Language For Given NFA, Then We Can Find DFA Easily   I Have Solved Many Problems Before In Previous Posts We Have Seen How To Construct DFA Directly In Previous Solved Problems.   I Want You To Identify The Language For The Given Nfa & For That NFA For That Language, You Have To Draw DFA. Very SImple.

If You Observe Above NFA 0,1 Is Going To q0 Then After 3 Zero's 000, It Is Going To Final State. Means String Should ContainThree Zero's(000) Continuously For Sure And Final State q3 Has Loop Of '0', It Means The String Should End With Three Zero's(000) String Compulsorily Should Contain 3 Zero's This Is The Language For This Given NFA.   DFA

  Explanation For DFA

If I Get '1' , I'll Be On q0 Only If I Get '0 On 'q0' I'll Go To 'q1' If I Get '0' On 'q1' I'll Go To 'q2' If I Get '0' On 'q2' I'll Go To 'q3'(FINAL STATE) Because I Should Reach Final State Once I Get 3 Zero's String If I '1' On 'q1','q2','q3' I'll Go To Initial State 'q0' And I'll Start Machine Again I Cant Go To Dead State If I Get '1' On q1,q2,q3 If I Go To Dead State I Can't Get Back And I Cant Start The Machine Again   Read the full article

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Theory Of Computation - Practice Questions On Language

If You Have the Ability To Think About A Problem These Problems Are Damn Eay For You, Lets Understand And Solve The Questions About Language In Theory Of Computation. Before Solving Questions I Hope You Understood About Language Which I Explained In Previous Post, If You Have Not Read, Click Here If You Know About Language Already, let's Solve The Questions On Language

  Q) Choose The Correct Statement From The Following. a) Every Regular Language Is Finite b) Every Non-Regular Language Is Finite c) Every Regular Language Is Infinite d) Every Non-Regular Language Is Infinite   Answer - d) Every Non-Regular Language Is Infinite Explanation - If You See The Language Hierarchy, Every Non-Regular Language Is Infinite

  Q) Which Of The Most Appropriate Answer For Finite Language. a) L Is Regular b) L Is CFL c) L Is CSL d) L Is REL   Answer -  L Is Regular   Explanation - Yes, L Is Regular Is The Most Appropriate Answer Because CFL, CSL, REL May Or May Not Be Appropriate.

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Theory Of Computation Practice Questions

Theory Of Computation Practice Questions Solved And Explained Briefly.  

Which Of The Following Statement Is Correct?

a) DFA Is More Efficient Than NFA b) NFA Is More Efficient Tham Than DFA c) DFA Is More Powerful Than NFA d) NFA Is More Powerful Than DFA   Answer Is a) Explanation 

E(NFA)=E(DFA) Expressive Power Of NFA Is Equal To DFA, The Languages Accepted By NFA Are Equal To The Languages Accepted By DFA. I Can Draw NFA For All RL. I Can Draw DFA For All RL.   Hence DFA Is More Efficient Because I'll Have Only One Router Or Way.  

2) Choose The Incorrect Statement From The Following.

a) DPDA Is More Efficient Than NPDA b) Capabilities Of DPDA & NPDA Are Same c) NPDA Is More Powerful Than DPDA. d) NOTA   Answer - b Explanation

E(NPDA)≥E(DPDA)  

3) For Which Of The Language An Automata Can Be Constructed In Both Deterministic And Non-Deterministic Mode To Accept That Language.

a) Regular b) CFL c) REL d) NOTA   Answer - a) & c)   Explanation a)E(NFA)=E(DFA)

b) E(NPDA) ≥ E(DPDA)

c) E(NTM)=E(DTM)

d) ×  

4) Choose Correct Statement From The Following

a) DFA Is More Powerful Than DPDA b) DPDA Is More Powerful Than DFA c) NFA Is More Powerful Than NPDA D) NOTA   Answer - b)   Explanation

a) FA Read the full article

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Power Of Alphabets In Automata

Power Of Alphabets (∑) In Automata, If ∈ Is An Alphabet Then ∑k Is The Set Of All The String From The Alphabet ∈ Of Length Exactly k. Example - ∑= {a,b}  // a & b are input alphabets We Are Talking About ∑k  (Sigma To The Power Of k)   ∑ 1= Set Of All The String Of Length 1. ={a,b} // Formed String   ∑ 2= Set Of All The String Of Length 2. ={aa,ab,ba,bb} // Formed Strings   ∑ 3= Set Of All The String Of Length 3. ={aa,ab,ba,bb,aab,aba,baa,bba} // Formed Strings   ∑ 0= Set Of All The String Of Length 0 (Empty Set) ={∈} // Empty Set Or Epsilon     Positive Closure Of  ∑ ∑+=∑1 ∪ ∑2∪ ∑3 ∪ ∑4...... Kleen Closure Of  ∑ ∑*= ∑0 ∪Σ1 ∪ ∑2∪ ∑3 ∪ ∑4...... // Kleen Closure Of  ∑ Is A ∪niversal Language  

To Remember The Above, Remember The Following Rules

(a)  ∑+ ⊂ ∑*   // Sigma To The Power Of Plus Is The Subset Of SigmaTo The Power Of Star (b) ∑* = ∑+ ∪ {∈} // Sigma To The Power Of Star Is Equal To Sigma To The Power Of Plus ∪nion Epsilon ( Empty Set) (c) ∑+ = ∑*- {∈} (d) ∑* ∪ ∑+ = ∑* (e) ∑* ∩ ∑+ = ∑+ (f) ∑* ∑+ = ∑+ (g) ∑* ∑* = ∑* (h) ∑+ ∑+ = ∑+ - ∑1             Read the full article

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Theory Of Computation Introduction

Theory Of Computation Is Also Known As Core Subject Of Computer Science. If You Are A Logical Thinker, If You Love To Learn Mathematics, This Subject Is Very Easy For You.   Let's Understand The Importance Of Theory Of Computation(TOC) In Computer Science And Basic Terminologies Used In Theory Of Computation.  

First We Have To Know What Is Mean By Symbol ?

You'll Be Seeing Lot Of Symbols In Your Day-to-Day Life, If You Observe The Keyboard You'll Be Seeing A-Z Alphabets And 0-9 Numbers. And You'll Be Seeing Special Characters, All These Are Considered As Symbols. Every Symbol On The Keyboard Has Specific Meaning. Symbols - A,B,C 0,1,2 @,#,$(,)

  What Is Alphabet?

We Represent Alphabets With " ∈ ", You May Have Seen This Symbol In Mathematics While Doing SUM. Here We Will Represent Alphabets With " ∈ " The Meaning Of " ∈ " Is A Particular Domain's Symbol. Alphabet As The Set Of Symbols Example - {a, b}, Which Are Always Finite  

What Is String?

String Is A Combination Of Alphabet, Any Number Of Times Is Called As String. Example - ∈ ={a,b}   // Converting Alphabet To String Set Of = {aa,ba,bb,aba,bba....}  // This Is String  

What Is Language?

A Language Is A Combination Of Strings On Given Certain Conditions.   Example - ∈ = {a,b} // Input Alphabet Using The Above Alphabet, You Have To Create String With The Length Of 2. L1=Set Of All The Strings Of Length 2 = {aa,ab,bb,ba} This Set Is Called As Finite Set, Any Set In Which We Can Count Elements Is Called Finite Set Because We Have Finite Number Of Elements In The Set. Using The Language We Created This Finite Set, Where The Set Contains Countable Elements, This Language Is Called As Finite Language.   Example 2 ∈ = {a,b} L2= Set Of All Strings Starts With ' a ' = { a,aab,aab,aba,abba,abab,.......} This Set Is Called As InFinite Set, Any Set In Which We Can Not Count Elements Is Called InFinite Set Because We Have InFinite Number Of Elements In The Set. Using The Language We Created This InFinite Set, Where The Set Contains UnCountable Elements, This Language Is Called As InFinite Language.   In This Post, We Have Understood About Symbol, Alphabet, String, Language.I Hope You Understood If You Have Any Doubts Comment Below.           Read the full article

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Theory Of Computation

Theory Of Computation Is Also Known As Core Subject Of Computer Science.If You Are A Logical Thinker, If You Love To Learn Mathematics, This Subject Is Very Easy For You. If You Have Puzzle Solving Ability This Subject Is Very Very Easy For You. In This Subject We Will Understand About Algorithm, Computation, Software & Hardware And The Purpose Of All These. Do You Know The Limitations Of Computer? Do You Think The Computer Can Solve Any Problem? Yes, There Are Problems Which Cannot Be Solved By Computer.  

Let's Understand The Purpose Of Theory Of Computation

I Want To Create A Mathematical Model For My Computations, Which Reflects From My Computer. What Is Computation? The Action Of Mathematical Calculation. The Task Which Can Be Performed By An Electronic Device, Like Calculator, Computer Etc. For Example - If I Want To Add 1+3, Using The Help Of Calculator I Can Solve This Or I Can Solve This Using My Computer. But, If I Want To Perform This Task Using Electronic Device, For Computation, I Have To Create Steps, I Have To Create Algorithm, So That I Can Solve This Problem.   If We Go Into The Past Years, In 1930's The Mathematicians, Logicians When They Were Trying To Understand The Meaning Of an A " Computation", A Central Question Asked Was Can Whether All Mathematical Problems Can Be Solved In Systematic Way. Then They Talked About Software And Hard Ware And Algorithms Etc. " Computation " Is The Word Which Came Out From Their Brain's. In Theory Of Computation, There Are Different Theories Came And Went Back In Their Brain.   In This Subject We Will Understand About These These Theories Under Theory Of Computation (i) Complexity Theory (ii) Computability Theory (iii) Automata Theory

Complexity Theory -

You Already Know About Complexity Theory, In Your Other Subjects. In Complexity Theory, I'll Categorize Problems Into Two Parts. If The Problem Is Easy, The Problem Will Be Called As Easy. If The Problem Is Hard, The Problem Will Be Called As Hard. But How We Can Categorize A "Problem" Is "Easy" Or " Hard", How Can We Say "This Problem Is Easy And This Problem Is Hard?" If The Problem Is Easy To Solve It Is Easy,If The Problem Is Hard To Solve It Is Called As Hard.This Will Not Happen In Computer Science. Easy For Example - If I Want To Search A Phone Number From A Phone Number Directory Which Has 'n' Number Of Phone Numbers.I'll Put A Searching Algorithm And I'll Get That Phone Number I Need.I'll Tell You The That Searching Algorithm Complexity And You Already Have Idea About Searching Algorithm's. One More Example - Imagine That I Have 1Lakh Numbers And I Want To Arrange That Numbers In To Ascending And Descending Order. And I'll Put A Sorting Algorithm And I'll It Into Ascending And Descending Order. Hard Example - If I Want To Create Time Table Lecturers And Subjects For Different Time, It Is Little Hard, The Problem Will Be Solved For Sure But I Cant Know The Exact Time,Those Type Problems Comes Under Hard Category.   In Complexity Theory I Have To Divide Problems In To Easy And Hard.  

 Computability Theory -

In Computability Theory I Should Divide Problem Into Two Categories.1) Solvable 2) Unsolvable The Problem Which I Can Solve By Showing A Mathematical Proof Comes Under 'Solvable' , The Problem Which I Cant Solve Using A Algorithm Or Any Of My Mathematical Methods Comes Under Unsolvable, Until Someone Solves It. For Example - Imagine I Have Problem Which I Cant Solve Using The Algorithms And Methods I Have Today In My Machine, The Problem May Be Solved After 50 Years But Today For Me It Is Unsolvable And The Problem Comes Under The List Called ' Unsolvable '  

Automata Theory

In Automata Theory I Should I Learn About Different Mathematical Models.Finite Automata,Context Free Grammer, Turning Machine Are Few Methods And There Are Lot More Methods In Automata Theory. These Three Are Mathematical Model Which Represents Computations.   Fnite Autometa,Context Free Grammer,Turing Machine Here I Have To Know Which Method Is Powerful Which Can Solve More Problems. If I Talk About Finite Automata, Finite Automata Has Some Restrictions,Where I Cannot Solve Multiple Problems.Some Problems Can Be Solved And Some Finite Autometa Cannot Solve. Turing Machine Can Solve Problem Which Can Be Solved By The Finite Automata And It Can Solve The Problem Which Cannot Be Solved By Finite Automata.   And Here I Have To Know "How A Problem Can Be Categorized" And I Should Know "Which Model Solves That Problem. We Will Understand Indetail About These Models And Categorization.   Basically Theory Of Computation My Main Focus Will Be On Automata Theory, Then We Will Discuss About The Computational Theory.       To Understand This Subject You Need To Have Logical Thinking,One Problem Will Not Depend On Other Problem, Each And Every Problem Is Unique In This Subject.                         Read the full article