sr1 toc intro handout

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CSE 211: Introduction to the Theory of Computation

Md. Shaifur Rahman

Dept. of CSEBangladesh University of Engineering & Technology

Week 1 Class 1

Shaifur (BUET) CSE211: TOC-Intro Week 1 Class 1 1 / 29



Outline

1 Necessity of CS Theory Courses

2 What is Theory of Computation?

3 Course Outline




Why Theory? (again!)

We don’t like theory, because:One has to memorize a lot of theorems, lemmas, proofs

It is inherently boring!

Where is the output?

How is it related to our career-path?We like non-theory courses,

We like programming in C, C++, JAVA, Python, Android etc.

The outcome is visible, projects done in these courses brighten our

resumeThe skills are necessary for the career-path of a developer

Achieving grades are comparatively easy :)




Theory vs. Non-theory

Difficult to separate theories from all courses!

Course Theory-bias

CSE 103 (Discrete Math) 90%

CSE 105 (Structured Programming Language) 30%

CSE 201 (Object Oriented Programming Language) 20%

CSE 203 (Data Structure) 80%CSE 205 (Digital Logic Design) 60%

In this semester:

Course Theory-bias

CSE 207 (Algorithms) 95%CSE 209 (Digital Electronics and Pulse Technique) 40%

CSE 211 (Theory of Computation) 100%

CSE207 is the fraternal brother of CSE211




The Power of Theory

Write a C program that finds all the factors of a 512-bit integer

For input 24, Output is: 1, 2, 3, 4, 6, 8, 12 and 24 , since 24 = 1× 23 × 3But wait! 512-bit number!So the input ranges from 1 to 2512 − 1 thats roughly 1.3× 10154

The highest input can be of 155-digits!

Input: 12345676449393993. . .

930049922 (155 digits)

3 hours gone, your C-program is not coming up with any answer?Your program worked perfectly for 1, 2, 10, 20 even 50 digit inputs!

So what’s wrong??

Is your program erroneous?

Is the computation power of the machine insufficient?

Is there a probem in the nature of the input?




The Power of Theory (Contd.)

Case: Your program is erroneousHow do you know? The program is not even stopping!It may indeed stop after 10 hours and show the correct output!

Case: Computation Power of the Machine is insufficientWe may use multi-core processors clocked at a very high speedlike: 10-core processor each core clocked at 3 GHzCan we afford that machine?What if the number is even bigger? Can the machine scale up to the

input?Caveat: Even with today’s fastest computer, the computation for a500-digit input will take 5 billion years!





Case: Nature of Input

Some numbers are too big or too smallAre you crazy? 500-digit number? That’s pretty big!So is:

the number of stars in the universe

the number of electrons in your body

the energy in electron volt when two stars collide

the number of combinatorial choices of pairings of almost half-million

genes in our cellSo, data can be very large or too small? You never can tell





Case: Nature of Computation

Can we break down the processing into multiple blocks and distribute

those among multiple computers?Can we predict the time required by analyzing the nature of input(input-domain) and the method of computation (algorithm)?

Can we identify certain problems as really hard problems?




Really Hard Problems

General Tile Fitting Problem

Given a finite set of tiles and an area on a plane, is it possible to cover thecomplete area using each type of tiles at least once?




Really Hard Problems (Contd.)

There are many problems that cannot be guaranteed to be solved by any

computer! These are called undecidable problems

David Hilbert

Entscheidungsproblem

Give an algorithm A that takes as input

a hypothesis H and answers True orFalse based on a finite set of axioms S A.In other word, Algorithm A decideswhether H logically follows from S A.

Hypothesis: A machine works withoutconsuming any energy.Decision: False (according to the set of axioms of Physics)





Alonzo Church & Alan Turing

Alonzo Church and Alan Turing independently proved respectively in 1936and in 1937 that

Hilbert’s Entscheidungsproblem is undecidable

This is known as Church-Turing Thesis.





More undecided problems:

Mortal Matrix Problem

Given a finite set of n× n matrices with integer entries, determinewhether they can be multiplied in some order, possibly with repetition, toyield the zero matrix.

Conjugacy problem

Given two words x and y of a Group G, whether there exists a z such that:y = zxz−1

*A group is an algebraic structure consisting of a set with one or moreoperators such that the result of applying the operators on one or moreelements of the is also an element in the set. For example, the set of integer I is a group.





Some problems are hard but still solvable i.e. NOT undecidable

(Euclidian) Travelling Salesman ProblemGiven a set of vertices and their coordinates, find a tour or cycle whichvisits each vertex exactly once and is of the minimum length.





Some problems are hard but still solvable i.e. NOT undecidable

Chromatic Number or Graph Coloring ProblemGiven a graph G = (V,E ), find the minimum number of different colorsneeded for this graph such that two adjacent vertices are not given thesame color.

Total three different colors are needed




R ll H d P bl (C d )




There is relationships among the hard solvable problems, we can comparetheir hardness!

Circuit satisfiability problem(Circuit-SAT) can be reducedto Travelling Salesman

Problem (TSP)If there is a good solution toCircuit-SAT, there is also agood solution to TSP


R ll H d P bl (C d )




Moral of the story

If the given problem is proved to be undecidable or unsolvable, don’tinvest your valuable effort to write program to solve it.

If the given problem is solvable but still hard, it is not your program’sfault that the computation is taking too much time.

If the given problem is solvable and not hard, and your program failsto solve it in the reasonable amount of time, only then invest effort

for improvement of your program!


Th f C i



Theory of Computation

Theory of computation

The branch that deals with whether and how efficiently problems can besolved on a model of computation, using an algorithm

The field is divided into three major branches:

Automata TheoryComputability Theory and

Computational Complexity Theory

There are many models of computation:

Turing MachineCombinatory logic

Markov Algorithm etc.

In our CS couses, we use Turing Machine model.


B h f Th f C t ti



Branches of Theory of Computation

Automata Theory

the study of abstract machines ( in our case, Turing Machine) andthe problems that can be solved using these machines

If the abstract machine cannot solve the problem, no computer onearth, no matter how powerful it is, can ever solve the problem

Topics include: Finite Automaton (FA), Push-down Automaton(PDA), Turing Machine (TM) etc.

studied in the theory of computation, compiler design, verificationsystems, robotics etc.




Branches of Theory of Computation (Contd )



Branches of Theory of Computation (Contd.)

Computational Complexity Theory

deals with classifying computational problems according to theirinherent difficulty

Two major aspects are considered: time complexity (how many stepsthe algorithm takes) and space complexity (how much memory isrequired by the program)

studies complexity classes: P , NP , NPC and relationships among

these classes


Course Outline



Course Outline

Course Name: CSE 211 (Theory of Computation)Credit Hour: 2.0Number of Class Tests: 3

Course Teachers:

Md. Shaifur Rahman (first seven weeks before mid-term)and

Sukarna Barua (after mid-term)


Class Outline



Class Outline

Week Day Topics

1 1 Introduction to TOC

2 Deterministic Finite Automaton (DFA)

2 1 DFA (Contd.)

2 Operators of Finite Automaton

3 1 Non-deterministic Finite Automaton (NFA)

2 Equivalence of DFA and NFA4

1 Regular Expressions2 Class Test-1

5 1 Regular Expressions (Contd.)

2 Pumping Lemma

6 1 Context-free Grammar (CFG)2 CFG (Contd.)

7 1 Overall Discussion

2 Class Test-2


Textbooks



Textbooks

Main textbook

Introduction to the Theory of Computation (3rd Edition)

byMichael Sipser


Textbooks (Contd )



Textbooks (Contd.)

Supporting textbooks

Introduction to Automata Theory,Languages & Computation

(3rd Edition)by

Hopcroft et. al.


Textbooks (Contd )



Textbooks (Contd.)

Supporting textbooks

Elements of the Theory of Computation(2nd Edition)

by

Lewis et. al.


Course Website



Course Website

http://sites.google.com/site/cse211february2013






Question?


sr1 toc intro handout

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