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Computation Theory

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Title: Computation Theory


1
Computation Theory
  • ??????

2
Content
  • Overview
  • The Limitation of Computing
  • Complexities
  • Textbooks Grading

3
Computation Theory
  • Overview

4
What is Computation?
  • Computation can be actually defined as finding a
    solution to a problem from given inputs by means
    of an algorithm.
  • What is an algorithm?
  • Are following two statements equivalent?
  • What can be computed?
  • What can be solved algorithmically?

5
Tools for Computation
  • Your grandparents (or parents)
  • pen and paper, or
  • chalk and slate, or
  • mentally,
  • sometimes with the aid of tables.
  • Now, electronic computers
  • The theory of computation began early in the
    twentieth century, before modern electronic
    computers had been invented.

6
Computation Models
  • Register Machines
  • Turing Machines
  • Markov Algorithmic Machines
  • Recursive Functions
  • ?-calculus

Equivalent in computational power
7
Church-Turing Thesis
Turing, Alan (1912-1954)
  • Every effective computation or algorithm can be
    carried out by a Turing Machine.

Effective Computation
Algorithm
Any computer program
Truing Machine
8
Computation Theory
  • The Limitation of Computing

9
What Can be Done by a Computer?
  • Problem Solvability

What computers can?
What computers can't?
10
Halting Problem
Given a description of an algorithm and a
description of its initial arguments, determine
whether the algorithm, when executed with these
arguments, ever halts.
11
Halting Problem
Unsolvable?
Given a description of an algorithm and a
description of its initial arguments, determine
whether the algorithm, when executed with these
arguments, ever halts.
12
Determination of Problems Solvability
Methods
  • Problem Reduction
  • Diagonalization Principle
  • Rices Theorem

13
Computation Theory
  • Complexities

14
Unsolvability vs. Solvability
15
Complexities
Can a solvable problem be solved in practical
sense?
Time Complexity
Space Complexity
16
P and NP
17
P and NP
Polynomial-Time Solvable
Nondeterministic Polynomial-Time Solvable
18
P and NP
P NP ?
No answer, now.
19
NP-Completeness
1. L ? NP
L ? NP-Complete iff
2. L ? NP
L
p.r
P NP can be concluded if you can find one L ?
NP-Complete being polynomial-time solvable, i.e.,
L ? P.
20
Some NP-Complete Problems
TSP (Traveling Salesman Problem)
21
Some NP-Complete Problems
Bounded Tiling Problem
22
More on NP-Completeness
  • How to prove a problem being NP-complete or
    NP-hard?
  • Problem Reduction
  • How to solve the problem if it is known to be
    NP-complete or NP-hard?
  • E.g., by heuristic algorithms

23
More Advance Topics
  • Molecular Computation Theory
  • Quantum Computation Theory

24
Computation Theory
  • Textbooks Grading

25
Textbooks
  • Programs, Machines and Computation An
    Introduction to the Theory of Computing (out of
    print)
  • Authors Keith Clark and Don Cowell
  • Publisher McGraw-Hill Companies
  • Elements of the Theory of Computation, 2/E
  • Authors Harry R. Lewis, Christos H.
    Papadimitriou and Christos Papadimitriou
  • Publisher Prentice Hall
  • Introduction to Automata Theory, Languages, and
    Computation, 2/E
  • Authors John E. Hopcroft, Rajeev Motwani and
    Jeffrey D. Ullman
  • Publisher Addison-Wesley
  • An Introduction to Formal Languages and Automata,
    3/E
  • Author Peter Linz
  • Publisher Jones and Bartlett

26
Grading
  • Homework 20
  • You have to hand in each assignment on time
  • In-Class Examination 20
  • Midterm Examination 30
  • Final Examination 30
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