Computation Theory

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Computation Theory

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Content

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

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Computation Theory

• Overview

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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?

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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.

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Computation Models

• Register Machines
• Turing Machines
• Markov Algorithmic Machines
• Recursive Functions
• ?-calculus

Equivalent in computational power
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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
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Computation Theory

• The Limitation of Computing

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What Can be Done by a Computer?

• Problem Solvability

What computers can?
What computers can't?
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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.
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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.
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Determination of Problems Solvability
Methods

• Problem Reduction
• Diagonalization Principle
• Rices Theorem

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Computation Theory

• Complexities

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Unsolvability vs. Solvability
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Complexities
Can a solvable problem be solved in practical
sense?
Time Complexity
Space Complexity
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P and NP
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P and NP
Polynomial-Time Solvable
Nondeterministic Polynomial-Time Solvable
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P and NP
P NP ?
No answer, now.
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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.
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Some NP-Complete Problems
TSP (Traveling Salesman Problem)
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Some NP-Complete Problems
Bounded Tiling Problem
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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

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More Advance Topics

• Molecular Computation Theory
• Quantum Computation Theory

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Computation Theory

• Textbooks Grading

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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

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Grading

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