CMPT 308

1
CMPT 308 Computability and Complexity Fall 2004
Instructor Andrei Bulatov, email
abulatov_at_cs.sfu.ca
TA Ramsay Dyer, email
rhdyer_at_cs.sfu.ca
Learning resources

• Prerequisites MACM 201
• Lectures MWF 1030 1120, in WMX
  3210 (37 lectures)
• Course Text Introduction to the Theory of
  Computation
• by Mike
  Sipser
• Instructors office hours W 1300
  1400 in ASB 10855, or by appointment
• Assignments 5 sets of exercises,
  solutions to the first one are due to Sept 29th
• TAs office hours TBA

Course web page www.cs.sfu.ca/abulatov/CMP
T308
Marking scheme

• 5 homework assignments, worth 6 each
• midterm, worth 30, and
• final exam, worth 40

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Computability and Complexity
1-1
Introduction
Introduction
Computability and Complexity Andrei Bulatov
3
Computability and Complexity
1-2
Fundamental Questions
A computer scientist might be expected to have
answers to some Fundamental questions such as

• What is a computer ?
• What problems can computers solve?
• Can these problems be classified ?

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Computability and Complexity
1-3
Problems
Given 2 collections of DNA sequences
and

• What is the shortest DNA sequence that contains
  all of
• and as
  subsequences?
• What is the shortest DNA sequence, formed by
  overlapping elements
• from these sets, which begins with
  and ends with ?
• Is there a set of indices
  such that
• Can each be paired with a distinct in
  such a way that and
• are 99 identical? How many ways
  can this be done?

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Computability and Complexity
1-4
Comments

• Three of these examples are problems that can
  be solved
• by a computer
• One is easy the other two are hard
• The other example cannot be solved by any
  known
• computer

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Computability and Complexity
1-5
More examples

• Computer viruses Write a program that prints
  its own
• source code
• Perfect virus detection software Write a
  program that detects
• whether any given program prints its own
  text
• Can mathematics be automatized? Does there
  exist a computer
• program that would distinguish true
  mathematical statements
• from false ones?

easy
impossible
impossible
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Computability and Complexity
1-6
Aims
This course is designed to enable you to

1. State precisely what it means for a problem to be
   computable, and show that some problems are not
   computable
2. State precisely what it means to reduce one
   problem to another, and construct reductions for
   simplest examples
3. Classify problems into appropriate complexity
   classes, and use this information effectively

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Computability and Complexity
1-7
Prehistory
1900
1928
1933
Hilberts program
Formalize mathematics and establish that

• Math is consistent a mathematical statement
  and its negation cannot ever both be proved
• Math is complete all true mathematical
  statements can be proved
• Math is decidable there is a mechanical rule
  to determine whether a given mathematical
  statement is true or false

Even for arithmetic at most one of the first two
properties can be reached
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Computability and Complexity
1-8
The Machine
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Computability and Complexity
1-9
Complexity Measures and Non-Determinism
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Computability and Complexity
1-10
Other Computational Models
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Computability and Complexity
1-11
Course Outline

• Turing Machine and other computational models
• Theory of Computability and Undecidable
  problems
• First Order Logic and Gödels Incompleteness
  Theorem
• Complexity Measures and Complexity classes
• Time Complexity, classes P and NP
• Space Complexity, classes L and PSPACE
• Probabilistic and Approximation algorithms
• Interactive Computation and Cryptography

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• Is the relation (a,b),(b,a) on a,b
  symmetric?
• Is the function f from 1,2,3 to a,b
  defined by f(1)a, f(2)b, f(3)a
  bijective?
• Can one make a list of all real numbers?
• Is the language natural
  regular?
• Is the graph
  bipartite?
• How many edges has a 6-vertex tree?
• Does the graph
• contain a 4-clique?
• Is a CNF?
• Is equivalent to x?
  y?