CMPT 308 — Computability and Complexity
Fall 2004
Instructor:
TA:
Andrei Bulatov, email: abulatov@cs.sfu.ca
Ramsay Dyer,
email: rhdyer@cs.sfu.ca
Learning resources:
• Prerequisites: MACM 201
• Lectures:
MWF 10:30 – 11:20, in WMX 3210 (37 lectures)
• Course Text: Introduction to the Theory of Computation
by Mike Sipser
• Instructor’s office hours: W 13:00 – 14:00 in ASB 10855, or by appointment
• Assignments: 5 sets of exercises, solutions to the first one are due to Sept 29th
• TA’s office hours: TBA
Course web page:
www.cs.sfu.ca/~abulatov/CMPT308
Marking scheme:
• 5 homework assignments, worth 6% each
• midterm, worth 30%, and
• final exam, worth 40%
Computability and Complexity
1-1
Introduction
Introduction
Computability and Complexity
Andrei Bulatov
Computability and Complexity
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 ?
1-2
Computability and Complexity
1-3
Problems
Given 2 collections of DNA sequences {a1 , a2 ,, ak } and
{b1 , b2 ,, bk }
1. What is the shortest DNA sequence that contains all of a1 , a2 ,, ak
and b1 , b2 ,, bk as subsequences?
2. What is the shortest DNA sequence, formed by overlapping elements
from these sets, which begins with a1 and ends with bk ?
3. Is there a set of indices i1 , i2 ,, in such that
ai1  ai2    aik  bi1  bi2    bik
4. Can each ai be paired with a distinct b j in such a way that ai and
b j are 99% identical? How many ways can this be done?
Computability and Complexity
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
1-4
Computability and Complexity
More examples
• Computer viruses: Write a program that prints its own
source code
easy
• Perfect virus detection software: Write a program that detects
whether any given program prints its own text
impossible
• Can mathematics be automatized? Does there exist a computer
program that would distinguish true mathematical statements
from false ones?
impossible
1-5
Computability and Complexity
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
1-6
Computability and Complexity
Prehistory
1900
1928 1933
Hilbert’s program
Gödel’s Incompleteness Theorem
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
1-7
Computability and Complexity
1-8
The Machine
1936 1937 1945 1951
Turing Machine
Undecidability of Logic
First digital computer
Recursive Functions
Markov’s algorithms
1960
Computability and Complexity
1-9
Complexity Measures and Non-Determinism
1960
1970 1971 1975
Complexity Measures, Birth of CC
Study of Complexity Measures
Main Complexity Classes
P=NP Question
Cook-Levin’s Theorem
NP-complete problems
Computability and Complexity
1-10
Other Computational Models
1970
1980 1985 1997
Shor’s integers factoring
New computational models introduced:
Probabilistic Algorithms
Approximation Algorithms
Interactive Algorithms
Cryptography
Quantum algorithms
Computability and Complexity
Course Outline
• Turing Machine and other computational models
• Theory of Computability and Undecidable problems
• First Order Logic and Gödel’s 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
1-11
1.
2.
3.
4.
5.
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 {( ab) n | n natural} regular?
Is the graph
bipartite?
6.
How many edges has a 6-vertex tree?
7.
Does the graph
8.
9.
contain a 4-clique?
Is ( x  y )  ( x  y ) a CNF?
Is x  ( x  y ) equivalent to x? y?