TDDD65
Introduction to the Theory of Computation
Lecture 1
Gustav Nordh
Department of Computer and Information Science
gustav.nordh@liu.se
2012-08-29
Outline - Lecture 1
Objectives and Motivation
Contents
Course Organization
Basic Mathematical Notions
Finite Automata
Objectives and Motivation
“Theoretical Computer Science (TCS) studies the inherent
powers and limitations of computation, that is broadly defined to
include both current and future, man-made and naturally arising
computing phenomena.”
Objectives and Motivation
“Theoretical Computer Science (TCS) studies the inherent
powers and limitations of computation, that is broadly defined to
include both current and future, man-made and naturally arising
computing phenomena.”
The objective of this course is to give an introduction to
1
Formal Languages and Automata
2
Computability
3
Complexity
Formal Languages and Automata
What can be computed with very limited memory?
Formal Languages and Automata
What can be computed with very limited memory?
Such memory restricted devices are all around us
Formal Languages and Automata
What can be computed with very limited memory?
Such memory restricted devices are all around us
Many applications within CS ranging from compiler
construction and text search to computer opponents in
video games
Computability
What can be computed at all?
Computability
What can be computed at all?
Understanding the fundamental limits of computation
including the limits of
and any future “computers”
Computability
What can be computed at all?
Understanding the fundamental limits of computation
including the limits of
and any future “computers”
There are fundamental problems that cannot be solved.
Complexity
What can be computed efficiently?
Complexity
What can be computed efficiently?
Understanding the fundamental limits of computation
including the limits of
and any future “computers”
Complexity
What can be computed efficiently?
Understanding the fundamental limits of computation
including the limits of
and any future “computers”
Some problems seems to require more resources to solve
than others, for example coming up with a correct proof for
a mathematical statement seems to take more time than
verifying the correctness of a proof
Complexity
What can be computed efficiently?
Understanding the fundamental limits of computation
including the limits of
and any future “computers”
Some problems seems to require more resources to solve
than others, for example coming up with a correct proof for
a mathematical statement seems to take more time than
verifying the correctness of a proof
One of the most important open questions in all of
mathematics
Contents
1
Formal Languages and Automata (6h)
Finite Automata and Regular Languages
Pushdown Automata and Context-free Languages
2
Computability (3h)
Turing machines and the notion of an Algorithm
Problems that cannot be solved (Undecidability)
3
Complexity (6h)
Big-O notation and analysis of algorithms
Time-complexity classes, P, NP
NP-completeness
Space complexity? P vs NP?
Course Organization
Course Book:
Michael Sipser. Introduction to the Theory of Computation,
(2nd ed.), Thomson 2006
Webpage: www.ida.liu.se/∼TDDD65
Lectures
Problem Solving Sessions
Homeworks (1 ECTS)
Exam (3 ECTS)
Course Organization: Homeworks (1 ECTS)
Two batches of obligatory homework
The grade is pass/fail and to pass (and get your 1 ECTS)
you need to pass both batches
To pass a batch of homeworks you need to make a real
effort on every problem and solve more than 2/3 of the
problems correctly
If you fail, you will have to wait and do the homeworks for
next years course
You need to sign up in webreg so that I can report your
grade (follow the link on the homepage)
Course Organization: Homeworks (1 ECTS)
Collaboration policy: “General discussions about the homework
problems is allowed (and encouraged). However, you must
write up your own solutions separately. If your solution is
heavily inspired by someone else’s ideas, please give proper
credit in your write-up to the people with whom you worked. If
you consult any reference material other than the textbook,
please note on your homework which sources you used for
each problem. You MUST understand all the solutions in your
write-up and be prepared to explain them to me!”
Course Organization: Exam (3 ECTS)
Written exam: October (January, August, October,...)
The grades are Fail/3/4/5 and will be your final grade for
the course
Basic Mathematical Notions: Strings and Languages
Basic Mathematical Notions: Strings and Languages
An alphabet is a finite set of symbols (denoted Σ, Γ)
A string over Σ is a finite sequence of symbols from Σ
Basic Mathematical Notions: Strings and Languages
An alphabet is a finite set of symbols (denoted Σ, Γ)
A string over Σ is a finite sequence of symbols from Σ
Example
1010 is a string over Σ1 = {0, 1}
theory is a string over Σ2 = {a, b, c, . . . , z}
Basic Mathematical Notions: Strings and Languages
An alphabet is a finite set of symbols (denoted Σ, Γ)
A string over Σ is a finite sequence of symbols from Σ
Example
1010 is a string over Σ1 = {0, 1}
theory is a string over Σ2 = {a, b, c, . . . , z}
The length of a string w (denoted |w|) is the number of
symbols it contains. |1010| = 4, |theory | = 6
The empty string (denoted ε) is the string of length 0
The concatenation of strings x and y is written xy .
x = slum and y = dog gives xy = slumdog.
x k denotes the concatenation of x with itself k times.
015 0 = 0111110, x 2 y = slumslumdog.
Basic Notions: Strings and Languages
A language is a set of strings over an alphabet
Σ∗ is the language consisting of all strings over Σ
Basic Notions: Strings and Languages
A language is a set of strings over an alphabet
Σ∗ is the language consisting of all strings over Σ
Union and intersection: A ∪ B = {x | x ∈ A or x ∈ B},
A ∩ B = {x | x ∈ A and x ∈ B}
Complement: A = {x ∈ Σ∗ | x ∈
/ A}
Concatenation: AB = {xy | x ∈ A and y ∈ B}
Basic Notions: Strings and Languages
A language is a set of strings over an alphabet
Σ∗ is the language consisting of all strings over Σ
Union and intersection: A ∪ B = {x | x ∈ A or x ∈ B},
A ∩ B = {x | x ∈ A and x ∈ B}
Complement: A = {x ∈ Σ∗ | x ∈
/ A}
Concatenation: AB = {xy | x ∈ A and y ∈ B}
Ak denotes the concatenation of A with itself k times
(note: A0 = {ε})
Star: A∗ = A0 ∪ A1 ∪ A2 ∪ . . .
Examples of Languages
Examples of Languages
the set of odd binary numbers
the set of prime numbers
the set of syntactically correct Java programs
the set of positive integer solutions to x n + y n = z n for
n>2
the set of true mathematical statements