1. Introduction
Chapter 0

2. What is this course about?
Giorgi Japaridze Theory of Computability
Subject:
The fundamental mathematical properties of computers
(hardware, software and certain applications).
Questions:
What does computation mean?
What can be computed and what can not?
How quickly?
With how much memory?
On which type of machines?

3. Why do you need this course?
0.b
Giorgi Japaridze Theory of Computability
To be able to call yourself an educated person
To expand your mind and abilities
To acquire conceptual tools for solving practical problems
To get the diploma

4. The three central areas of TOC
Giorgi Japaridze Theory of Computability
The main question in TOC:
What are the fundamental capabilities and limitations of computers?
Each of the three central areas of TOC focuses on this question
but interprets it differently.
Automata Theory:
What can be computed with different sorts of weak machines, such as
Finite automata,
Pushdown automata, etc.?
Computability Theory:
What can be computed with the strongest possible machines, such as
Turing machines?
Complexity Theory:
How efficiently can things be computed, in particular, in how much
Time,
Space?

5. Set --- any collection of distinct objects.
Sets
0.d
Giorgi Japaridze Theory of Computability
Set --- any collection of distinct objects.
E={2,4,6,8,…}, or
E={x | x is a positive integer divisible by 2}, or
E={x | x=2k for some positive integer k}, etc.
Describing a set:
Set-related terminology and notation:
a E “a is an element of E”, or “a is in E”
aE “a is not an element of E”, or “a is not in E”
S T “S is a subset of T”
i.e. every element of S is also an element of T
ST “the intersection of S and T ”
i.e. the set of the objects that are both in S and T
ST “the union of S and T ”
i.e. the set of the objects that are in either S or T or both
 “the empty set”
P(S) “the power set of S”
i.e. the set of all subsets of S

6. Sequences, tuples, Cartesian products
Giorgi Japaridze Theory of Computability
A sequence is a finite or infinite list.
E.g.: 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,… is a sequence of natural numbers
An n-tuple is a sequence with n elements.
E.g.: (5,2) tuple (pair)
(3,0,3) tuple (triple) !
{1,2,2} = {1,2} = {2,1}, but
(1,2,2)  (1,2)  (2,1)
The Cartesian product of sets S and T is defined by
ST={(s,t) | sS and tT}
Similarly,
S1  S2  ...  Sn={(s1,s2,…,sn) | s1S1, s2S2, …, snSn}

7. Functions
0.f
Giorgi Japaridze Theory of Computability
Function f from set A to set B --- assignment of a unique element f(a)B to each aA
A B
the range of f
the domain of f f a b c 1 2 3 4
f: A  B
the type of f
N --- natural numbers: {0,1,2,…} R --- rational numbers: {0, 5, 8.6, 1/3, etc.}
If x,y always take values from N, what are the types of f,g,h?
f(x)=2x g(x)=x/ h(x,y)=x+y
f: g: h:

8. String over  --- a finite sequence of symbols from .
Strings and languages
0.g
Giorgi Japaridze Theory of Computability
Alphabet --- a finite set of objects called the symbols of the alphabet.
E.g.:  = {a,b,…,z}
 = {0,1}
 = {0,1,$}
String over  --- a finite sequence of symbols from .
E.g.: x = is a string over .
|x|= “the length of x is 5”.
The empty string is denoted . ||=0.
Concatenation xy of the strings x and y ---
the result of appending y at the end of x. k
xk xx…x
Language over  --- a set of strings over .

9. What else you need to know:
Giorgi Japaridze Theory of Computability
You are expected to have basic knowledge and experience with:
Predicates and relations
Graphs
Boolean logic
Mathematical definitions and proofs