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Introduction. Episode 0. 0.a. Giorgi Japaridze Theory of Computability. What is TOC (Theory of Computation) about?. Subject: The fundamental mathematical properties of computers (hardware, software and certain applications). Questions: What does computation mean?

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introduction
Introduction

Episode 0

what is toc theory of computation about

0.a

Giorgi JaparidzeTheory of Computability

What is TOC (Theory of Computation) about?
  • 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?
the three central areas of toc

0.b

Giorgi JaparidzeTheory of Computability

The main question in TOC:

What are the fundamental capabilities and limitations of computers?

The three central areas of TOC

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?
slide4

0.c

Giorgi JaparidzeTheory of Computability

Set --- any collection of distinct objects.

Sets
  • E={2,4,6,8,…}, or
  • E={x | xis a positive integer divisible by 2}, or
  • E={x | x=2kfor some positive integerk},etc.

Describing a set:

Set-related terminology and notation:

aE---“a is an element ofE”, or “ais inE”

aE ---“ais not an element ofE”, or “ais not inE”

ST--- “Sis asubsetofT”

i.e. every element ofSis also an element ofT

ST---“theintersection ofSandT ”

i.e. the set of the objects that are both inSandT

ST---“theunionofSandT ”

i.e. the set of the objects that are in eitherSorT or both

--- “theemptyset”

P(S) --- “thepower setofS”

i.e. the set of all subsets ofS

sequences tuples cartesian products

0.d

Giorgi JaparidzeTheory of Computability

Sequences, tuples, Cartesian products

Asequenceis 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

Ann-tupleis a sequence withnelements.

E.g.: (5,2) --- 2-tuple (pair)

(3,0,3) --- 3-tuple (triple)

!

{1,2,2} = {1,2} = {2,1}, but

(1,2,2)  (1,2)  (2,1)

TheCartesianproductof setsSandTis defined by

ST={(s,t) | sSandtT}

Similarly,

S1 S2 ...  Sn={(s1,s2,…,sn) | s1S1, s2S2, …, snSn}

functions

0.e

Giorgi JaparidzeTheory of Computability

Functionf from set A to set B --- assignment of a unique element f(a)B to each aA

Functions

AB

the rangeof 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)=2xg(x)=x/2 h(x,y)=x+y

f: g: h:

N  N

N  R

NN  N

strings

0.f

Giorgi JaparidzeTheory of Computability

Alphabet --- a finite set of objects called thesymbolsof the alphabet.

E.g.:  = {a,b,…,z}

 = {0,1}

 = {0,1,$}

Strings

Stringover--- a finite sequence of symbols from.

E.g.:x = 01110 is a string over.

|x|=5 ---“thelengthof x is 5”.

Theempty stringis denoted. ||=0.

Concatenationxyof the stringsxandy---

the result of appendingy at the end ofx.

k

xk --- xx…x