Introduction

1. Introduction Chapter 0

2. 0.a Giorgi JaparidzeTheory of Computability What is this course 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?

3. 0.b Giorgi JaparidzeTheory of Computability Why do you need this course? • 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. 0.c 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?

5. 0.d 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

6. 0.e 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}

7. 0.f 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:

8. 0.g Giorgi JaparidzeTheory of Computability Alphabet --- a finite set of objects called thesymbolsof the alphabet. E.g.:  = {a,b,…,z}  = {0,1}  = {0,1,$} Strings and languages 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 Language over  --- a set of strings over .

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