Introduction Chapter 0

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?

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

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?

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

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) --- 2-tuple (pair) (3,0,3) --- 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}

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/2 h(x,y)=x+y f: g: h:

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 = 01110 is a string over . |x|=5 --- “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 .

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