Introduction Episode 0

What is TOC (Theory of Computation) 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? 0.a

The three central areas of TOC Automata Theory: Computability Theory: Complexity Theory: 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. What can be computed with different sorts of weak machines, such as Finite automata, Pushdown automata, etc.? What can be computed with the strongest possible machines, such as Turing machines? How efficiently can things be computed, in particular, in how much Time, Space? Giorgi Japaridze Theory of Computability 0.b

Sets Giorgi Japaridze Theory of Computability 0.c 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. 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 Describing a set: Set-related terminology and notation:

Sequences, tuples, Cartesian products Giorgi Japaridze Theory of Computability 0.d 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, S 1  S 2 ...  S n ={(s 1,s 2,…,s n ) | s 1  S 1, s 2  S 2, …, s n  S n } !

Functions Giorgi Japaridze Theory of Computability 0.e Function f from set A to set B --- assignment of a unique element f(a)  B to each a  A 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? the type of f the range of f the domain of f f: A  B A B abcabc f f(x)=2x g(x)=x/2 h(x,y)=x+y f: g: h: N  N  N N  RN  N

Strings Giorgi Japaridze Theory of Computability 0.f 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|=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. x k --- xx…x k