Fundamental of TOC

1. Theory of computation

2. COMPLEXITY THEORY
 The main question asked in this area is “What makes some problems
computationally hard and other problem easy?”
 What makes some problems computationally hard and others easy?
 Basing upon the properties
 Basing upon the difficulty level set
 Problem can be hard in worst case scenario

3. COMPUTABILITY THEORY
 Determining whether a mathematical statement is true or false.
 The theoretical models that were proposed in order to understand solvable and
unsolvable problems led to the development of real computers.
 The theories of computability and complexity are closely related. In complexity
theory, the objective is to classify problems as easy ones and hard ones;
whereas in computability theory, the classification of problems is by those that
are solvable and those that are not.

4. AUTOMATA THEORY
 Automata theory deals with the definitions and properties of mathematical
models of computation.
 Automata Theory deals with definitions and properties of different types of
“computation models”.
 Examples of such models are:
 Finite Automata. These are used in text processing, compilers, and hardware
design.
 Context-Free Grammars. These are used to define programming languages and
in Artificial Intelligence.
 Turing Machines. These form a simple abstract model of a “real” computer,
such as your PC at home.

5. MATHEMATICAL NOTIONS AND TERMINOLOGY

6. SETS
 A set is a collection of well-defined objects.
 The set of natural numbers is N = {1, 2, 3, . . .}.
 The set of integers is Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}.
 The set of rational numbers is Q = {m/n : m ∈ Z, n ∈ Z, n 6= 0}.
 The set of real numbers is denoted by R.
 If A and B are sets, then A is a subset of B, written as A ⊆ B, if every element of
A is also an element of B. For example, the set of even natural numbers is a
subset of the set of all natural numbers. Every set A is a subset of itself, i.e., A ⊆
A. The empty set is a subset of every set A, i.e., ∅ ⊆ A.
 If B is a set, then the power set P(B) of B is defined to be the set of all subsets of
B: P(B) = {A : A ⊆ B}. Observe that ∅ ∈ P(B) and B ∈ P(B).

7. SETS
 If A and B are two sets, then
a) Their union is defined as A ∪ B = {x : x ∈ A or x ∈ B},
b) Their intersection is defined as A ∩ B = {x : x ∈ A and x ∈ B},
c) Their difference is defined as A B = {x : x ∈ A and x ∉B},
d) The Cartesian product of A and B is defined as A × B = {(x, y) : x ∈ A and y ∈
B},
e) The complement of A is defined as A = {x : x ∉A}.
 A binary relation on two sets A and B is a subset of A × B.

8. FUNCTION
 A function f from A to B, denoted by f : A → B, is a binary relation R, having
the property that for each element a ∈ A, there is exactly one ordered pair in R,
whose first component is a.
 We will also say that f(a) = b, or f maps a to b, or the image of a under f is b.
 The set A is called the domain of f, and the set {b ∈ B : there is an a ∈ A with
f(a) = b} is called the range of f.
 A function f : A → B is one-to-one (or injective), if for any two distinct elements
a and a ′ in A, we have f(a) 6= f(a ′ ).
 The function f is onto (or surjective), if for each element b ∈ B, there exists an
element a ∈ A, such that f(a) = b; in other words, the range of f is equal to the
set B. A function f is a bijection, if f is both injective and surjective.

9. EQUIVALENCE RELATION
 A binary relation R ⊆ A × A is an equivalence relation, if it satisfies the
following three conditions:
a) R is reflexive: For every element in a ∈ A, we have (a, a) ∈ R.
b) R is symmetric: For all a and b in A, if (a, b) ∈ R, then also (b, a) ∈ R.
c) R is transitive: For all a, b, and c in A, if (a, b) ∈ R and (b, c) ∈ R, then also (a,
c) ∈ R.

10. GRAPH
 An undirected graph, or simply a graph, is a set of points with lines connecting
some of the points. The points are called nodes or vertices, and the lines are
called edges, as shown in the following figure.
 The number of edges at a particular node
is the degree of that node.
 An edge from a node to itself, called a self-loop.
 In a graph G that contains nodes i and j,
the pair(i, j)represents the edge that
connects i and j.
 The order of i and j doesn’t matter in an undirected graph, so the pairs (i, j) and
(j, i) represent the same edge.
 Formal description of the graph ({1, 2, 3, 4, 5}, {(1, 2), (2, 3), (3, 4), (4, 5), (5, 1)})
and ({1, 2, 3, 4}, {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)})

11. GRAPH
 When we label the nodes and/or edges of a graph, which then is called a labeled
graph.
 We say that graph G is a subgraph of graph H if the nodes of G are a subset of
the nodes of H, and the edges of G are the edges of H on the corresponding
nodes.

12. GRAPH
 A path in a graph is a sequence of nodes connected by edges. A simple path is a
path that doesn’t repeat any nodes. A graph is connected if every two nodes
have a path between them.
 A path is a cycle if it starts and ends in the same node. A simple cycle is one that
contains at least three nodes and repeats only the first and last nodes.
 A graph is a tree if it is connected and has no simple cycles. A tree may contain
a specially designated node called the root. The nodes of degree 1 in a tree,
other than the root, are called the leaves of the tree.
Path in a Graph Cycle in a Graph Tree

13. GRAPH
 A directed graph has arrows instead of lines, as shown in the following figure.
The number of arrows pointing from a particular node is the outdegree of that
node, and the number of arrows pointing to a particular node is the indegree.
 In a directed graph, we represent an edge from i to j as a pair (i, j). The formal
description of a directed graph G is (V,E), where V is the set of nodes and E is
the set of edges.
 {1,2,3,4,5,6}, {(1,2),(1,5),(2,1),(2,4),(5,4),(5,6),(6,1),(6,3)}
 A path in which all the arrows point in the same direction as its steps is called a
directed path. A directed graph is strongly connected if a directed path connects
every two nodes.

14. SYMBOLS & ALPHABETS
 Symbols are an entity or individual objects, which can be any letter, alphabets
or any picture.
 Ex 1,a,b,#
 Alphabets are a finite set of symbols.
 It is denoted by Σ.
 Ex Σ={a,b}, Σ={A,B,C}, Σ={0,1,2}

15. STRING
 A string is a finite collection of symbols from the alphabet. The string is
denoted by “w”.
 If Σ={a,b} , string generated from Σ are {aa,ab,aaa,bb,ba,aba....}.
 A string with zero occurrence of symbols is known as an empty string. It is
represented by ε.
 The number of symbols in a string is called length of the string. It is denoted by
|w|.
 Ex w=010, length of string |w|=3.

16. LANGUAGE
 A language is a collection of appropriate string. A language which is formed
over Σ can be finite or infinite.
 L1={set of string of length 2}={aa,bb,ba,ab} finite language
 L2={set of all strings starts with ‘a’}={a,aa,aab,abb,abab} infinite language

17. BOOLEAN
 The Boolean values are 1 and 0, that represent true and false, respectively. The
basic Boolean operations include
a) Negation (or NOT ), represented by ¬,
b) Conjunction (or AND), represented by ∧,
c) Disjunction (or OR), represented by ∨,
d) Exclusive-or (or XOR), represented by ⊕,
e) Equivalence, represented by ↔ or ⇔,
f) Implication, represented by → or ⇒.