LECTURE 2THEORY OF COMPUTATION

Yasir Imtiaz Khan

GRAPHS Set of points with the lines connecting some of the

points (also called simple graph). The points are called nodes or vertices and the

lines are called edges.

Numbers of edges at a particular node is the degree of that node.

G = (V, E)

GRAPHS CONTINUE…..Path: in a graph is a sequence of nodes

connected by edges.Simple Path: is a path that does not

repeat nodes.Connected Graph: if every two nodes

have a path between them.Cycle: A path is a cycle if it starts and ends

with same node.Simple Cycle: contains at least three

nodes and repeats only the first and last nodes

GRAPHS CONTINUE….Tree: if it is connected and has no simple

cyclesDirected Graph: If it has arrows instead

of linesStrongly Connected: if a directed path

connects every two nodes.

Language: a set of strings

String: a sequence of symbols from some alphabet

Example: Strings: cat, dog, house Language: {cat, dog, house}

zcba ,,,, Alphabet:

Languages are used to describe computation problems

},17,13,11,7,5,3,2{ PRIMES

},6,4,2,0{ EVEN

}9,,2,1,0{ Alphabet:

ALPHABETS AND STRINGS

abbawbbbaaavabu

ba,

baaabbbaabaabbaaba

Example Strings

Example Alphabet:

An alphabet is a set of symbols

A string is a sequence of symbols from the alphabet

String variables

}9,,2,1,0{ Decimal numbers alphabet

102345 567463386

}1,0{Binary numbers alphabet

100010001 101101111

STRING OPERATIONS

m

nbbbvaaaw

21

21

bbbaaaabba

mn bbbaaawv 2121

Concatenation

abbabbbaaa

12aaaw nR

naaaw 21 ababaaabbb

Reverse

bbbaaababa

STRING LENGTH

Length:

Examples:

naaaw 21nw

124

aaaabba

PROOFSTheorem

Mathematical statements proved true.

Lemmas Assist in other proof so we proof

Corollaries Related statements are true (Conclude other

things)

PROOF BY CONTRADICTION In a proof by contradiction we assume, along

with the hypotheses, the logical negation of the result we wish to prove and then reach some kind of contradiction.

That is, if we want to prove "If P, Then Q", we assume P and Not Q.

EXAMPLE (PROOF BY CONTRADICTION) Theorem. There are infinitely many prime

numbers. Proof. Assume to the contrary that there are

only finitely many prime numbers, and all of them are listed as follows: p1, p2 ..., pn.

Consider the number q = p1p2... pn + 1. The number q is either prime or composite. If we divided any of the listed primes pi into q, there would result a remainder of 1 for each i = 1, 2, ..., n. Thus, q cannot be composite. We conclude that q is a prime number, not among the primes listed above, contradicting our assumption that all primes are in the list p1, p2 ..., pn.

PROOF BY INDUCTION Mathematical induction: is a method of

mathematical proof typically used to establish that a given statement is true of all natural numbers.

Base Case Inductive Step

16

Theorem:

For all n>=1.

Proof #1: (by induction on n)

Basis:n = 1

1 = 1

2)1(

1

nnin

i

2)11(11

1

i

i

17

Inductive hypothesis:Suppose that for some k>=1.

Inductive step:We will show that

by the inductive hypothesis

It follows that for all n>=1. �

2)1(

1

kkik

i

2)2)(1(1

1

kkik

i

)1(1

1

1

kiik

i

k

i

)1(2)1(

kkk

2)1(2)1(

kkk

2)2)(1(

kk

2)1(

1

nnin

i

EXAMPLE PROOF BY INDUCTION

AUTOMATA THEORY Deals with the properties of computation

models.

Abstract Model of digital computer so it should have features like

MemoryControl UnitALUInputOutput