THEORY OF COMPUTATION COMPLETE NOTES PART 1


What is TOC?

In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory, computability theory and computational complexity theory.

In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. There are several models in use, but the most commonly examined is the Turing machine.

Automata theory

In theoretical computer science, automata theory is the study of abstract machines (or more appropriately, abstract ‘mathematical’ machines or systems) and the computational problems that can be solved using these machines. These abstract machines are called automata.

This automaton consists of

  • states (represented in the figure by circles),
  • and transitions (represented by arrows).

As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function (which takes the current state and the recent symbol as its inputs).

Uses of Automata: compiler design and parsing.

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Introduction to formal proof:

Basic Symbols used :

U – Union

∩- Conjunction

ϵ – Empty String

Φ – NULL set

7 negation

‘ – compliment

= > implies

 

 

Additive inverse: a+(-a)=0

Multiplicative inverse: a*1/a=1

Universal set U={1,2,3,4,5} Subset A={1,3}

A’ ={2,4,5}

Absorption law: AU(A ∩B) = A, A∩(AUB) = A

 

De Morgan’s Law:

(AUB)’ =A’ ∩ B’

(A∩B)’ = A’ U B’

Double compliment

(A’)’ =A

A ∩ A’ = Φ

Logic relations:

a € b = > 7a U b

7(a∩b)=7a U 7b

 

Relations:

Let a and b be two sets a relation R contains aXb. Relations used in TOC:

Reflexive: a = a

Symmetric: aRb = > bRa

Transition: aRb, bRc = > aRc

If a given relation is reflexive, symmentric and transitive then the relation is called equivalence relation.

 

Deductive proof: Consists of sequence of statements whose truth lead us from some initial statement called the hypothesis or the give statement to a conclusion statement.

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Additional forms of proof:

Proof of sets

Proof by contradiction

Proof by counter example

 

Direct proof (AKA) Constructive proof:

If p is true then q is true

Eg: if a and b are odd numbers then product is also an odd number.

Odd number can be represented as 2n+1 a=2x+1, b=2y+1

product of a X b = (2x+1) X (2y+1)

= 2(2xy+x+y)+1 = 2z+1 (odd number)

 

 

 

 

 

Proof by contrapositive:

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Proof by Contradiction:

 

H and not C implies falsehood.

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Be regarded as an observation than a theorem.

 
For any sets a,b,c if a∩b = Φ and c is a subset of b the prove that a∩c =Φ Given : a∩b=Φ and c subset b

Assume: a∩c   Φ

Then

= > a∩b  Φ = > a∩c=Φ(i.e., the assumption is wrong)

 

Proof by mathematical Induction:

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