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BCS 303
THEORY OF COMPUTATION (3-1-0)
Cr.-4
Module – I
(10 Lectures)
Introduction to Automata: The Methods Introduction to Finite Automata, Structural
Representations, Automata and Complexity. Proving Equivalences about Sets, The
Contrapositive, Proof by Contradiction, Inductive Proofs: General Concepts of Automata
Theory: Alphabets Strings, Languages, Applications of Automata Theory.
Finite Automata: The Ground Rules, The Protocol, Deterministic Finite Automata: Definition
of a Deterministic Finite Automata, How a DFA Processes Strings, Simpler Notations for
DFA’s, Extending the Transition Function to Strings, The Language of a DFA
Nondeterministic Finite Automata: An Informal View. The Extended Transition Function, The
Languages of an NFA, Equivalence of Deterministic and Nondeterministic Finite Automata.
Finite Automata With Epsilon-Transitions: Uses of ∈-Transitions, The Formal Notation for an
∈-NFA, Epsilon-Closures, Extended Transitions and Languages for ∈-NFA’s, Eliminating ∈Transitions.
Module – II
(10 Lectures)
Regular Expressions and Languages: Regular Expressions: The Operators of regular
Expressions, Building Regular Expressions, Precedence of Regular-Expression Operators,
Precedence of Regular-Expression Operators
Finite Automata and Regular Expressions: From DFA’s to Regular Expressions, Converting
DFA’s to Regular Expressions, Converting DFA’s to Regular Expressions by Eliminating States,
Converting Regular Expressions to Automata.
Algebraic Laws for Regular Expressions:
Properties of Regular Languages: The Pumping Lemma for Regular Languages, Applications
of the Pumping Lemma Closure Properties of Regular Languages, Decision Properties of
Regular Languages, Equivalence and Minimization of Automata,
Context-Free Grammars and Languages: Definition of Context-Free Grammars, Derivations
Using a Grammars Leftmost and Rightmost Derivations, The Languages of a Grammar,
Parse Trees: Constructing Parse Trees, The Yield of a Parse Tree, Inference Derivations, and
Parse Trees, From Inferences to Trees, From Trees to Derivations, From Derivation to Recursive
Inferences,
Applications of Context-Free Grammars: Parsers, Ambiguity in Grammars and Languages:
Ambiguous Grammars, Removing Ambiguity From Grammars, Leftmost Derivations as a Way
to Express Ambiguity, Inherent Anbiguity
Module – III
(10 Lectures)

Pushdown Automata: Definition Formal Definition of Pushdown Automata, A Graphical
Notation for PDA’s, Instantaneous Descriptions of a PDA,
Languages of PDA: Acceptance by Final State, Acceptance by Empty Stack, From Empty Stack
to Final State, From Final State to Empty Stack
Equivalence of PDA’s and CFG’s: From Grammars to Pushdown Automata, From PDA’s to
Grammars
Deterministic Pushdown Automata: Definition of a Deterministic PDA, Regular Languages
and Deterministic PDA’s, DPDA’s and Context-Free Languages, DPDA’s and Ambiguous
Grammars
Properties of Context-Free Languages: Normal Forms for Context-Free Grammars, The
Pumping Lemma for Context-Free Languages, Closure Properties of Context-Free Languages,
Decision Properties of CFL’s
Module –IV
(10 Lectures)
Introduction to Turing Machines: The Turing Machine: The Instantaneous Descriptions for
Turing Machines, Transition Diagrams for Turing Machines, The Language of a Turing
Machine, Turing Machines and Halting
Programming Techniques for Turing Machines, Extensions to the Basic Turing Machine,
Restricted Turing Machines, Turing Machines and Computers,
Undecidability: A Language That is Not Recursively Enumerable, Enumerating the Binary
Strings, Codes for Turing Machines, The Diagonalization Language
An Undecidable Problem That Is RE: Recursive Languages, Complements of Recursive and RE
languages, The Universal Languages, Undecidability of the Universal Language
Undecidable Problems About Turing Machines: Reductions, Turing Machines That Accept the
Empty Language. Post’s Correspondence Problem: Definition of Post’s Correspondence
Problem, The “Modified” PCP, Other Undecidable Problems: Undecidability of Ambiguity for
CFG’s
Text Book:
1. Introduction to Automata Theory Languages, and Computation, by J.E.Hopcroft,
R.Motwani & J.D.Ullman (3rd Edition) – Pearson Education
2. Theory of Computer Science (Automata Language & Computations), by K.L.Mishra &
N. Chandrashekhar, PHI

MODULE-I
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.
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.
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)

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