AUTOMATA AND
LANGUAGE THEORY
CS342 - Fall 2023
Lecture 1 : Overview & Finite Automata (FA)
Dr. Fawzya Ramadan
Fawzya.ramadan@fayoum.edu.eg
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Overview
• This course will focus on definitions and properties of
fundamental mathematical models of computation
(Automata Theory).
• We will be interested in both the inherent capabilities and
limitations of these computational models, as well as their
relationships with formal languages.
• The course covers three broad areas:
• (1) Regular Languages
• (2) Context-free Languages
• (3) Turing machines
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Automata Theory
• Finite automatons (FA):
• Used in text processing, compilers, and hardware design.
• It is the heart of many electromechanical devices such as
automatic doors, elevators, dishwashers, calculators,…etc.
• Context-Free Grammars (CFG):
• Used to describe the syntax of essentially every modern
programming language
• Every modern complier uses CFG concepts to parse programs.
• Describing natural languages.
• Turing Machines:
• These form a simple abstract model of a “real” computer, such as
your PC at home.
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Automata Theory
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• Lectures:
Main Textbook
“Introduction to
the Theory of
Computation”
• Wednesday 01:00 - 03:00 pm at Hall 2
• Section:
• Eng. Asmaa Seoudy
• Thursday 11:00 - 01:00 pm at Hall 5
• Thursday 01:00 - 03:00 pm at Hall 5
• Office hours (Tentative):
• Monday 09 am - 11 am
• Course Classroom
• Class Room Link:
https://classroom.google.com/c/NTkzODMwMjI4
NzQz?cjc=lnrhtio
• Class Room Code: lnrhtio
• All announcements, lecture slides, assignments,
grades… etc. will be published on the course
classroom, so you have to check regularly for
updates.
• Reference Books
• P. Linz. Introduction to Formal Languages and
Automata
• J. Hopcroft, R. Motwani, and J. Ullman.
Introduction to Automata Theory, Languages,
and Computation
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Grading
• Assessment Methods:
• Sheets & Quizzes:
• Midterm Exam :
20%
• Final Exam :
60%
• Notes:
• All assignments are individual.
20%
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Course Policies
• Slides :
• Will usually be uploaded on the web the morning of the class.
• The slides are for MY convenience and for helping you recollect the
material covered.
• They are not a substitute for, or a comprehensive summary of, the
textbook.
• Resources:
• We will follow the textbook closely.
• There are more resources than you can possibly read – including
books, lecture slides and notes.
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Course Policies (Cont.)
• Assignments Submission:
• All assignments will be handed out in class section time.
• NO late submissions will be permitted.
• All students must follow submission date.
• Late submissions can be accepted only in strong circumstances but
with loss of some credit.
• Plagiarism:
• Will be dealt with very strictly
• Many students find it helpful to consult their peers while doing
assignments and this is expected. However, it is not acceptable
practice to pool thoughts and produce common answers.
• Students who allow their assignments to be copied are as guilty as
those who copy and will be treated accordingly.
• Copying solutions from the Internet or books or any other public
sources without explicit citations is prohibited!
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Course Topics
• Regular Languages and their descriptors:
• Finite automata
• Nondeterministic finite automata (NFA)
• Regular expressions (RE)
• Closure properties of regular languages.
• Context-free languages and their descriptors:
• Context-free grammars (CFG)
• Pushdown automata (PDA)
• Turing Machines (TM)
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Course Schedule
Topics
Readings
Course Overview & Introduction
Ch 0
Finite Automata : DFA
Ch 1.1
Finite Automata : NFA
Ch 1.2
NFA Equivalence to DFA
Ch 1.2
Regular Expression : RE
Ch 1.3
Midterm
-
Context-Free Grammar and Languages: CFG &
CFL
Ch 2.1
Push Down Automata : PDA
Ch 2.2
Pumping Lemma
Ch 2.3
Turing Machine
Ch 3.1
Revision
-
Quiz
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Courses Depend on CS342
• CS441: Compiler Construction (Mandatory)
• CS467: Theory of Computation (Elective)
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Our Course…
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Overview of Languages
• Sentences is the basic building block of languages
• Sentence = Syntax + Semantics
• Grammar is the study of the structure of a sentence
• Ex.:
<sentence> ::= <noun phrase> <verb> <noun
phrase>
<noun phrase> ::= <article> < noun>
“A person entered the room”
Can you draw the derivation tree of the above
sentence?
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Natural vs. Formal Languages
Natural Language
Formal Language
• Rules comes after the language
• Developed with strict rules
• Imprecise
• Precise
• Ambiguous
• Unambiguous
• Hardly be processed by machine
• Can be processed by machine
• Highly flexible
• Unfamiliar notations
• No special learning effort needed
• Initial learning effort needed
• Ex. English, French, Arabic,
• Ex. C++, Java, Python, …etc
…etc.
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Mathematical Preliminaries (Sets)
• A set is a collection of objects called “elements” or
“members”
• Set membership  , non-membership 
• A = {1, 2, 3}  1  A
• B = {train, bicycle, bus, airplane}  ship  B
• Represented as :
• D = { 2, 4, 6, …}
• D = { j : j > 0 , and j = 2k for some k > 0}
• D = { j : j is non-negative and even }
• Finite set is a set with finite number of elements, while
Infinite set is a set with infinite number of elements.
• C = {a, b, c, d, e, f, g, …., z}  finite set
• D = {2, 4, 6, 8, ….}  infinite set
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Mathematical Preliminaries (Sets)
• Set operations: Union, Intersection, and Complement
• Subset: A is subset of B  A  B
• Proper set A proper subset is one that contains a few elements of the
original set whereas an improper subset, contains every element of the
original set along with the null set.
• For example, if set A = {2, 4, 6}, then,
•
•
•
•
•
Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.
Proper Subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}
Improper Subset: {2,4,6}
Proper Symbol : A ⊂ B
Improper symbol : A  B
• Power set (P) is a set of all subsets of a set including the empty set 
and the set itself.
• Example: A = {0, 1} , What is P(A)?
• Empty set () is a set with zero members (i.e. is subset of any set)
• ={}
• Neither order nor repetition of members matter in sets.
• {1, 2, 3} and {3, 2, 1} and {1, 2, 3, 2} are all the same set.
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Mathematical Preliminaries (Sequence)
• A sequence of objects is a list of these objects in some
order ( e.g.: function parameters )
• The sequence 7, 21, 57 is ( 7, 21, 57 )
• Set vs. Sequence:
• (7, 21, 57) and (21, 7, 57) are different sequences
• {7, 21, 57} and {21, 7, 57} are same sets
• (7, 7, 57) and (7, 57) are different sequences
• {7, 7, 57} and {7, 57} are same sets
• Finite sequences are often called tuples.
• A sequence with k elements is a k-tuple.
• (7, 21, 57) is a 3-tuple and (1, 5) is a 2-tuple (pair)
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Mathematical Preliminaries (Functions
and Relations)
• Function is an object that sets an input-output
relationship. Takes an input and produces an output.
• Also called a mapping, maps a given input to an output.
• f(a) = b  f maps a to b
• Domain (D): is a set of possible inputs.
• Range (R): is a set of possible outputs.
• f : D  R , f is a function with domain D and range R
• Example: function add is written as f : Z x Z  Z
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Mathematical Preliminaries (Functions
and Relations)
• Functions can be described in different ways:
• Procedure Table: 1-D if input is one argument, or 2-D if the domain
is a Cartesian product of two sets.
• A predicate or property is a function whose range is
{TRUE, FALSE}.
• A relation is a property whose domain is a set of k-tuples.
• Example: “less than” is a relation with 2-tuple input (binary relation)
• For binary relation, aRb means aRb = TRUE
• Binary relation R is an equivalence relation if:
• R is reflexive , if for every a, aRa
• R is symmetric, if for every a and b, aRb implies bRa
• R is transitive, if for every a, b, and c, aRb and bRc implies aRc
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Alphabets
• Alphabet () is defined to be any nonempty finite set
whose members are the symbols of this alphabet.
• Examples:
•  = { 0, 1 }
•  = { a, b, c, d, e, f, g, h, I, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}
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Strings
• A string (w ) over an alphabet is a finite sequence of
symbols from that alphabet.
• Example:
• 101011 is a string over the  = {0,1}
• cat is a string over the  = { a, b, c, d, e, f, g, h, I, j, k, l, m, n, o, p,
q, r, s, t, u, v, w, x, y, z}
• If w is a string over , then the length of w, |w |, is the
number of symbols it contains.
• Example: | 101011 | = 6
• The empty string () is the string of length zero.  ||= 0
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Languages
• Language (L) is a set of strings.
• Language L over alphabet  is a set of strings over 
• Example: Language of even a‟s over the alphabet  = {a, b} is:
• L = { aa, aaaa, aaaaaa, ... , ak, ...} , where k is an even number.
• L = { ak | k is an even number }
• * is the set of all strings over  (i.e. any language L   * ).
• Simple Languages:
• L = All words in the American Heritage Dictionary
• L = *, where  = {a,b}
• L = {x   *:|x| = 3}  {aaa, aab, aba, abb, baa, bab, bba, bbb}
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Complicated Languages
• The set of strings x  {a, b}* such that x has more a‟s than
b‟s.
• The set of strings x  {0, 1}* such that x is the binary
representation of a prime number.
• All „C‟ programs that do not go into an infinite loop.
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Be Careful to Distinguish…
•   The empty string (a string)
•   The empty set (a set, possibly a language)
• {} The set containing one element, which is the empty
string (a language)
• {} The set containing one element, which is the empty
set (a set of sets, maybe a set of languages)
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Automaton
• Automaton is a machine which automatically performs a
range of functions according to a predetermined set of
coded instructions.
• Automata are distinguished by the temporary memory.
Temporary
Memory
Automaton
CPU
Input Memory
Program Memory
Output Memory
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Finite Automata (FA)
• No temporary memory
• Ex: Vending machine with small computing power
Finite Automata
CPU
Input Memory
Program Memory
Output Memory
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Pushdown Automata (PDA)
• Temporary memory is Stack
• Ex: Programming languages (medium computing power)
Stack
Push , Pop
Pushdown Automata
CPU
Input Memory
Program Memory
Output Memory
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Turing Machine
• Temporary memory is Stack
• Ex: Algorithms (highest computing power)
Random Access
Memory
Turing Machine
CPU
Input Memory
Program Memory
Output Memory
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FINITE
AUTOMATA (FA)
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Overview
 Automaton
• a self-operating machine that can move
automatically.
• a machine or control mechanism designed
to follow a precise sequence of
instructions.
 Finite Automata (FA) is a model
of computer with an extremely
limited amount of memory.
 FA devices examples:
• Elevator controller.
• Household appliances controllers,
such as dishwasher and electronic
thermostats.
• Automatic door controller.
Pinocchio
Automaton
(Wikipedia)
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Finite Automata (FA)
• Consider one-way automatic door.
• The door has two pads that can sense when someone is
standing on them, a front and rear pad.
• State Diagram:
Neither
Rear
Both
Front
Rear
Both
Front
closed
open
Neither
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Finite Automata (FA)
• Finite Automata (FA) is the simplest machine that can
recognize an infinite language with a finite number of
states (a finite memory).
• FA is a : “Read once”, “no write” procedure.
• A FA is made up of states and transitions, and as it sees a
symbol or letter of input, it makes a transition to another
state taking the current state and symbol as input.
• The transition function of an FA has two classes:
• Deterministic  DFA
• Nondeterministic  NFA
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Finite Automata (FA)
• The automata receives an input string and produces an
output which is either accept or reject.
• The process proceeds as follows:
1.
2.
3.
4.
Start at the start state of the machine.
Receive symbol by symbol from the input string from left to write.
Read each symbol and make a transition from state to another.
Produce the output after reading the last symbol, accept if the
current state is an accept state and reject otherwise.
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Finite Automata (FA) – State Diagram
• State Diagram of a FA, consists of states and
transitions.
• One of the states is a start state, indicated by an arrow
pointing to it from nowhere.
• One or more states are accept (final) states, indicated by
a double circle.
0
1
1
q1
0
q3
q2
0,1
A finite automaton called Ml that has three states
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Finite Automata (FA) Example
A finite automaton Ml
• What will be the output when feeding Ml with the strings:
“1101” , “0100”, and “101000” ?
• What type of strings that Ml accepts or rejects?
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Formal Definition of FA
• Formal definition resolves any uncertainties about what is
allowed in a FA.
• The formal definition says that a finite automaton is a 5tuple (quintuple) of five objects:
1.
2.
3.
4.
5.
Set of states (Q)
Input alphabet ()
Rules for moving – transition function ()
Start state (q0)
Accept states (F)
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Formal Definition of DFA
• Formal definition of a DFA is:
M = ( Q ,  ,  , q0 , F ) where
Q
is a finite set of states.

is a finite set of symbols, called the alphabet.
 : Q X   Q is a transition function that defines the rules
for moving.
q0  Q is the start state.
F  Q is a set of accept (final) states.
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Questions about DFA
• FA were allowed to have 0 accept states?
• FA must have exactly one transition exiting every state for
each possible input symbol?
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Questions about DFA (Answers)
• FA were allowed to have 0 accept states?
Yes, as F can be the empty set 
• DFA must have exactly one transition exiting every state
for each possible input symbol?
Yes, as  specifies exactly one next state for each
possible combination of a state and an input symbol
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Formal Definition of DFA (Example)
• How can we formally define this automaton?
1.
2.
3.
4.
5.
Q = {q1 , q2 , q3}
 = {0, 1}
 is described as:
q1 is the start state
F = {q2}
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Language Recognized by FA
• Language of a FA machine M is denoted by L(M).
• L(M) is the set A of all string that machine M accepts, i.e.
L(M) = A
• We say that: M recognizes A
• Example:
L(M) = A, where
A = { w | w contains at least one 1 and an even number of
0s follow the last 1}
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Language Recognized by FA
• How many string can a machine accept and how many
language can it recognize?
A machine may accept several strings, but it recognize
only one language.
• What if a machine accepts no string, what language does
it recognize?
The machine recognizes the empty language ()
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DFA Example 1
• Give state diagrams of DFAs recognizing the following
language.
 = {0, 1}
• L1 = {w | w contains the substring 001}
1
q1
0,1
0
0
1
q2
0
q3
1
q4
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DFA Example 2
• Give state diagrams of DFAs recognizing the following
language.
 = {0, 1}
• L2 = {w | w contains either 101 or 11 as a substring}
0
q1
0,1
1
0
1
q2
q4
1
0
q3
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END OF LECTURE!
See you next week…
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