theory of automata
ASSIGNMENT 1
(FA21-BCS-011) AMNA SOHAIB
(FA21-BCS-014) ARSHIA AZMAT
(FA21-BCS-017) FAHAD KHAN
(FA21-BCS-020) FASIHA ARSHAD
(FA21-BCS-041) MOMINA AMJAD
Theory of automata
Assignment 1
Q1: What is the link of Automata Theory with the Theory of Computation?
Automata theory and the theory of computation are two closely related fields of computer science
that deal with the study of computation and computational devices. The theory of computation
analyzes the computational problems that these devices can solve, whereas automata theory studies
abstract computing devices.
The fundamental theoretical foundation for the study of computation is provided by automata theory.
It provides a framework for understanding the capabilities and limitations of various computing
devices, such as finite-state machines, pushdown automata, and Turing machines. Algorithms that are
able to effectively solve problems are developed by utilizing these abstract models of computation to
evaluate the computational complexity of a variety of issues.
In contrast, the study of the computational problems that can be solved by these abstract computing
devices is the primary focus of the theory of computation. This includes the study of the complexity of
various computational problems, the creation of effective problem-solving algorithms, and algorithm
analysis. In addition, the theory of computation provides a framework for understanding the
connection that exists between various computational problems and the various devices that are
capable of solving them.
To understand the theory of computation, formal languages and automata theory must be studied.
For example, the study of regular languages and finite-state machines is crucial for understanding basic
string manipulation and parsing algorithms, while the study of context-free languages and pushdown
automata is essential for understanding compilers and programming language design.
In conclusion, automata theory and the theory of computation are two closely related fields that are
essential for understanding the fundamental principles of computer science. Automata theory
provides a theoretical foundation for the study of computation, while the theory of computation
focuses on the study of computational problems and the algorithms that can solve them. Both fields
are critical for the design and analysis of efficient algorithms and the development of new computing
technologies.
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Q2: What is the role of Alan Turing in Theory of Automata?
Alan Turing played a significant role in the development of the theory of automata. His work on the
concept of Turing machines, which are abstract models of computation, laid the foundation for the
study of computational complexity and the limits of computability. Turing machines are essentially
mathematical models that can simulate the logic of a computer program and are capable of performing
any computation that can be done by a modern electronic computer.
Turing's work on the theory of automata was not limited to the concept of Turing machines. He also
introduced the idea of a universal machine, which is a machine capable of simulating any other
machine. This concept played a key role in the development of computer science, as it paved the way
for the development of the first electronic computers. In addition to his theoretical work on the theory
of automata, Turing also made significant contributions to the war effort during World War II by
developing the Bombe.
Overall, Alan Turing's contributions to the theory of automata and computer science were
groundbreaking and significant. His work on the concept of Turing machines and the limits of
computability laid the foundation for the study of computational complexity and the development of
modern electronic computers.
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Q3: What are the different applications of Automata Theory? Mention at least 5.
AUTOMATA THEORY
Automata theory is a branch of theoretical computer science that deals with the study of abstract
machines and computational models, which are used in problem solving. Understanding the
fundamental concepts of computing as well as the characteristics and constraints of various
computational models is the core goal of automata theory.
Automata theory includes the study of formal languages, which are sequences of symbols that have a
specific syntax and grammar. These languages can be represented by mathematical models called
automata. There are different types of automata, such as Finite Automata, Pushdown Automata,
Turing Machines, and Cellular Automata. These models vary in their computational power and
complexity and can be used to solve different types of problems in various fields.
APPLICATIONS
Here are five different applications of Automata Theory:
•
COMPILER DESIGN:
Automata Theory is widely used in the design of compilers and interpreters for programming
languages. It is used in the development of two essential components of a compiler:
1. Syntax Analyzer: a parser responsible for checking the correctness of syntax according to
some rules.
2. Lexical Analyzer: a scanner that is used to break down the input source code into a
sequence of tokens.
Use of Automata Theory in compiler design ensures development of efficient lexical and syntax
analysers that can recognize the input source code and produce an error-free parse tree that can
be used to generate the machine code. The efficient use of automata theory in compiler design
helps to produce fast and reliable compilers that can handle large and complex programs.
•
NATURAL LANGUAGE PROCESSING:
Automata Theory plays a crucial role in natural language processing, especially in the areas of
speech recognition and machine translation. It is used to model the structure and syntax of natural
language and to design algorithms for language recognition and translation. It provides a powerful
set of tools for developing models and algorithms for various tasks in natural language processing.
Some specific uses of automata theory in NLP:
1.
2.
3.
4.
Part-of-Speech tagging
Morphological analysis
Sentiment Analysis
Machine Translation
•
SOFTWARE VERIFICATION AND TESTING:
Automata Theory is used to verify and test the correctness and reliability of software systems. It is
used to model software behaviour and to design algorithms for detecting and eliminating errors
and bugs in software systems.
•
DATABASE SYSTEMS:
Automata Theory is used in the design of database systems and data management systems. It is
used to model the structure and behaviour of databases and to design algorithms for data retrieval
and manipulation. It helps in the processes of:
1. Query optimization
2. Transaction Processing
3. Data Mining
4. XML Processing
Automata theory provides a rich set of tools and techniques for analysing, processing, and
managing data in various contexts, and it has important applications in database systems and
related fields.
•
ROBOTICS:
The design of robots and autonomous systems require the use of Automata theory to some extent.
It is used to model the behaviour and decision-making processes of robots and to design
algorithms for robot control and navigation. Some examples include:
1. Planning: represent the possible plans and schedules of the robot, and can help optimize
them based on various criteria, such as time, resource utilization, or energy efficiency.
2. Finite State Machines: used to model the behaviour of robots and other autonomous
systems.
3. Model Checking: verifying correctness and safety of a system.
CONCLUSION
Automata Theory is a very vast concept having several applications in various fields. It provides a
powerful set of tools and techniques for problem solving and has important applications in various
domains of research and development.
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Q4: What Regular Expressions could solve the following real-world problems.
i. Admin can search students through their registration numbers. To match the registration
number with the record a generalized regular expression will help the application. Possible
patterns of registration numbers include FA10-BCS-008, SP11-BSE-001, FA11-BDS-009,
SP20-BCY-087, FA21-BAI-002, FA10-RCS-011, and SP14-PCS-007.
Regular expression:
[SP|FA] [0-9] [0-9] [-] [BCS|BSE|BDS|BCY|BAI|RCS|PCS] [-] [0-9]3
ii. Provide regular expressions for the following.
a) To match a yahoo email address
For a specific yahoo email, the email is supposed to end with ‘@yahoo.com’. The part before
that can consist of numbers, alphabets, and/or some characters such as a period, hyphen, and
an underscore.
[A-Za-z0-9 ._-]* @yahoo.com
b) IP address
As an IP address can be from 0 to 255 not more than that, hence:
zeroTo255 -> ((0-9){1, 2} | (0|1) (0-9){2} | 2[0-4](0-9) | 25[0-5])
first condition means any 1 or 2 digit number from 0-9. Second condition means if the first
digit is 0 or 1, then last two can be any from 0-9. Third condition says that if the first digit is 2,
second must be from 0-4 and third from 0-9. Last condition says that if the first two digits are
25 then the third must be from 0-5. Combining this to make the regular expression for the IP
Address:
zeroTo255 + "\\." + zeroTo255 + "\\." + zeroTo255 + "\\." + zeroTo255
c) HTTP URL’s with .com domain
For a URL, it must start with either http or https and must include ‘://’. After that it can consist
of numbers, alphabets, and special characters. This is for the domain part and the next set of
characters is for the query part. The length of the domain can only be from 2 to 256. After this
a period follows which is followed by 2 to 6 characters part of the top domain. The next set of
characters are for the query.
(http|https)://) [a-zA-Z0-9@:%._\\+~#?&//=]{2,256} \\.[a-z]{2,6} ([-a-zA-Z09@:%._\\+~#?&//=]*)
d) C variable
For a c variable it can only start with an alphabet or an underscore, nothing else but can
consist of numbers also. Hence, the regular expression becomes:
(A-Za-z|_)* (A-Za-z0-9_)*
Q5.
a) How the pacman game can be represented using finite automata (FA)? Consider the below
mentioned case study of the pacman game for designing FA. Mention all the 5-tuples of FA,
draw the state transition table and state diagram for this problem. For those unfamiliar with
the gameplay, Pac-Man requires the player to navigate through a maze, eating pellets and
avoiding the ghosts who chase him through the maze. Occasionally, Pac-Man can turn the
tables on his pursuers by eating a power pellet, which temporarily grants him the power to
eat the ghosts. When this occurs, the ghosts’ behaviour changes, and instead of chasing PacMan they try to avoid him.
The ghosts have following behaviours or exist in these states:
• Randomly wander the maze
• Chase Pac-man, when he is within line of sight
• Flee Pac-man, after Pac-Man has consumed a power pellet
• Return to the central base to regenerate if dead
States of the system are
• Pac-Man eats the Ghost , He will keep finding the dots and power pellets for eating them
• Ghost eats the Pac-Man, Game ends
State transition table
State diagram
b) How can we represent the working of an automatic washing machine using Finite
state automata?
c) What would be the possible accepted strings for the following languages.
i.
Language of all strings having length greater than 3 and prefix bb over
∑={a,b}
L = {bbaa, bbaab, bbab, bbaba, bbabb, bbbaa, bbbab, bbbba, bbbbb, … }
ii.
L = {a n b n | n N},  = {a,b}
L = {ab, aabb, aaabbb, aaaabbbb, aaaaabbbbb, aaaaaabbbbbb, aaaaaaabbbbbbb,
aaaaaaaabbbbbbbb }
iii.
Language of all strings having even number of 0’s and odd number of 1’s
over ∑={0,1}
L = {1, 001, 00111, 00001, 1100111, 10101, 000111011,… }
iv.
Language of all strings having exactly two a’s and more than three b’s over
∑={a,b}
L = {abbba, bababbbb, aabbbbbbbb, bbabbbab, abbbbbbba, … }
v.
Language of PALINDROME over ∑={a,b} with length not greater than 5.
L = { ε, a, b, aa, bb, aba, bab, abba, baab, bbb, … }
d) Prove that for all sets S,
a) (S+ ) * = (S* ) *
S+ = All strings excluding ε
S* = All strings including ε
(S+)* = ε and all strings in S+
= ε and all strings in (except ε, all string)
= ε and all string
= S*
(S*)* = ε and all strings in S*
= ε and all string in (ε and all strings)
= ε and all string
= S*
Hence proved: (S+)* = (S*)*
b) (S+ )+ = S +
(S+)+ = except ε, all string in S+
= except ε, all String in (except ε, all string)
=except ε, all string
=S+
Hence proved (S+)+ = S+
e) Let S = {ab, bb} and T = {ab, bb, bbbb}. Show that S* is a subset of T*
S* = {ε, ab, bb, abab, bbbb, abbbb, bbbbbb, abbbbbb, bbbbbbbb, …}
T* = {ε, ab, bb, bbbb, abbbb, bbbbbb, abbbbbb, bbbbbbbb, abbbbbbbb, …}
Since every element of S* is also an element of T*, all of the strings in S* contain only the
letters "a" and "b", which are also the only letters in T*, we can say that S* is a subset of T*.
Now, show that T* is a subset of S*:
Since every element of T* is also an element of S*, we can observe that every element of T*
can be generated using only the letters "a" and "b", which are also the only letters in S*. Any
element of T* that contains more than one "a" can be obtained by concatenating the string
"ab" with some element of S*, while any element of T* that contains only one "a" can be
obtained by concatenating the string "a" with some element of S*, we can say that T* is a
subset of S*.
Therefore, since S* and T* are both subsets of each other, we can say that S* = T*.
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