Short Notes on Automata Theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science and discrete mathematics (a subject of study in both mathematics and computer science). The word automata (the plural of automaton) comes from the Greek word αὐτόματα, which means "self-making".
Finite Automata: Deterministic And Non-deterministic Finite Automaton (DFA)Mohammad Ilyas Malik
The term "Automata" is derived from the Greek word "αὐτόματα" which means "self-acting". An automaton (Automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically.
Short Notes on Automata Theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science and discrete mathematics (a subject of study in both mathematics and computer science). The word automata (the plural of automaton) comes from the Greek word αὐτόματα, which means "self-making".
Finite Automata: Deterministic And Non-deterministic Finite Automaton (DFA)Mohammad Ilyas Malik
The term "Automata" is derived from the Greek word "αὐτόματα" which means "self-acting". An automaton (Automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
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1. NORTH WESTERN UNIVERSITY
Computer Science & Engineering
Course Titel: Compiler Design
Course Code: CSE-4103
Presentation on: REGULAR EXPRESSION
Submitted by:
Name : MD. MONIRUL ISLAM
ID: 20151116010
Submitted to:
Name : SAJIB CHATTERJEE
Lecturer
Computer Science & Engineering
North Western University , Khulna
2. INDEX:
* General Introduction
* Equivalence of FA and RE
* DFA to RE: State Elimination
* Converting a RE to an Automata
* Algebra for Regular Expression’s
* References.
3. General Introduction
► Regular expression is an important notation for specifying patterns. Each pattern
matches a set of strings, so regular expressions serve as names for a set of strings.
Programming language tokens can be described by regular languages.[1]
4. R is a regular expression if it is:
1. a for some a in the alphabet , standing for the language {a}
2. ε, standing for the language {ε}
3. Ø, standing for the empty language
4. R1+R2 where R1 and R2 are regular expressions, and + signifies
union (sometimes | is used)
5. R1R2 where R1 and R2 are regular expressions and this signifies
concatenation
6. R* where R is a regular expression and signifies closure
7. (R) where R is a regular expression, then a parenthesized R is
also a regular expression
This definition may seem circular, but 1-3 form the basis
Precedence: Parentheses have the highest precedence, followed by *,
concatenation, and then union.
5. RE Examples
• L(001) = {001}
• L(0+10*) = { 0, 1, 10, 100, 1000, 10000, … }
• L(0*10*) = {1, 01, 10, 010, 0010, …} i.e. {w | w has exactly a single 1}
• L()* = {w | w is a string of even length}
• L((0(0+1))*) = { ε, 00, 01, 0000, 0001, 0100, 0101, …}
• L((0+ε)(1+ ε)) = {ε, 0, 1, 01}
• L(1Ø) = Ø ; concatenating the empty set to any set yields the empty set.
• R ε = R
• R+Ø = R
• Note that R + ε may or may not equal R (we are adding ε to the language)
• Note that RØ will only equal R if R itself is the empty set.
6. Equivalence of FA and RE
Finite Automata and Regular Expressions are equivalent. To show this:
- we can express a DFA as an equivalent RE
- we can express a RE as an ε-NFA. Since the ε-NFA can be converted to
a DFA and the DFA to an NFA, then RE will be equivalent to all the
automata we have described.
7. DFA to RE: State Elimination
Eliminates states of the automaton and replaces the edges with regular
expressions that includes the behavior of the eliminated states.
Eventually we get down to the situation with just a start and final node, and
this is easy to express as a RE
8. State Elimination
Consider the figure below, which shows a generic state s about to be eliminated.
The labels on all edges are regular expressions.
To remove s, we must make labels from each qi to p1 up to pm that include the
paths we could have made through s.
q1
qk
s
p1
pm
.
.
.
.
.
.
R1m
R11
SQ1
QK
Rkm
Rk1
P1
Pm
q1
qk
p1
pm
.
.
.
R11+Q1S*P1
Rkm+QkS*Pm
.
.
.
Rk1+QkS*P1
R1m+Q1S*Pm
9. DFA to RE via State Elimination
Starting with intermediate states and then moving to accepting states, apply the state
elimination process to produce an equivalent automaton with regular expression labels
on the edges.
The result will be a one or two state automaton with a start state and accepting state.
If the two states are different, we will have an automaton that looks
like the following:
Start
S
R
T
U
Figure 01 : (R+SU*T)*SU*
10. If the start state is also an accepting state, then we must also perform a state elimination
from the original automaton that gets rid of every state but the start state. This leaves the
following:
Start
R
Figure 02: R*.
If there are n accepting states, we must repeat the above steps for each accepting states
to get n different regular expressions, R1, R2, … Rn. For each repeat we turn any other
accepting state to non-accepting. The desired regular expression for the automaton is
then the union of each of the n regular expressions: R1 R2… RN
11. DFARE Example
Convert the following to a RE
Figure 03: Convert to RE
First convert the edges to RE’s:
3Start 1 2
1 1
0
0
0,1
3Start 1 2
1 1
0
0
0+1
12. Eliminate State 1:
To: 3Start 1 2
1 1
0
Figure 04: Convert to RE
0+1
3Start 2
11
0+10 0+1
Note edge from 33
Figure 05: (0+10)*11(0+1)*
13. Converting a RE to an Automata
we can convert an automata to a RE. To show equivalence we must also go the other
direction, convert a RE to an automaton.
We can do this easiest by converting a RE to an ε-NFA
-Inductive construction
-Start with a simple basis, use that to build more complex parts of the NFA
Basis:
R=a
R=ε
a
ε
R=Ø
17. Algebraic Laws for RE’s
Just like we have an algebra for arithmetic, we also have an algebra for regular
expressions.
-While there are some similarities to arithmetic algebra, it is a bit different with regular
expressions.
18. Algebra for RE’s
Commutative law for union:
L + M = M + L
Associative law for union:
(L + M) + N = L + (M + N)
Associative law for concatenation:
(LM)N = L(MN)
Note that there is no commutative law for
concatenation,
i.e. LM ML
The identity for union is:
L + Ø = Ø + L = L
The identity for concatenation is:
L ε = ε L = L
The annihilator for concatenation is:
ØL = LØ = Ø
Left distributive law:
L(M + N) = LM + LN
Right distributive law:
(M + N)L = LM + LN
Idempotent law:
L + L = L
19. Laws Involving Closure
(L*)* = L*
i.e. closing an already closed expression does not change the language
Ø* = ε
ε* = ε
L+ = LL* = L*L
more of a definition than a law
L* = L+ + ε
L? = ε + L
more of a definition than a law
20. References
[1] Compiler Design - Regular Expressions ,tutorialspoint.
[2] Wikipedia.
[3] Alfred V. Aho, Monica S. Lam, Ravi Sethi, Jeffrey D. Ullman-Compilers -
Principles, Techniques, and Tools-Pearson_Addison Wesley (2006).