Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Fuzzy Computing
Prof. Dr. Hani Mahdi
Computer and System Engineering Department
Fall 2013
1
Agenda
Motivation for Fuzzy Computation
Basic Concepts
Fuzzy Systems
MOTIVATIONS FOR FUZZY
COMPUTATION
Motivation for Fuzzy Computation
Motivations
Fuzzy Knowle
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Theory of Computation
Prof. Dr. Hani Mahdi
Computer and System Engineering Department
Fall 2013
1
Different References
Essentials of Theoretical Computer Science
http:/www.cs.uky.edu/~lewis/educ.html
Copyrighted, doesn't cost money to read, but
otherwise
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Rough Sets Basics
Prof. Dr. Hani Mahdi
Computer and System Engineering Department
Fall 2013
Very Important References
Chapter 11 Rough Set Theory in
An Introduction to the Mathematics of Uncertainty
by Mark J. Wierman, August 20, 2010
http:/duck.creighto
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Second Quiz
CSE 600  Theoretical Foundation of CSE
BN:
Name in Arabic:
1. Consider the following NFSA:
Date: May 18, 2013
1
0
q0
0,1
0,1
1
q1
q2
0
1
d Give the equivalent Deterministic FSA (the transition table and the state diagram.)
e Put the equival
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Ain Shams University
Theoretical Foundation of Computer and Systems Engineering
Faculty of Engineering
Course Code: CSE600
Computer and Systems Engineering Department
Second Term 2012/2013
Prof. Dr. Eng. Hani M. K. Mahdi
M. Sc. and Ph. D. Course
Date: Jun
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Second Quiz
CSE 600  Theoretical Foundation of CSE
Date: May 17, 2014
BN:
Name in Arabic:
1. Consider the following NFSA: Accepted Language
1
0,1
0
q0
q1
1
a Give the equivalent Deterministic FSA (the transition table and the state diagram.)
b Put the
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Second Quiz
CSE 600  Theoretical Foundation of CSE
Date: December 21, 2013
BN:
Name in Arabic:
1. Construct a Turing Machine to compute the function f(x) = the integer part of (x/2).
The basic idea is to mark the most left 1 and go to the most right 1 (t
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Second Quiz
CSE 600  Theoretical Foundation of CSE
BN:
Name in Arabic:
1. Consider the following NFSA:
Date: May 18, 2013
1
0
q0
0,1
0,1
1
q1
q2
0
1
a Give the equivalent Deterministic FSA (the transition table and the state diagram.)
b Put the equival
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
First Quiz
CSE 600  Theoretical Foundation of CSE
BN:
Name in Arabic:
1. For Odd Numbers p. 368, 371
Date: November 19, 2011
For Even Numbers p. 368, 370,
Answers:
Odd Numbers:
a1 (2 + x 3x 2 8 x 3 ) + a2 (1 + x + x 2 + 5 x 3 ) + a3 (3 4 x 2 7 x 3 ) = 0
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Second Quiz
CSE 600  Theoretical Foundation of CSE
BN:
Name in Arabic:
1. Consider the following FSA:
Date: December 22, 2012
a Reduce the above FSA Draw the state diagram
b Put in Standard Form
c Write the language accepted
aA ABC ABCD ABCD ABCD
All
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
First Quiz
CSE 600  Theoretical Foundation of CSE
Date: March 17, 2012
BN:
Name in Arabic:
1. Determine if the following S is linearly independent
For Odd Numbers
For Even Numbers
p. 368, 370, 371, 372
For Odd Numbers
2 1
0 4
4 2 0 0
a1
+ a2 1 2 + a3
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
First Quiz
CSE 600  Theoretical Foundation of CSE
BN:
Name in
Arabic:
Date: November 3, 2012
3
1. In vector space C , compute the vector representation B (v) for the basis B and vector v
For Odd Bench Numbers
For Even Bench Numbers
2 1 3
11
B = 2 , 3
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
First Quiz
CSE 600  Theoretical Foundation of CSE
Date: November 2, 2013
BN:
Name in Arabic:
1. For the following matrix A
a. Determine the column space and the null space of A. b. State the rank and the nullity of A
4 5 1 1
4 5
2 1 7 7 1 1 4 5 1 1
3 4
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
First Quiz
CSE 600  Theoretical Foundation of CSE
Date: April 6, 2013
BN:
Name in Arabic:
1. Determine whether b is in the column space of A, and if so, express b as a linear combination of the column
vectors.
For Odd Bench Numbers
For Even Bench Numbers
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Logic
Logic is used to represent a simple fact.
Logic defines the ways of putting symbols
together to form sentences that represent
facts.
A Sentence is either Proposition or NOT a
Proposition.
39
Fuzzy Logic (FL)
Fuzzy Logic (FL) is derived from fuzz
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Rough Sets Basics
Prof. Dr. Hani Mahdi
Computer and System Engineering Department
Spring 2013
Very Important References
Chapter 11 Rough Set Theory in
An Introduction to the Mathematics of Uncertainty
by Mark J. Wierman, August 20, 2010
http:/duck.creigh
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
The Approximations Defintions
Let T = (U, A) and let B A and X U .
We can approximate X using only the
information contained in B by constructing
the Blower and Bupper approximations of
X, denoted B X and B X respectively, where
B X = cfw_x  [ x ]B X
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
NONDETERMINISTIC FINITE STATE
MACHINES (NDFSA)
40
NonDeterministic Finite State Machines
Types
Multiple Transitions
Null Transitions
Multiple Initial States
Completing the Nulls
Removing the Nulls
Combining Multiple Transitions
A Worked Example
41
M
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Rough Sets Basics
Prof. Dr. Hani Mahdi
Computer and System Engineering Department
Fall 2012
Very Important References
Chapter 11 Rough Set Theory in
An Introduction to the Mathematics of Uncertainty
by Mark J. Wierman, August 20, 2010
http:/duck.creighto
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Linear Algebra
Prof. Dr. Hani Mahdi
Computer and System Engineering Department
Fall 2013
1
Linear Algebra and its Applications
2
Linear Algebra  Tasks
Systems of Linear Equations
Vectors
Vector Spaces
Eigenvalues
Linear Transformations
Representations
3
Theoretical Foundations of Computer and Systems Engineering
ENGINEERIN 600

Winter 2016
Chapter V
Chapter V Vectors
VO Vector Operations
LC Linear Combinations
SS Spanning Sets
LI Linear Independence
LDS Linear Dependence and Spans
O Orthogonality
44
VO Vector Operations
VEASM Vector Equality, Addition, Scalar
Multiplication
VSP Vect