EMU, COMPUTER ENGINEERING DEPARTMENT 1999/2000 ACADEMIC YEAR, SPRING SEMESTER

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EMU, COMPUTER ENGINEERING DEPARTMENT1999/2000
ACADEMIC YEAR, SPRING SEMESTER

• C M P E 586
• Software Implementation of Fuzzy Systems
• PREPARED BY DR. KONSTANTIN DEGTIAREV
• FEBRUARY/JUNE 2000
• Slides use the material of books and journal
  papers

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 1
Software Implementation of Fuzzy Systems

• ? ? ? Reference
• G.J.Klir, U.H.St.Clair, Bo Yuan. Fuzzy Set
  Theory. Foundations Applications, Prentice Hall
  PTR, 1997
• B.Kosko. Fuzzy Engineering, Prentice Hall, 1997
• T.J.Ross. Fuzzy Logic with Engineering
  Applications, McGraw-Hill, 1995
• J.Yen, R.Langari. Fuzzy Logic. Intelligence,
  Control, and Information, Prentice Hall, 1999
• L.-X.Wang. A Course in Fuzzy Systems and Control,
  Prentice Hall, 1997
• W.Pedrycz (ed.). Fuzzy Modelling. Paradigms and
  Practice (Int. Series in Intelligent
  Technologies), Kluwer Academic Publ., 1996
• J.Yen. Fuzzy Logic - A Modern Perspective // IEEE
  Transactions on Knowledge and Data Engineering,
  vol.11, 1, January/February 1999
• L.A.Zadeh. The Birth and Evolution of Fuzzy Logic
  // Int. Journal on General
  Systems, vol.17, 1990, pp.95-105

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 2
Software Implementation of Fuzzy Systems

• ? ? ? Section1 Outline
• Introduction. Uncertainty, Imprecission and
  Vagueness
• Fuzzy Systems. Brief History of Fuzzy Logic.
  Foundation of Fuzzy Theory.
• Fuzzy Sets and Systems. Fuzzy Systems in
  Commercial Products
• Research fields in Fuzzy Theory
• the discussion of these topics takes
  approximately 4 lecture hours. One example is
  explained (CubiCalc? and fuzzyTECH? software
  packages are used)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 3
Software Implementation of Fuzzy Systems

• Most of the phenomena we encounter everyday are
  imprecise - the imprecision may be associated
  with their shapes, position, color, texture,
  semantics that describe what they are
• Fuzziness primarily describes uncertainty
  (partial truth) and imprecision
• The key idea of fuzziness comes from the
  multivalued logic Everything is a matter of
  degree
• Imprecision raises in several faces, e.g. as a
  semantic ambiguity

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 4
Software Implementation of Fuzzy Systems

• By fuzzifying crisp data obtained from
  measurements, FL enhances the robustness of a
  system
• Imprecision raises in several faces - for
  example, as a semantic ambiguity
• the statement the soup is HOT is ambiguous,
  but not fuzzy
• e.g. 20º,80º

Definition of the domain of discourse
Transaction to Fuzziness
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 5
Software Implementation of Fuzzy Systems

• The word fuzzy can be defined as imprecisely
  defined, confused, vague
• Humans represent and manage natural language
  terms (data) which are vague. Almost all answers
  to questions raised in everyday life are within
  some proximity of the absolute truth

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 6
Software Implementation of Fuzzy Systems

• Probability theory is one of the most traditional
  theories for representing uncertainty in
  mathematical models
• Nature of uncertainty in a problem is a point
  which should be clearly recognized by engineer -
  there is uncertainty that arises from chance,
  from imprecision, from a lack of knowledge, from
  vagueness, from randomness
• probability theory deals with the expectation of
  an event (future event, its outcome is not known
  yet), i.e. it is a theory of random events

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 7
Software Implementation of Fuzzy Systems

• Fuzziness deals with the impression of meaning of
  concepts expressed in natural language - it is
  not concerned with events at all
• Fuzzy theory handles nonrandom uncertainty

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 8
Software Implementation of Fuzzy Systems

• As it is stated by L.Zadeh, in many cases there
  is more to be gained from cooperation than from
  arguments over which methodology is best
• Many situations cover both kinds of uncertainty
• assume the weather forecast - tomorrow slight
  rains are highly probable

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 9
Software Implementation of Fuzzy Systems

• The principle of incompatibility (L.Zadeh, 1973)
• As the complexity of a system increases, our
  ability to make precise and yet significant
  statements about its behavior diminishes until a
  threshold is reached beyond which precision and
  significance (or relevance) become almost
  mutually exclusive characteristics

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 10
Software Implementation of Fuzzy Systems

• Intimate connection between fuzziness and
  complexity (L.A.Zadeh)
• a new approach to system analysis approximate
  and yet effective means of describing the
  behavior of systems which are too complex or too
  ill-defined to admit of precise mathematical
  analysis

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 11
Software Implementation of Fuzzy Systems

• A new approach to system analysis a departure
  from the conventional quantitative techniques of
  system analysis
• A new paradigm to develop approximate solutions
  that are both cost-effective and highly useful
• a Fuzzy System (FS) is defined as a system with
  operating principles based on fuzzy information
  processing and decision making
• There are several ways to represent knowledge,
  but the most commonly used has a form of rules
• IF (premise)A THEN (conclusion)B

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 12
Software Implementation of Fuzzy Systems

• From a knowledge representation viewpoint, a
  fuzzy IF-THEN rule is a scheme for capturing
  knowledge that involves imprecision - if we know
  a premise (fact), then we can infer another fact
  (conclusion)
• A fuzzy system (FS) is constructed from a
  collection of fuzzy IF-THEN rules
• Acquisition of knowledge captured in IF-THEN
  rules is NOT a trivial task (expert knowledge,
  systems measurements, etc.)
• The building blocks for fuzzy IF-THEN
  rules are FUZZY SETS

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 13
Software Implementation of Fuzzy Systems

• The rule
• IF the air is cool THEN set the motor speed to
  slow
• has a form
• IF x is A THEN y is B,
• where fuzzy sets cool and slow are labeled
  by A and B, correspondingly
• A and B characterize fuzzy propositions about
  variables x and y
• Most of the information involved in human
  communication uses natural language terms that
  are often vague, imprecise, ambiguous by their
  nature, and fuzzy sets can serve as the
  mathematical foundation of natural language

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 14
Software Implementation of Fuzzy Systems

• A Fuzzy Set is a set with a smooth boundaries
• Fuzzy Set Theory generalizes classical set
  theory to allow partial membership
• Fuzzy Set A is a universal set U is determined by
  a membership function ?A(x) that assigns to each
  element x?U a number A(x) in the unit interval
  0,1
• Universal set U (Universe of Discourse) contains
  all possible elements of concern for a particular
  application
• Fuzzy set has a one-to-one correspondence
  with its membership function

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 15
Software Implementation of Fuzzy Systems

• Fuzzy set A is defined as
• A (x, A(x)) , x?U, A(x)?0,1
• A(x) Degree(x?A) is a grade of membership of
  element x?U in set A

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 16
Software Implementation of Fuzzy Systems

• The membership functions themselves are NOT fuzzy
  - they are precise mathematical functions once a
  fuzzy property is represented by a membership
  function, nothing is fuzzy anymore
• Suppose U is the interval 0,85 representing the
  age of ordinary human beings, and the linguistic
  term young as a function of age (value of the
  variable age) can be defined as
• see the graphical representation on the next
  slide
• !! pay attention to the usage of the symbol
  /

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 17
Software Implementation of Fuzzy Systems

• If U is a set of integers from 1 to 10 (
  U1,2,,10 ), then small is a fuzzy subset of
  U, and it can be defined using enumeration
  (summation notation)
• A small 1/11/20.85/30.75/40.5/50.3/60.
  1/7

Universe of discourse U is continuos
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 18
Software Implementation of Fuzzy Systems

• In the previous example elements of U (universal
  set) with zero membership degrees are not
  included into enumeration
• A notion of a fuzzy set provides a convenient way
  of defining abstraction - a process which plays a
  basic role in human thinking and communication
• All theories that use the basic concept of fuzzy
  set can be called in a whole Fuzzy Theory
• Rough classification of Fuzzy Theory can be
  depicted as follows note that dependencies
  between the branches are not shown

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 19
Software Implementation of Fuzzy Systems
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 20
Software Implementation of Fuzzy Systems

• The idea of Fuzzy Sets appeared in 1964
  L.A.Zadeh (Professor of the University of
  California at Berkeley) We need a radically
  different kind of mathematics, the mathematics of
  fuzzy or cloudy quantities which are not
  described in terms of probability distributions
• The paper Fuzzy Sets (Zadeh L.A., Information
  and Control, vol.8, pp.338-353, 1965) first used
  the word fuzzy to mean vague in technical
  literature
• criticized by academic community the idea caused
  a development of fuzzy set theory foundation
  (1965-1980)
• academic research work stimulates first
  industrial applications of fuzzy systems
  (1977-1990) - cement kiln controller (Denmark),
  train control system (Sendai subway, Japan),
  digital and analog fuzzy chips (USA, Japan)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 21
Software Implementation of Fuzzy Systems

• Currently, the application fields of fuzzy
  systems cover signal processing, communications,
  expert systems, medicine, business/finance,
  control (industrial processes and consumer
  electronics),
• widening of collaboration between universities
  and industry, fuzzy boom (1987-present) Japan
  ? Europe ? USA
• 1992 1st IEEE International Conference on Fuzzy
  Systems
• appearance of software companies (INFORM,
  Aptronix,etc.)
• Fuzzy Logic Toolbox for MATLAB was released in
  1994
• Courses on fuzzy sets and systems in Universities
  curricula
• Engineering consists largely of recommending
  decisions based on insufficient information....
  It is essential that these students be exposed to
  ways of treating uncertainty and vagueness. This
  also requires that existing faculty utilize these
  methods
• (Colin Brown, conference of NAFIPS)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 22
Software Implementation of Fuzzy Systems

• Appearance of the new computational paradigms and
  intensification of research in certain areas
  (genetic algorithms/evolutionary strategies,
  neural networks)
• L.A.Zadeh introduced a term soft computing (1992)
• ------------ EXAMPLE 1 -------------
• Fuzzy Toolbox Demo (MATLAB)
• by Dr.R.Babuška (Delft University of Technology,
  The Netherlands)
• If-Then Rules. Fuzzy reasoning (example)
• Word 97 document (preliminary explanations)
• end of the Section 1

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 23
Software Implementation of Fuzzy Systems

• ? ? ? Section 2 Outline
• Mathematical Background of Fuzzy Systems.
  Classical (crisp) vs. Fuzzy Sets. Representation
  of Fuzzy Sets
• Types of Membership Functions. Basic concepts
  (support, singleton, height, ?-cut, convexity).
  Fuzzy Set Operations
• S- and T-norms. Properties of Fuzzy Sets. Sets as
  points in Hypercubes. Cartesian Product. Crisp
  and Fuzzy Relations
• Linguistic variables and hedges. Membership
  function design (shape analysis)
• the discussion of these topics takes
  approximately 10 lecture hours. Examples are
  explained using CubiCalc, fuzzyTECH and FL
  Toolbox packages

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 24
Software Implementation of Fuzzy Systems

• Fuzzy Systems F ?n ? ?p use m rules to map
  vector input x to vector or scalar outputs F(x)
• Fuzzy (Rule-based) Systems make use of linguistic
  variables in their antecedents and consequents
• Linguistic variables can be naturally represented
  by fuzzy sets and logical connectives of these
  sets

input X
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 25
Software Implementation of Fuzzy Systems

• A classical (crisp) set A in the universe of
  discourse U can be defined in three ways
• - by enumerating (listing) elements (often
  called list or extensional definition)
• - by specifying the common properties of
  elements (intensional or rule definition)
• the notation A x P(x) means that set A is
  composed of elements x such that every x has the
  property P(x)
• - by introducing a zero-one membership function
  (characteristic or indicator definition)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 26
Software Implementation of Fuzzy Systems

• Crisp set is a set with precise boundary, and
  classical set theory is founded on the idea that
  we can make crisp, exact distinctions between two
  groups, i.e. between those individuals (elements)
  that are definitely in the result set (group 1),
  and those that are definitely outside it (group
  2)
• The basic operations on classical sets (A and B
  are crisp sets in the universe of discourse U)
• complement divides the universal set U into
• 2 (two) parts

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 27
Software Implementation of Fuzzy Systems

• Fundamental properties of the basic operations
  (these properties are also encountered in
  propositional logic)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 28
Software Implementation of Fuzzy Systems

• Fundamental properties satisfy to a principle of
  duality replacing of empty set, U, ?, ? with
  U, empty set, ?, ?, respectively, brings again
  valid property
• The notion of membership in fuzzy sets becomes a
  matter of degree (number in the closed interval
  0,1)
• Membership of an element from the universe in
  fuzzy set is measured by a function that attempts
  to describe vagueness and ambiguity

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 29
Software Implementation of Fuzzy Systems

• Membership functions can be represented (a)
  graphically, (b) in a tabular or list form, (c )
  analytically and (d) geometrically (as a points
  in the unit cube)
• Geometrical representation for two-element
  universal set U (x1,x2) has a following
  vizualization

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 30
Software Implementation of Fuzzy Systems

• see the previous figure Vertices (0,0),
  (0,1), (1,0) and (1,1) represent all crisp sets
  that can be defined for the universal set U, e.g.
  the point (1,0) corresponds to the crisp set x1
  (element x2 has no membership)
• Membership functions can be symmetrical or
  asymmetrical, and the most commonly used forms
  are triangular, trapezoidal, Gaussian and bell
  (the first two dominate in applications due to
  simplicity and computational efficiency)
• Membership functions are typically defined on
  one-dimensional universes, and in most cases, the
  membership function appears in the continuos form
• Fuzzy Toolbox Demo (MATLAB)
• by Dr.R.Babuška (Delft University of Technology)
• FuzzyTECH and CubiCalc (explanations)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 31
Software Implementation of Fuzzy Systems

• The height of a fuzzy set A is the highest
  (maximum) value of its membership function, i.e.
  height(A)
• If a fuzzy set has a height 1, then it is called
  a normal fuzzy set in contrast, if height(A) lt
  1, the fuzzy set is said to be subnormal
• A subnormal set is a fuzzy set that contains only
  elements with partial (lt1) membership
• In most of applications fuzzy sets are normal,
  and during the reasoning process usually
  subnormal fuzzy sets are generated
• A set of all elements of the universal set U
  whose degree of membership in a fuzzy set A is
  nonzero is called the support of a fuzzy set A,
  i.e. supp(A)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 32
Software Implementation of Fuzzy Systems

• A set of all elements x of the universal set U
  with a property ?A(x) 1 (A is a fuzzy set) is
  called the core of a fuzzy set A (core(A))

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 33
Software Implementation of Fuzzy Systems

• A fuzzy set whose support is a single point in
  the universe of discourse U is called a fuzzy
  singleton
• Each fuzzy set A is associated with a family of
  crisp subsets of A - their elements have such
  membership degrees that they are restricted to a
  crisp subset of 0,1
• A crisp set A? that contains those x?U for which
  is called an ?-cut of a fuzzy
  set A
• The general property of ?-cuts for any fuzzy
  set A and two values ?1, ?2 ?0,1 that satisfy
  to the condition ?1lt ?2 the following is true

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 34
Software Implementation of Fuzzy Systems

• Fuzzy sets may be completely characterized by
  their ?-cuts
• (decomposition
  theorem of fuzzy sets)
• Example (Lecture
  hours)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 35
Software Implementation of Fuzzy Systems

• Consider a fuzzy set A which is represented
  analytically in the universe of discourse U
  5,15 as follows

Example
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 36
Software Implementation of Fuzzy Systems

• step 1 Several particular values of ? are
  chosen from the unit interval 0,1 - they are
  0.1, 0.3, 0.5, 0.7 and 0.9
• step 2 converting each of the ?-cuts A? to
  fuzzy sets for each x?U using the formula
• (fuz_set?) ??A?(x)

Sometimes the theorem is referred as resolution
principle (approximate representation of
membership function)
Example
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 37
Software Implementation of Fuzzy Systems

• Fuzzy set A is convex if for any elements x1, x2
  and x3 from the universal set U, the relation x1lt
  x2lt x3 implies that
• General property the intersection of two convex
  sets produces a convex set
• ? Convexity and ?-cuts

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 38
Software Implementation of Fuzzy Systems

• Generalization of set operations to fuzzy sets is
  not obvious
• Operations on fuzzy sets are crucial to the fuzzy
  inference process
• In the rule IF (A or B) THEN C the true value
  of C is the true value of the disjunction
  (operation or)
• Assume two fuzzy sets A and B are defined on the
  universe of discourse U - three basic operations
  can be represented as follows
• Fuzzy set A is equal to fuzzy set B if and only
  if
• ?A(x) ?B(x), ?x?U

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 39
Software Implementation of Fuzzy Systems

• Fuzzy sets overlap with their complements (an
  element
• may partially belong to both fuzzy set and
  sets complement)
• In contrast, classical (crisp) sets never
  overlap with their
• complements

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 40
Software Implementation of Fuzzy Systems

• Two fundamental laws of Classical Set theory -
  law of Excluded Middle and
  law of Contradiction
• ? are violated in Fuzzy Set
  Theory (!!)
• Standard fuzzy operations are quite adequate in
  many practical applications of FS, but they do
  not utilize the real expressive power of fuzzy
  sets (what are the other possibilities that may
  satisfy the requirements of practice? )
• In practice, algebraic sum (1) and algebraic
  product (2) are used for a definition of union
  and intersection of two fuzzy sets, respectively

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 41
Software Implementation of Fuzzy Systems

• General notation
• Operator s is called an s-norm if it satisfies to
  the following axioms for any x, y, z and w
  ?0,1
• Some of the operators (s-norms) that model
  (i.e. extend) fuzzy union

see the next slide...
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 42
Software Implementation of Fuzzy Systems

• (S1) Drastic sum
• (S2) Hamacher sum
• (S3) Dubois-Prade class
• (S4) Yager class
• Note for arbitrary fuzzy sets A and B
  membership values x and y stand for ?A(x) and
  ?B(x), correspondingly

Continuation t-norms (triangular norms)
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 43
Software Implementation of Fuzzy Systems

• Operator t is called an t-norm (triangular norm)
  if it satisfies to the following axioms for any
  x, y, z and w ?0,1
• Some of the operators (t-norms) that model
  (extend) fuzzy intersection
• (T1) Drastic product
• (T2) Hamacher product

More still to come...
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 44
Software Implementation of Fuzzy Systems

• (T3) Dubois-Prade class
• (T4) Yager class
• Self-studying exercise Prove that the Yager
  t-norm (class T4) converges to the min operator
  when the parameter ? is in the infinite limit
• An important properties of s-norms and t-norms
  can be summarized as follows

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 45
Software Implementation of Fuzzy Systems

• s-norms are bounded below by max (standard fuzzy
  union) and bounded above by drastic sum (S1)
• t-norms are bounded below by drastic product (T1)
  and bounded above by min (standard fuzzy
  intersection)
• s-norms (a set of fuzzy disjunction operators)
  are often called triangular conorms or shortly,
  t-conorms
• The alternative forms of operators AND and OR are
  called compensatory operators (they compensate
  the
• strictness of min and max operators proposed
  by
• L.A.Zadeh)

?
?
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 46
Software Implementation of Fuzzy Systems

• Operator c is called a fuzzy complement if it
  satisfies to the following axioms for any x and y
  ?0,1
• Some of the operators that model (extend) fuzzy
  complement
• (C1) Sugenos complement
• (C2) Yagers complement
• Demonstration (MATLAB environment)

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 47
Software Implementation of Fuzzy Systems

• Main types of membership functions (MF)
• (a) Triangular MF is specified by 3 parameters
  a,b,c
• (b) Trapezoidal MF is specified by 4 parameters
  a,b,c,d

More to come...
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 48
Software Implementation of Fuzzy Systems

• (c) Gaussian MF is specified by 2 parameters
  a,?
• (d) Bell-shaped MF is specified by 3 parameters
  a,b,?
• (e) Sigmoidal MF is specified by 2 parameters
  a,b

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 49
Software Implementation of Fuzzy Systems
Image form Neuro-Fuzzy and Soft Computing
(J.-S.R.Jang, C.-T.Sun, E.Mizutanani -
supplementary slides
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 50
Software Implementation of Fuzzy Systems

• Assume X and Y are two arbitrary classical sets.
  The Cartesian product of sets X and Y is a set of
  all ordered pairs (xi,yj), xi?X, yj?Y that is
• Suppose X x1, x2, x3, x4 , Y y1, y2, y3
  the set XxY consists of 12 ordered pairs
  (xi,yj), i1,2,3,4, j1,2,3 - in this case, a
  graphical representation (as nodes of a grid) is
  convenient
• Generalization of Cartesian product to n
  arbitrary classical sets X1, X2,, Xn

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 51
Software Implementation of Fuzzy Systems
If the cardinality of the set X is n(X) and the
cardinality of the set Y is n(Y), then the
cardinality of the Cartesian product (set of
elements) is n(XxY) n(X)n(Y)
See the previous slide...
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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 52
Software Implementation of Fuzzy Systems

• Omitting superscripts, an ordered sequence of n
  elements (x1, x2, , xn) is called an ordered
  n-tuple
• A subset of the Cartesian product
  is called an n-ary relation built
  over domains (sets) X1,X2,, Xn - if n2, the
  binary relation on X1 and X2 (from X1 to X2) can
  be formally defined as a set of ordered pairs in
  X1xX2 that is
• where P(x1,x2) is a property to which each pair
  (x1,x2)?? satisfies
• Example Suppose that both X1 and X2 are sets
  of real
• numbers ?5,20, i.e. X1
  X2 ?5,20. The binary relation ?
  ? ?(X1,X2) less than has

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CMPE 586
Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 53
Software Implementation of Fuzzy Systems

• the following formal analytical representation
• Graphical form of the binary relation ?
• A binary relation ? can be also represented by

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Software Implementation of Fuzzy Systems

• means of membership function
• (arbitrary n-ary relation is a mapping
• ?(X1,X2,, Xn) X1xX2x xXn ? 0,1 )
• If a set X1xX2 is finite, then the values of
  function ?? can be collected into a relational
  matrix
• Relations are intimately involved in logic,
  approximate reasoning, rule-based systems, etc.
• A rule IF x is A THEN y is B describes a
  relation between the variables x and y - as
  implication A ? B, rule expresses a
  mapping (subset of Cartesian product)
  between input and output domains

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 55
Software Implementation of Fuzzy Systems

• A fuzzy relation generalizes the concept of
  classical (crisp) relation introducing a degree
  of membership for each ordered n-tuple (x1,
  x2,, xn) in
• For 2D case it can be defined as follows
• Examples
• a) x1 is close to x2 (both x1 and x2 are
  numbers)
• b) if x1 is medium, then x2 is high (x1 is an
  observed state,
• whereas x2 is a result state or action)
• c) x1 is similar to x2 (x1 and x2 can be
  objects, human beings, properties)

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 56
Software Implementation of Fuzzy Systems

• Formally, fuzzy relation in
  can be defined as a fuzzy set
• where is a mapping
• In different sources notations and can
  be used for crisp and fuzzy relations,
  correspondingly (if it is clear from the contents
  which of two relations is used, sign can be
  dropped)
• means membership
  degree of the ordered n-tuple in the fuzzy
  relation , where
• , or a degree to
  which fuzzy relation holds true for objects
  
• Example
  (Lecture hours)

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 57
Software Implementation of Fuzzy Systems

• Suppose that X1 and X2 are line segments 0,50
  and 20,40, respectively, on the set of real
  numbers . A fuzzy relation
  x1 is approximately equal to x2 may be
  defined by the membership function

Example
Fuzzy relations enhance our capability to deal
with relational concepts expressed in a natural
language !
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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 58
Software Implementation of Fuzzy Systems

• A fuzzy relation x1 is much larger than x2 may
  be defined by the membership function

Membership values line segment 0,1
Fuzzy relations are also fuzzy sets, and
fundamental properties of fuzzy sets hold for
fuzzy relations as well
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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 59
Software Implementation of Fuzzy Systems

• Fuzzy set operations (complements, unions,
  intersections) are applicable to fuzzy relations
  too

Sugenos complement (parameter ? 2)
Standard complement
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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 60
Software Implementation of Fuzzy Systems

• Operations on fuzzy sets defined on different
  universal domains produce a multidimensional
  fuzzy set (the following shows graphically the
  result of operation )

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Software Implementation of Fuzzy Systems
IF x is Ai ...
Example (Lecture hours)
THEN y is Bi
Contour plot
(MATLAB 5.2 environment)
Fuzzy sets Ai and Bi have Gaussian and
bell-shaped forms
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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 62
Software Implementation of Fuzzy Systems

• A fuzzy rule is described formally by a fuzzy
  relation between antecedent and consequent, and
  it defines a partition (fuzzy relation ) in
  the space (Cartesian product). The union
  of all partitions
• forms a fuzzy graph (term introduced by
  L.A.Zadeh) of a fuzzy model. The more partitions
  (patches) the more accurate description of the
  functional dependency Yf(X), where f is a crisp
  function
• Fuzzy system approximates a crisp function f by
  means of patches the approximation is uniform,
  and it allows specification of error level
  (accuracy) in advance

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Software Implementation of Fuzzy Systems

• A patch covering leads to major serious problem
  of fuzzy systems exponential explosion of rules
  (number of rules) as a result of increasing the
  number of input variables
• Fuzzy Approximation Theorem (B.Kosko, L.A.Zadeh)

A function can be approximated to any prescribed
accuracy provided that sufficient fuzzy rules
are available S.Wu and M.J.Er
References 1. L.Wang. Fuzzy Systems are
Universal Approximators // Proc.
Int.Conf.Fuzzy Syst., 1992
2. B.Kosko. Fuzzy Systems as Universal
Approximators // IEEE Transactions on
Computers, vol.43, 11, 1994
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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 64
Software Implementation of Fuzzy Systems

• In a framework of fuzzy rules as conjunctions (IF
  x is Ai THEN y is Bi ? Ai ? Bi ? Ai ? Bi
• (functional relationship between input and
  output on a base of linguistic terms, i.e. Ai and
  Bi, i 1,n)
• (A) Mamdani implication
• (correlation-minimum)
• (B) Larsen implication
• (correlation-product dilution of
  membership values that are both small)
• 
  
  cases (A) and (B)
• Compound fuzzy propositions (e.g. x1 is Ai1 and
• x2 is Ai2) are interpreted as fuzzy
  relations

Demonstration (MATLAB environment)
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Software Implementation of Fuzzy Systems

• (1) x1 is Ai1 and x2 is Ai2
  ,
• where t 0,1?0,1 ? 0,1 is a t-norm
• (2) x1 is Ai1 or x2 is Ai2
  ,
• where s 0,1?0,1 ? 0,1 is a s-norm
• As an implication, fuzzy rules represents human
  abilities of imprecise reasoning
• Fuzzy rule is an implication between fuzzy
  propositions
• IF ltfuzzy propositiongtAi THEN ltfuzzy
  propositiongtBi (Ai ? Bi )
• Fuzzy rule describes an entailment of Bi by Ai
  (Ai entails Bi)
• Fuzzy logic can be considered as a generalization
  
• of a classical binary and multivalued logic

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 66
Software Implementation of Fuzzy Systems

• The foundation of a fuzzy implication rule is
  the narrow sense of fuzzy logic
• In the classical propositional calculus (algebra
  of propositions), implication P?Q, where P and Q
  are two simple propositions, is logically
  equivalent (gives the same truth table values) to
• (1) and
• (2)
• As Ai and (or) Bi have fuzzy predicates,
• the implication becomes a fuzzy implication
• Two-valued logic

forms (1) and (2)
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2000 Slide 67
Software Implementation of Fuzzy Systems

• For the crisp propositions P and Q the
  implication P?Q becomes global, i.e. the truth
  table covers all possible combinations of 0/1
  values of P and Q)
• Fuzzy implication can be considered
  as a local one, i.e. it gives a large truth value
  only when both and have large truth
  values
• Having a rule IF x is Ai THEN y is Bi, we may
  propose a following interpretation
• IF x is Ai THEN y is Bi ELSE ltnothinggt
• (each rule covers a local part of the whole
  working space)
• Implication as a conjunction Ai ? Bi ? Ai ? Bi
  (Mamdani (min) and Larsen (product) slide 64
  implications are the most widely used in fuzzy
  systems and fuzzy control)

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2000 Slide 68
Software Implementation of Fuzzy Systems

• Fuzzy rules can be considered as not being local
  as well, and this fact opens a way to
  classification of fuzzy rules as
• (A) fuzzy mapping rules (FMR)
• (B) fuzzy implication rules (FIR)
• Case (A) FMR describe a functional dependency
  between systems inputs and outputs by means of
  linguistic terms (a totality of rules is
  represented by a fuzzy graph). A collection of
  fuzzy mapping rules are often called fuzzy model
• Case (B) FIR describe an implication logic
  relationship between two fuzzy propositions that
  use linguistic terms and hedges (generalization
  of two-valued logic)

similarities

• Both types of rules
• are represented as a fuzzy relations between
  antecedent and consequent parts
• use compositional rule of inference as an
  inference engine

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Software Implementation of Fuzzy Systems
differences

• 
  !
• A major stages in fuzzy modeling are as follows
• 1. fuzzy partition
• 2. mapping of regions to local models
• 3. fusing of local models into a global model
• 4. defuzzification (quantitative summary of
  possibility
  distribution of models output)
• The representatives of fuzzy implication families
  (FI) are shown on the next slide (form FI1 is
  also refered in
• the literature as Dienes-Rescher implication)

• But
• the semantics of relations is different
• different operators are used in compositional
  rule of inference
• two types of rules behave the same in the case
  when antecedent parts of rules are satisfied, but
  the behavior is different if antecedents are not
  satisfied

Fuzzy Rule-Based Modeling
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Prepared Dr.Konstantin Degtiarev, February-June
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Software Implementation of Fuzzy Systems

• Selected fuzzy implication (FI) operators
• (FI1) Zadehs classical maximum FI
• (FI1) Zadehs classical maximum FI
• (FI2) Lukasiewicz implication
• (FI3) Godelian FI
• (FI4) Standard sequence implication
• (it is shown
  on the next slide)

equivalent to (FI1) when
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Software Implementation of Fuzzy Systems

• Standard sequence implication
• Although some have been more extensively used,
  there is no such thing as the fuzzy implication
  operator and any practical application should
  consider different alternatives, checking the
  effectiveness of each one (A.Kandel, R.Pacheco,
  A.Martins, S.Khator. The foundations of
  rule-based computations in fuzzy models )
• (fuzzy implication operators (1) Zadehs, (2)
  Lukasiewicz, (3) Godelian)
• PS. Godelian fuzzy implication is also referred
  in the literature
• as Brouwerian implication )

Demonstration (MATLAB environment)
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Software Implementation of Fuzzy Systems

• The relationship between implications
• Two questions of interest
• (1) What are the criteria for choosing a
  combination of fuzzy operators ?, ? and ? in
  order to obtain a certain form of fuzzy
  implication? For example, Lukasiewicz
  implication (FI2) is a member of a family that
  generalizes a material implication of a classical
  logic (disjunction ?) Yagers s-norm with ?
  equal to 1 and (complement ?) standard
  complementation 1-
• (2) For the fuzzy implication what
  we mean
• by equivalence to and
  ?

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 73
Software Implementation of Fuzzy Systems

• When a fuzzy relation is NOT finite, and it is
  defined in n-dimensional space, we have to
  specify it analytically (through appropriate
  formula)
• A fuzzy relation on a finite Cartesian product
  X1xX2 is usually represented by a fuzzy
  relational matrix with elements taken from 0,1
• Example Assume X1 a1, a2, a3 , X2 b1,
  b2, b3, b4
• are two sets of cities.
  The relational concept close (distance
  expressed in Km) is represented by the
  (3,4)-matrix

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 74
Software Implementation of Fuzzy Systems

• Two kinds of questions we can keep in mind
  ???
• a) what is the degree that particular pair of
  cities are considered to be close to each other?
  (membership function of a fuzzy set)
• b) what is a possibility that a short distance
  (closeness to each other) corresponds to a
  specific pair of city ai ( ) and city bj
  ( )? (possibility distribution of a
  short distance (closeness))
• Composition of relations
• Composition of two crisp binary relations R1 and
  R2 requires their compatibility

Crisp case
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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 75
Software Implementation of Fuzzy Systems

• Relation TR1R2 (composition of two crisp
  relations) consists of those pairs (x,z) ,x?X,
  z?Z, of the Cartesian product XxZ that via the
  given relations R1 and R2 share at least one
  element y?Y
• Example Assume that X 20,40 , Y 0,50
  ,
• Z 10,40 X,Y,Z ? ?
  (set of real numbers). Two crisp relations R1 ?
  XxY and R2 ? YxZ are defined (shown on the next
  slide), and it is required to find their
  composition
• 
  - relation T
• (as a result of compositional operation) is a
  subset
• of the Cartesian product XxZ

(see the next slide for the graphical
representation)
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Software Implementation of Fuzzy Systems
Cartesian product XxY
line zy
relation R2 ? YxZ
Both relations are defined in two-dimensional
Euclidean space
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Prepared Dr.Konstantin Degtiarev, February-June
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Software Implementation of Fuzzy Systems

• Two common forms of composition operation
• a) max-min composition
• or in terms of 0/1 membership functions
• b) max-product (in general, max-star)
  composition
• For example, max-product composition has a
  following form

sign ? denotes any t-norm
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Software Implementation of Fuzzy Systems

• It is stressed that max-min method of
  composition effectively expresses the approximate
  and interpolative reasoning used by humans when
  they employ linguistic propositions for deductive
  reasoning (T.Ross. Fuzzy Logic with Engineering
  Applications, McGraw-Hill, 1995 N.Vadiee.
  Fuzzy rule based expert systems I, 1993)
• A main goal of fuzzy logic is to form a
  foundation for reasoning (inference) with
  imprecise propositions such reasoning is called
  approximate reasoning
• Consider a given rule IF x is A THEN y is B,
  where A and B are fuzzy sets (fuzzy propositions,
  fuzzy predicates), and a fact x is A (A and A
  are not necessarily identical). The result
  produced by fuzzy inference engine
• y is B A?R, where R is a fuzzy relation
  which
• represents an implication (x is A) ? (y is B)

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Prepared Dr.Konstantin Degtiarev, February-June
2000 Slide 79
Software Implementation of Fuzzy Systems

• An ordinary-language term represented by a fuzzy
  set is called linguistic value
• A linguistic variable can be considered as a
  composition of a symbolic variable and a numeric
  variable
• Linguistic variable is a fundamental element in
  human knowledge representation
• Transition from crisp mathematics to fuzzy
  mathematics by means of fuzzy set theory has
  allowed mathematical representations to become
  compatible with expressions in natural language
• Linguistic hedges are special linguistic terms by
  which other (primary) linguistic terms are
  modified
• Linguistic hedge (modifier) may be interpreted as
  an unary operator that modifies
  the meaning of a fuzzy set

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Prepared Dr.Konstantin Degtiarev, February-June
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Software Implementation of Fuzzy Systems

• Some of the most commonly used operators (hedges)
  and their functions are as follows

Be cautious when treating the meaning of NOT
Functions of hedges
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Software Implementation of Fuzzy Systems

• Brief comments on the previously mentioned
  functions
• (A) Concentration function keeping the
  original shape (form) of a membership function,
  shrink it over the universe of discourse, and a
  level of concentration can be adjusted by
  changing a value of power (gt1) applied to
  membership values)
• (B) Dilution function opposite to
  concentration function it results in
  spreading of membership function over universal
  set through changing a power (lt1) of MF values
• (C) Contrast intensification changes slightly
  a shape of membership function, widening MF for
  possibility values gt 0.5 and narrowing it when
  the latter is ? 0.5
• (D) Negation (not mentioned above) mirrors
  imaging
• of MF with respect to ?(x)0.5

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Software Implementation of Fuzzy Systems

• 
  Example (Lecture hours)

Negation
Contrast intensification
Concentration dilution
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Software Implementation of Fuzzy Systems

• A definition of linguistic variable proposed by
  L.A.Zadeh (The concept of a linguistic variable
  and its application to approximate reasoning
  I,II, Information Sciences,8,pp.199-251,301-357)
  can be formulated as follows
• A linguistic variable is characterized by
• (a) its name N
• (b) a set L of linguistic values it can
  take
• (c) universal set U (physical domain)
  in which it is defined
• (d) a rule R that associates each
  linguistic value of L with a fuzzy
• set in U
• Some observations on MF shape analysis
• (1) The location and granularity (number) of
  MFs are the two relatively more important (from
  the standpoint
• of affecting performance of the fuzzy inference

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Software Implementation of Fuzzy Systems

• algorithm) issues compared to the shape of each
  member- ship function
• (2) The shape of MF characterizes uncertainty in
  the fuzzy variable in general, a high level of
  detail in shape design is considered as a
  conceptual error
• (3) Most of applications nowadays use simple
  convex membership functions (due to computational
  simplicity and relative easiness of
  implementation in hardware, the commonly
  practical are piece-wise-linear forms,
  e.g. triangular and trapezoidal MFs)
• (4) In most cases heights of membership
  functions
• of antecedent variables are equal to 1.0 (normal
• sets) if the heights of MFs in consequent part
  of
• the rules is less than 1.0, then it may cause

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Software Implementation of Fuzzy Systems

• so-called paralysis of implication results

(5) Overlapping is an important design
consideration each antecedent MF should
overlap only with imme- diate neighboring
membership functions, i.e.
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Software Implementation of Fuzzy Systems

• formally it can be expressed as
  . Overlap of MFs
  determines a degree of cooperation (or, switching
  degree) between corresponding rules
• (6) Each consequent MF represents one rule a
  height of MF determines the strength of
  contribution from each rule, and MFs location
  affects the actual decision value
• (7) In general, shape modifications of
  antecedent MF (compared to those of consequent
  ones) produce more significant effects on the
  output behavior
• (8) Adjustment (symmetric changes in overlap) of
  all consequent MFs do not produce significant
  changes of outputs behavior
• end of the Section 2

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Software Implementation of Fuzzy Systems

• ? ? ? Section 3 Outline
• Basic Principles of Inference in Fuzzy Logic
  (Entailment, Conjunction, Generalized Modus
  Ponens, Generalized Modus Tollens). Fuzzy IF-THEN
  rules. Canonical form
• Fuzzy Systems and Algorithms. Approximate
  Reasoning
• Fuzzy Inference Engines. Graphical Techniques of
  Inference. Fuzzification/Defuzzification
• Fuzzy System Design and its Elements (conceptual
  model). Design Options
• the discussion of these topics takes
  approximately 8?10 lecture hours. Examples are
  explained using fuzzyTECH, FL Toolbox packages
  and
• MATLAB demonstrations

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Software Implementation of Fuzzy Systems

• Contrariwise , continued Tweedledee, if it
  was so, it might be and if it were so, it would
  be but as it isnt , it aint. Thats logic
  (Lewis Carroll, Through the Looking Glass)
• In the propositional logic the inference can be
  depicted as follows
• If the resulting premises are both true, then the
  conclusion is also true (the truth conclusion is
  inferred or deduced from truth premises we call
  it a valid deduction). For example, the following
  form is invalid

• P and Q are propositions (variables)
• sign ? represents the relation if-then
• symbol (therefore) is placed before
  conclusion

premise 1 IF P THEN Q premise 2
not P
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Prepared Dr.Konstantin Degtiarev, February-June
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Software Implementation of Fuzzy Systems

• Among basic inference rules (the premises
  logically imply rules conclusions) we can
  mention the following
• (a) Modus Ponens (b)
  Modus Tollens
• (Lat. ? method of affirming) (Lat. ?
  method of denying)
• (c) Hypothetical Syllogism (many mathematical
  arguments contain a chain of if-then statements)
• A test for validity of Modus Tollens is as
  follows

The implication connective (?) is especially
important as a basis of fuzzy implication rules
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Software Implementation of Fuzzy Systems

• premises are conjuncted in the antecedent part
  of implication, and conclusion form its
  consequent part

How validity is checked?
Fuzzy Logic (FL) generalizes the notion of
truth values in classical logic, and provides a
background for reasoning (inferencing) when
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Software Implementation of Fuzzy Systems

• corresponding conditions are only partially
  satisfied
• Approximate reasoning (original rule IF x is A
  THEN y is B)
• 1. Possibility distribution of the variable x
• 2. Implication possibility from x to y
• possibility
  distribution of y
• Both fuzzy implication and fuzzy mapping rules
  use a compositional rule of inference for
  calculation of output results, but there are
  still some differences
• If proposition P is described by set A ? X, and
  proposition Q is described by set B ? Y, then the
  classical implication P?Q can be represented by
  the relation R as follows
• its
  schematical
• representation (Venn diagram, Figure 1)

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Software Implementation of Fuzzy Systems

• is as follows
• A rule IF x is A THEN y is B can be expressed
  as a compound conditional statement IF x is A
  THEN y is B ELSE y is Nothing (no action). In
  general,
• expression IF x is A THEN y is B ELSE y is C

See comments related to Figure 2 below
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Software Implementation of Fuzzy Systems

• can be expressed as a disjunction
• IF x is A THEN y is B ? IF x is THEN
  y is C
• In the logic of compound statements this can be
  written as
• 
  , where S is a proposition
• described by set C ? Y (see Venn diagram,
  Figure 2)
• Figure 1 (shaded area truth domain of the
  implication P?Q)
• Figure 2 (shaded area truth domain of the form
  , where
  )
• Consider a given rule IF x is A THEN y is B,
  where A and B are fuzzy sets (fuzzy propositions,
  fuzzy predicates) defined on universes X and Y,
  correspondingly,
• and a fact x is A (A and A are not
  necessarily
• identical)

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Software Implementation of Fuzzy Systems

• Having a rule and a fact, what can we say about
  conclusion (consequent) B ?
• Generalized Modus Ponens
• Rule (premise 1) x is A ? y is B
• Fact (premise 2) x is A
• Infer (result produced by fuzzy inference
  engine) y is B
• B A ? R, where R is a fuzzy relation
  (associations between the elements of