Fuzzy Logic Introduction

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Fuzzy Logic - Introduction
Adriano Cruz NCE e IM/UFRJ Adriano_at_nce.ufrj.br

• Computers are useless, they can only give you
  answers.
• Pablo Picasso

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Introduction

• Adriano Cruz
• NCE-IM UFRJ
• adriano_at_ufrj.br
• Light travels faster than sound. That is the
  reason why some people look brighter until they
  start talking.
• Linux Journal

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Bibliography 1

• J. Yen, R. Langari, Fuzzy Logic Intelligence,
  Control and Information, Prentice Hall, 1999
• J. R. Jang, C. Sun, E. Mizutani, Neuro-Fuzzy and
  Soft Computing A Computational Approach to
  Learning and Machine Intelligence, Prentice Hall,
  1997
• Slides and notes http//equipe.nce.ufrj.br/adrian
  o/fuzzy/bibliogr-ic.htm
• C. von Altrock, Fuzzy Logic NeuroFuzzy
  Applications Explained, Prentice Hall PTR, 1995

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Bibliography 2

• H. T. Nguyen, E. A. Walker, A First Course in
  Fuzzy Logic, Chapman Hall/CRC, 2000
• Bart Kosko, Fuzzy Thinking, Harper Collins
  Publishers, 1994, ISBN 0-00-654713-3
• L. H. Tsoukalas, R. E. Uhig, Fuzzy and Neural
  Approaches in Engineering, John Wiley and Sons,
  Inc, 1997

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Summary

• Introduction
• Fuzzy Sets
• Fuzzy Set Operations
• Fuzzy Systems

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Artificial Intelligence?

• AI is the activity of providing such machines as
  computers with the ability to display behaviours
  that would be regarded as intelligent if it were
  observed in humans (R. McLeod)?
• AI is the study of agents that exist in an
  environment, perceive and act. (S. Russel and P.
  Norvig)?

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Artificial Intelligence?

• AI emphasizes symbolic processing
• Acts on higher levels of intelligence
• AI seeks to understand

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Computational Intelligence

• Acts on lower levels of Intelligence
• Uses learning extensively
• Pattern recognition and heuristics play important
  roles

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Computational Intelligence

• Fuzzy Logic
• Artificial Neural Networks
• Evolutionary Systems
• Swarm Intelligence
• Hybrid Systems

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Computational Intelligence

• Fuzzy Logic
• Artificial Neural Networks
• Evolutionary Systems
• Swarm Intelligence
• Hybrid Systems

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Fuzzy Logic

• Logic that deals mathematically with imprecise
  information usually employed by humans.
• Multi-valued logic that extends Boolean logic
  usually employed in computer science.

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Fuzzy Logic

• Used to alleviate difficulties in developing and
  analysing complex control systems.
• Function approximator
• Decision systems

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Fuzzy Logic

• Who is greater than 1.80 m?
• Who is tall?
• Who weighs more than 100 kg?
• Who is heavy?
• The driver was heavy and tall.

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Computational Intelligence

• Fuzzy Logic
• Artificial Neural Networks
• Evolutionary Systems
• Swarm Intelligence
• Hybrid Systems

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Artificial Neural Networks

• Computational models that try to emulate the
  structure of the human brain wishing to reproduce
  at least some of its flexibility and power.
• ANN consist of many simple computing elements
  usually simple nonlinear summing operations
  highly connected by links of varying strength.

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ANNs

• ANNs are able to learn from examples.
• Function approximators.
• Solutions not always correct.
• ANNs are able to generalize the acquired
  knowledge.

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Neurons
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Neural Networks
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Structure
Inputs
Input layer
Weight Matrix 1
Weight Matrix 2
Hidden layer
Output layer
Outputs
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Training

• Weight values change during the training process
• Values are presented at the inputs and outputs
  are compared to the desired values.
• Wrong outputs cause weights to change in order to
  reduce the error
• Process is repeated with different inputs till
  the ANN is able to give the correct answers
• Hopefully the ANN will be able to give the
  correct answer even to inputs that were not
  trained.

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Computational Intelligence

• Fuzzy Logic
• Artificial Neural Networks
• Evolutionary Systems
• Swarm Intelligence
• Hybrid Systems

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Evolutionary Systems

• ES are global search and optimization algorithms
  modelled from natural genetic principles such as
  natural selection.
• They are stochastic searching methods.
• Good solutions will survive and be combined by
  the natural selection process.
• At the end the most fit will survive.

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The Metaphor

• The metaphor that lays behind GAs is the natural
  selection.
• The problem of each species in the nature is seek
  for the best adaptations in order to survive in a
  hostile environment that is in constant
  modification.

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Adaptation

• The sets of characteristics of an individual,
  that distinguishes from everybody else, defines
  its survival capacity.
• These characteristics are determined by its
  genetic material.

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Mechanisms

• The competition for scarce resources makes the
  apts survive and reproduce.
• Through reproduction the genes from individuals
  are transmitted to their descendants.
• This continuous process of selection and
  reproduction of the best individuals may conduct
  to more adpated individuals.

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GA Flux
begin
Randomly
Initial Population
Mutation
Current generation
Next Generatio
Selects Parents
Evaluates
Generates Sons
Crossing
OK?
No
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Computational Intelligence

• Fuzzy Logic
• Artificial Neural Networks
• Evolutionary Systems
• Swarm Intelligence
• Hybrid Systems

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Swarm Intelligence

• Swarm Intelligence (SI) is the property of a
  system whereby the collective behaviours of
  (unsophisticated) agents interacting locally with
  their environment cause coherent functional
  global patterns to emerge.
• SI provides a basis with which it is possible to
  explore collective (or distributed) problem
  solving without centralized control or the
  provision of a global model.

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Characteristics of a swarm

• Distributed, no central control or data source
• No (explicit) model of the environment
• Perception of environment, i.e. sensing
• Ability to change environment.

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Motivations

• Robust nature of animal problem-solving
• simple creatures exhibit complex behaviour
• behaviour modified by dynamic environment.
• Emergent behaviour observed in
• bacteria
• ants
• bees
• ...

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Ant Colonies

• Ants are behaviourally unsophisticated
  collectively perform complex tasks.
• Ants have highly developed sophisticated
  sign-based stigmergy
• communicate using pheromones
• trails are laid that can be followed by other
  ants.
• Stigmergy is a method of indirect communication
  in a self-organising emergent system where its
  individual parts communicate with one another by
  modifying their local environment.

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Computational Intelligence

• Fuzzy Logic
• Artificial Neural Networks
• Evolutionary Systems
• Swarm Intelligence
• Hybrid Systems

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Hybrid Systems

• Each intelligent technique has its particular
  strengths and weakness and cannot be applied to
  universally to every problem.
• Mixing together these techniques systems improve
  the quality of the solutions and allows
  application to different tasks.

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History
EA
FL
ANNs
AI
40s
43 Neuron Model
47 Cybernetics
50s
57 Perceptron
56 AI
Adaline - Madaline
60s
65 Fuzzy Sets
60 Lisp
74 Back- Propagation
70s
Genetic Algorithm
74 Fuzzy Control
Expert Systems
80 Self orgazing map 82 Hopfield 83 Boltzmann
Mach
85 Fuzzy modelling (TSK model)?
80s
Immune modelling
Genetic programming
90s
Neuro-Fuzzy modelling
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Why do we reason as we do?
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Aristotle

• Macedonian philosopher who lived
• between 384 e 322 AC
• Studied under Plato in the Academy
• Creator of formal logic
• His father Nichomachus was court physician to
  King Amyntas
• Associates the spirit of observation and a
  classification instinct
• He was considered during the middle ages the
  philosopher
• He shaped much of the western mind.

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Limitations of the Aristotles Logic

• Objects are separated on very clear categories
• One object either belongs to a category or
  another
• Either you are or not
• Helps to separate objects into well defined
  categories.

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Aristotle X Buddha

• Everything must either be or not be, whether in
  the present or in the future.
• Aristotle
• I have not explained that the world is eternal or
  not eternal. I have not explained that the world
  is finite or infinite.
• The Buddha

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Why fuzzy logic?

• Every language is vague.
• All traditional logic habitually assumes that
  precise symbols are being employed. It is
  therefore not applicable to this terrestrial
  life, but only to an imagined celestial one.
• Everything is vague to a degree you do not
  realize till you have tried to make it precise.
• Bertrand Russel

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Why fuzzy logic?

• As far as the laws of Mathematics refer to
  reality, they are not certain and as far as they
  are certain, they do not refer to reality.
• Albert Einstein

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How to classify?

• Happy people
• Small rooms
• High temperatures
• Faster cars
• High tax rates
• High people

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To be or not to be?

• Bertrand Russel, while trying to formalize
  Mathematic had difficulties due to the liars
  paradox.
• I am lying.
• If Eubulides statement was true, then he is
  lying when he says I am lying and so he isn't,
  i.e. his statement is false.
• If his statement is false, then he isn't lying
  when he tells us he is, and so his statement is
  true.

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Answer To be and not to be.

• Consider the set of all sets that are not members
  of its own set. Is it a member of this set?
• If it is a member then it is not, but if it is
  not then it is.

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The Detractors

• Fuzzy theory is wrong, wrong, and pernicious.
  What we need is more logical thinking, not less.
  The danger of fuzzy logic is that it will
  encourage the sort of imprecise thinking that has
  brought us so much trouble. Fuzzy logic is the
  cocaine of the science.
• Prof. William Kaham - U. Cal - Berkeley

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The Detractors

• Fuzzification is a kind of scientific
  permissiveness. It tends to result in socially
  appealing slogans unaccompanied by the discipline
  of hard scientific work and patient observation.
• Prof. Rudolf Kalam - U. Florida - Gainesville

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The Beginning

• Lotfy Zadeh. Fuzzy Sets, Information na
  Control, 1965
• Principle of Incompatibility
• As the complexity of a system increases, our
  ability to make precise yet significant
  descriptions about its behaviour diminishes until
  a threshold is reached beyond which precision and
  significance (or relevance) become almost
  mutually exclusive characteristics.
• Lofty Zadeh

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Fuzzy Thinking
No
No
Yes
Yes
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Fuzzy Thinking

• If the interest rate is high and the deficit is
  high then there will be a recession
• If rush hour then diminish the interval between
  busses
• If the tyre skids then loose the brake a bit
• If the soil is very dry then water it for very
  long time

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Fuzzifying
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Fuzzy Systems
X
YF(X)?
Function F(x) is unknown
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Approximation of Functions
patches
Y
X
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Fuzzy Aproximation Theorem

• Patches are pieces of knowledge about a problem
• Every patch corresponds to a rule or proposition
• If the speed is high then step on the break

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Fuzzy Aproximation Theorem

• An additive fuzzy system FX-gtY uniformly
  approximates fX-gtY if X is compact and f is
  continuous.
• Bart Kosko

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Fuzzy Systems
Rules
Sets
Operators
Data Management
Deffuzzifier
Fuzzyfier
Inference Engine
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Advantages

• Use rules that express imprecision of the real
  world.
• Easy to understand, test and maintain
• Easy to be prototyped
• Robust. They operate even when there is lack of
  rules or wrong rules.
• Need less rules
• Parallel evaluation of rules
• Accumulate evidences in favour and against

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Disadvantages

• Need more tests and simulation
• Do not learn easily
• Difficult to establish correct rules
• Lack of precise mathematical model

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Commercial Products

• Sendai subway 16 stations and 13,5 km route,
  designed by Hitachi
• Washing machines that measure weight, saturation
  time and water clarity in order to program cycles
• Portable camcorders with automatic focus and
  anti-jitter
• Vacuum cleaners that measure air dust to set
  suction power
• Microwave ovens that measure temperature,
  humidity, weight of food to set time and power.

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Commercial Products

• Sugeno designed a voice controlled system to
  operate an unmanned helicopter
• Anti-Lock Braking Systems Nissan, Mitsubishi.
  Honda, Mazda, Hyunday, BMW, Bosch and Peugeot
• Suspension, transmission and fuel injector
  systems are usual.
• Hitachi uses approximately 150 rules to trade in
  Japanese bonds and futures
• Yamaichi Securities uses hundreds of rules to
  manage a stock fund
• Anaesthesia Control and Fuzzy Data Analysis for
  Cardio-Anaesthesia

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Products
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Questions?

• Is fuzzy logic probability ?
• Find a fuzzy product description.
• Find fuzzy development tools.
• Fuzzy Logic is a multi values logic. Find other
  examples.