Computing with Words using Mathematica

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Computing with Words using Mathematica
ECE 1001

• Marian S. Stachowicz
• Laboratory for Intelligent Systems
• University of Minnesota, Duluth

October 1, 2002
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Authors

• Jonathan Andersh, undergraduate student from
  Laboratory for Intelligent Systems, University of
  Minnesota, Duluth
• Lance Beall, undergraduate student from
  Laboratory for Intelligent Systems, University of
  Minnesota, Duluth
• Catherine Marie Charlton, undergraduate student
  from Music Department, Cornell University,
  Ithica, author of the Strange Attractor, CD
  Album of Piano Improvisations,
• Marian S. Stachowicz, Professor from Laboratory
  for Intelligent Systems, University of Minnesota,
  Duluth

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What are Fuzzy Sets?
Problem 1 Given the set U 1,2,3,4,5,6,7,8,9,1
0,11,12, describe the set of prime numbers.
A u in U u is a prime number
The elements of the set are defined unequivocally
as A 2,3,5,7,11
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Problem 2 Now using the same set U, suppose we
want to describe the set of small numbers.
M u in U u is a small number
Now, it is not so easy to define the set. We can
use a sharp transition like the following,
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Or an alternative way to define the set would be
to use a smooth transition.
Characteristic Function
A U --gt 0,1
Membership Function
M U --gt 0,1
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The set of small numbers which we described
earlier are represented as a fuzzy set in the
following manner in the Fuzzy Logic Pack.
Small FuzzySet1,1,2,1,3,.9,4,.6,5,.4
,6,.3, 7,.2,8,.1,9,0,10,
0,11,0,12,0,
UniversalSpace -gt 0,12
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Fuzzy set theory provides a strict mathematical
framework in which vague conceptual
phenomena can be precisely and rigorously studied.
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Fuzzy Areas of Application

• Fuzzy Logic Control
• Fuzzy Modeling
• Fuzzy Arithmetic

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Fuzzy Logic Conrol
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Fuzzy Modeling
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Fuzzy Arithmetic
A1, A2, A1A2
A1, A2, A1-A2
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Where is Fuzzy Logic Used?

• Linear and Nonlinear Process Control
• Robotics, Automation, Tracking
• Consumer Electronics
• VCRs, Digital High Definition Television,
  Microwave Ovens, Cameras,etc.
• Pattern Recognition
• Image Processing, Machine Vision
• Decision Making
• Sensor Fusion, Risk Analysis
• Financial Systems
• Information Systems
• Data Base Management, Information Retrieval
• Data Analysis
• Meteorology
• Art and Music

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Contributions

• Professor Lotfi A. Zadeh of the University of
  California at Berkeley who is known as the
  Father of Fuzzy Logic, first introduced fuzzy
  logic and fuzzy set theory in 1965 as a means to
  model the uncertainty of natural language.
• Another person who has strongly supported our
  efforts is Professor George J. Klir, of the State
  University of New York at Binghampton. Professor
  Klir has authored many books, including the book
  which we use as the text for our fuzzy sets
  course.
• Finally, we would also like to acknowledge the
  help and support of Wolfram Research, Inc.
  Wolfram Research employees helped us design and
  develop the Fuzzy Logic Pack, and they took care
  of its production and distribution.
• Mathematica is a registered trademark of Wolfram
  Research, Inc.

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References

• 1 L.A. Zadeh, "Fuzzy sets", Information and
  Control, vol.8, pp. 338-353, 1965
• 2 George S. Klir and Tina Folger, Fuzzy Sets,
  Uncertainty, and Information. Prentice Hall,
  Englewood Cliffs, New Jersey, 1988.
• 3 H.J. Zimmermann, Fuzzy Set Theory and Its
  Applications, Second Edition, Kluwer Academic
  Publishers, Boston, MA, 1991.
• 4 M.S. Stachowicz and Lance E. Beall, Fuzzy
  Logic Pack for use with Mathematica, Wolfram
  Research, Inc. Champaign, IL 61820-7237,1995
• 5 C.M. Charlton, Strange Attractor, CD Album
  of Piano Improvisations, Orange Moon Production,
  Inc. 19672 Stevens Creek Blvd., 178, Cupertino,
  CA 95014