Fuzzy Logic: An Introduction

1
Fuzzy Logic An Introduction

• common sense?
• often used to solve problems
• often have vague/ambiguous/imprecise info
• fuzzy logic is
• not logic that is fuzzy
• rather, logic that describes fuzziness
• a system of logic in which a statement can be
  true, false, or any of a continuum of values in
  between
• fuzzy logic does not result in fuzzy output!

2
Fuzzy Logic

• many things not binary rather, degrees
• eg temperature, height, speed, distance, beauty
  all come on a sliding scale anddepend on
  context
• eg The motor is running hot.
• aka multi-valued logic

3
Set Theory

• well known, well defined
• an element is in a set or not
• but, many items in human discourse cannot be
  represented with traditional set theory
• large profit
• high pressure
• moderate temperature
• to represent these in set theory, we must
  quantize

4
Quantize?

• many decision-making andproblem solving
  situations are not easily(or correctly)
  quantized
• we use imprecise information all the time
• fuzzy set theory
• attempts to model human reasoningby using
  approximate and uncertain informationto make
  decisions

5
Applications

• subway control (Japan)?
• automatic transmissions
• linguistics
• washing machines adjust to each load
• cameras
• helicopters anyone can fly
• air conditioners cool, slow
• cancer treatment
• life expectancies

6
History

• 1930s Jan Lukasiewicz (Polish philosopher)?
• extended truth values 0,1 to real values 0..1
• if someone's height is 181cm, is that really
  tall?
• inexact reasoning possibility theory
• 1965 Lotfi Zadeh Fuzzy Sets
• formalized the theory
• extended set theory
• Azerbaijan, Iran, Russia, US (Berkeley)?

7
Fuzzy Logic

• based on
• degrees of membership,degrees of truth
• uses the
• continuum of logical values between 0
  (completely false) and 1 (completely true)
• acknowledges that
• things can be partly true and partly false at
  the same time
• shades of grey

8
Fuzzy Set

• fuzzy set A of universe X is defined by function
  µA(x) called the membership function of set A
• µA(x) X ? 0, 1, where
• µA(x) 1 if x is totally in A
• µA(x) 0 if x is not in A
• 0 lt µA(x) lt 1 if x is partly in A.
• allows a continuum of possible choices For any
  element x of universe X, membership function
  µA(x) equals the degree to which x is an element
  of set A.
• This degree, a value between 0 and 1, represents
  the degree of membership, also called membership
  value, of element x in set A.

9
Fuzzy Set Representation

• Initially determine the membership
  functionse.g tall example, we obtain fuzzy
  sets of tall, short and average
• Therefore, the universe of discourse consists of
  three sets short, average and talle.g. someone
  who is 184 cm tall is a member of the average
  men set with a degree of membership of 0.1, and
  at the same time, also a member of the tall set
  with a degree of 0.4

10
Advantages?

• simple rules
• model complex functions easily
• continuous adjustments/transitions (no jerks)?
• more human-like
• reduces space requirements
• e.g. fuel injection systems
• instead of a huge table of values,store a few
  rules and compute fuzzy sets
• produces exact results from imprecise
  (continuous) data

11
Probability vs Fuzzy Sets

• probability
• likelihood of something occurring
• fuzzy sets
• degree of membership
• e.g. sick vs not sick