Zadehs Vision From Traditional . . . of Going from Fuzzy Zadehs - - PowerPoint PPT Presentation

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 7 Go Back Full Screen Close Quit

Zadeh’s Vision

• f Going from Fuzzy

to Computing With Words: from the Idea’s Origin to Current Successes to Remaining Challenges

Vladik Kreinovich

Department of Computer Science University of Texas at El Paso El Paso, TX 79968, USA vladik@utep.edu

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 7 Go Back Full Screen Close Quit

1. The Origins of Fuzzy Techniques: Reminder

• Some experts are very skilled in medical diagnostics,

control, etc.

• Ideally: every patient should be diagnosed by the best

doctor.

• Problem: the best doctor does not have time to see all

patients.

• Solution: incorporate the expert knowledge into an au-

tomatic system that everyone can use.

• Problem: experts often describe their knowledge by us-

ing imprecise (“fuzzy”) words from natural language.

• Examples: expert rules include conditions like “if a

tumor is small”, “if a car is far away and going fast”.

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 7 Go Back Full Screen Close Quit

2. Traditional Fuzzy Techniques (cont-d)

• Fuzzy logic is a technique for transforming imprecise

expert rules into precise decision, precise control, etc.

• Main idea: since we are not sure whether x is small,

assign a degree of smallness to different values x.

• In the computer: everything is represented as 0s and

1s; e.g., “true” is 1, “false” is 0.

• We want degrees intermediate between 0 and 1, so it

is natural to use numbers from [0, 1].

• Elicitation: polling (probability-type), Likert scale, etc.
• Need to combine degrees: what is the degree to which

a car is far away and going fast?

• Ideal solution: ask the expert about all possible com-

binations of distance and speed.

• Problem: there are too many combinations to ask about.

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 7 Go Back Full Screen Close Quit

3. t-norms, t-conorms, etc.

• Problem (reminder): we need to estimate degrees of

A & B etc., and we cannot simply elicit them.

• Solution: we estimate the degree of A & B based on

degrees of A and B: d(A & B) = f&(d(A), d(B)).

• Details: requirements like A & B ≡ B & A and

A & (B & C) ≡ (A & B) & C lead to t-norms.

• Problem: there exist many different t-norms that sat-

isfy all these requirements.

• Details: different t-norms lead to different recommen-

dations.

• t-norms are selected empirically (if selected at all :-), so

that the elicited d(A & B) is the closest to f&(d(A), d(B)).

• Example: medically best t-norm (MYCIN) turned out

to be not appropriate for geophysics.

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 7 Go Back Full Screen Close Quit

4. From Traditional Fuzzy Logic to More Ade- quate Implementations of Computing With Words

• Problem: for the same statement, different experts pro-

duce different degrees.

• Traditional fuzzy logic: uses one of these degrees – or,

e.g., their average.

• Problem: an expert is not sure about his or her degree
• f belief in a statement: 71 or 72 on a scale 0–100?
• Traditional fuzzy logic: if an expert selects between 7

and 8 on 1–10 scale, use 7.5.

• More adequate representation of expert uncertainty:

– use range of possible values (interval-valued ap- proach); – also indicate degrees to which different values from the range are possible (general type-2 approach).

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 7 Go Back Full Screen Close Quit

5. Zadeh’s Vision

• In many applications: the outcome is an imprecise con-

clusion: e.g., the patient most probably has a flu.

• How this is done now:

– we start with words from natural language; – we transform them into numbers (intervals, etc.); – we process these numbers; and – we transform the resulting number into a natural language word describing the conclusion.

• Why we use numbers: only because we know how to

process numbers.

• Zadeh’s idea: cut the middleman:

– start with words, – process words, – produce the words as a result.

The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. From Traditional . . . Zadeh’s Vision Zadeh’s Vision: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 7 Go Back Full Screen Close Quit

6. Zadeh’s Vision: Challenges

• Ideally: we should operate directly with words.
• Example: we should be able to add small and medium

and get – what?

• This is the gist of numerous Zadeh’s examples like

– most Swedes are tall, – Johannes is a Swede, – what is the probability that Johannes is very tall?

• Challenges: we are still far from this vision.