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Hello,

A couple of weeks ago there was a discussion in this group following a

question of Andrej Albert on the difference between two models of

approximate reasoning. Continuing this discussion I will consider a small

example, and show that the choice of the model should be dependent on

the interpretation of the rules, and moreover, that for some interpretations

of the rules neither of the two models is appropriate.

Suppose we have a set of rules of the form "IF A THEN B" where A is a set

of symptoms and B is a set of diseases. The intention of the rules is to obtain

a diagnosis of the disease(s) of a patient, given information about the symptoms

this patient exhibits. We will only consider crisp sets here, taking the point of

view that a model for approximate reasoning should produce reasonable results

also when only crisp sets are considered.

Given a rule "IF A THEN B" where A is a set of symptoms and B is a set of

diseases and also given a set A' of symptoms, the resulting set B' of diseases

will depend on the meaning of the rules. A rule might be applicable if the

patient

exhibits ANY of the symtoms, or if the patient exhibits ALL the symtoms.

The result of the rule might mean that the patient suffers from ALL diseases from

the set, or that the disease is ANY of the diseases of the set.

This leads to four possible interpretations of a rule "IF A THEN B":

Interpretation 1 : IF the patient exhibits all symptoms from A THEN the patient

has

all diseases of set B

Interpretation 2 : IF the patient exhibits all symptoms from A THEN the disease

of this patient belongs to set B

Interpretation 3 : IF the patient exhibits anyl symptoms from A THEN the patient

has

all diseases of set B

Interpretation 4 : IF the patient exhibits any symptoms from A THEN the disease

of this patient belongs to set B

Now consider the case where there are 3 symtoms and 3 diseases, and

symptoms and diseases both are denoted by the numbers 1,2 and 3.

Consider the rule : IF {1,2} THEN {1}

We consider two input sets of symptoms: {1} and {3}.

The following table contains for both input sets and all four interpretations of

the

rule the desired output set of diseases.

Int. 1 Int. 2 Int. 3 Int. 4

Input {1} | {} {1,2,3} {1} {1}

Input {2} | {} {1,2,3} {} {1,2,3}

This table is computed as follows:

The desired result is {1} ( the right hand-side of the rule), in case the rule

is

applicable, i.e. for input {1} and interpretations 3 and 4 of the rule.

The desired result is {} if the rule is not applicable and the result denotes the

diseases of the patient.

The desired result is {1,2,3} if the rule is not applicable, and the result

denotes the

possible diseases of the patient.

Now let us compare this table with the results obtained from both approximate

reasoning models. Both models employ the formula

B'(y) = sup_x min (A'(x), J(A(x),B(y)))

In the Mamdani case, J is a t-norm, with the properties J(1,1)=1 and

J(0,1)=J(1,0)=J(0,0)=0.

In the second case (I call it the implication case), J is an implication

operator, with

the properties J(1,0)=0 and J(0,0)=J(0,1)=J(1,1)=1.

The result of the computation is:

Mamdani Implication

Input {1} | {1} {1}

Input {2} | {} {1,2,3}

So the Mamdani model produces the results which are desired in interpretation 3,

and the implication model produces the results which are desired in

interpretation 4.

One easily verifies that this holds true for other rules and inputs also.

So my conclusion for this example is : In case the interpretation of the rules is

interpretation 3, use the Mamdani model. In case the interpretation of the rules

is interpretation 4, use the implication model. In case the interpretation of the

rules

is interpretation 1 or interpretation 2, do not use any of these two models,

since

neither of them produces the desired results in case of crisp sets.

My conclusion in general is: In a practical situation, a simple test using crisp

sets can

determine which of the two models of approximate reasoning is appropriate, or

show

that neither of them is appropriate.

I look forward to receive your opinions!

greetings,

Pim van den Broek.

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<!doctype html public "-//w3c//dtd html 4.0 transitional//en">

<html>

Hello,

<p>A couple of weeks ago there was a discussion in this group following

a

<br>question of Andrej Albert on the difference between two models of

<br>approximate reasoning. Continuing this discussion I will consider a

small

<br>example, and show that the choice of the model should be dependent

on

<br>the interpretation of the rules, and moreover, that for some interpretations

<br>of the rules neither of the two models is appropriate.

<p>Suppose we have a set of rules of the form "IF A THEN B" where A is

a set

<br>of symptoms and B is a set of diseases. The intention of the rules

is to obtain

<br>a diagnosis of the disease(s) of a patient, given information about

the symptoms

<br>this patient exhibits. We will only consider crisp sets here, taking

the point of

<br>view that a model for approximate reasoning should produce reasonable

results

<br>also when only crisp sets are considered.

<br>Given a rule "IF A THEN B" where A is a set of symptoms and B is a

set of

<br>diseases and also given a set A' of symptoms, the resulting set B'

of diseases

<br>will depend on the meaning of the rules. A rule might be applicable

if the patient

<br>exhibits ANY of the symtoms, or if the patient exhibits ALL the symtoms.

<br>The result of the rule might mean that the patient suffers from ALL

diseases from

<br>the set, or that the disease is ANY of the diseases of the set.

<br>This leads to four possible interpretations of a rule "IF A THEN B":

<p>Interpretation 1 : IF the patient exhibits all symptoms from A

THEN the patient has

<br>all diseases of set B

<br>Interpretation 2 : IF the patient exhibits all symptoms from

A THEN the disease

<br>of this patient belongs to set B

<br>Interpretation 3 : IF the patient exhibits anyl symptoms from

A THEN the patient has

<br>all diseases of set B

<br>Interpretation 4 : IF the patient exhibits any symptoms from

A THEN the disease

<br>of this patient belongs to set B

<p>Now consider the case where there are 3 symtoms and 3 diseases, and

<br>symptoms and diseases both are denoted by the numbers 1,2 and 3.

<p>Consider the rule : IF {1,2} THEN {1}

<br>We consider two input sets of symptoms: {1} and {3}.

<br>The following table contains for both input sets and all four interpretations

of the

<br>rule the desired output set of diseases.

<p><tt>

Int. 1 Int. 2 Int. 3 Int. 4</tt>

<br><tt>Input {1} | {} {1,2,3}

{1} {1}</tt>

<br><tt>Input {2} | {} {1,2,3}

{} {1,2,3}</tt>

<p>This table is computed as follows:

<br>The desired result is {1} ( the right hand-side of the rule),

in case the rule is

<br>applicable, i.e. for input {1} and interpretations 3 and 4 of the rule.

<br>The desired result is {} if the rule is not applicable and the result

denotes the

<br>diseases of the patient.

<br>The desired result is {1,2,3} if the rule is not applicable, and the

result denotes the

<br>possible diseases of the patient.

<p>Now let us compare this table with the results obtained from both approximate

<br>reasoning models. Both models employ the formula

<p><tt>B'(y) = sup_x min (A'(x), J(A(x),B(y)))</tt>

<p>In the Mamdani case, J is a t-norm, with the properties J(1,1)=1 and

<br>J(0,1)=J(1,0)=J(0,0)=0.

<br>In the second case (I call it the implication case), J is an implication

operator, with

<br>the properties J(1,0)=0 and J(0,0)=J(0,1)=J(1,1)=1.

<p>The result of the computation is:

<p><tt>

Mamdani Implication</tt>

<br><tt>Input {1} | {1} {1}</tt>

<br><tt>Input {2} | {}

{1,2,3}</tt>

<p>So the Mamdani model produces the results which are desired in interpretation

3,

<br>and the implication model produces the results which are desired in

interpretation 4.

<br>One easily verifies that this holds true for other rules and inputs

also.

<p>So my conclusion for this example is : In case the interpretation of

the rules is

<br>interpretation 3, use the Mamdani model. In case the interpretation

of the rules

<br>is interpretation 4, use the implication model. In case the interpretation

of the rules

<br>is interpretation 1 or interpretation 2, do not use any of these two

models, since

<br>neither of them produces the desired results in case of crisp sets.

<p>My conclusion in general is: In a practical situation, a simple test

using crisp sets can

<br>determine which of the two models of approximate reasoning is appropriate,

or show

<br>that neither of them is appropriate.

<p>I look forward to receive your opinions!

<p>greetings,

<p>Pim van den Broek.

<br>

<br> </html>

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