# Re: Consistency of fuzzy rules

From: WSiler@aol.com
Date: Mon Oct 15 2001 - 12:34:11 MET DST

• Next message: Yaochu Jin: "Re: Consistency of fuzzy rules"

In a message dated 10/4/01 12:23:34 AM Central Daylight Time,
ftrjin@mail.ipm.net writes:

> This might be an issue of consistency of fuzzy rules. Roughly, two fuzzy
> rules
> mare inconsistent in the following situations
>
> i) The same condition, completely different consequents
> e.g.
> R1: If x1 is A1 and x2 is A2, then y is Positive Large
> R2: If x1 is A1 and x2 is A2, then y is Negative Large

I see no reason why different consequents mean necessarily that the rules are
inconsistent. We have permitted different consequents for many years. A great
ambiguities to be handled with great ease. The program should be able to
detect contradictions to be identified and handled in subsequent program
steps. For example, our program for classification of regions in an
echocardiogram has these two rules:

IF (in Data xbar is Left and ybar is Center) THEN region is RV;
IF (in Data xbar is Left and ybar is Center) THEN region is RARV;

This is, in fact, a contradiction, since the consequents are mutually
exclusive. A subsequent rule to resolve this contradiction, bringing in
additional data, rules out RARV if RA is present in the same frame:

IF (in Data frame is <FR> and region is RARV and region is RA)
IF (in Data frame is <FR> and region is RV )
THEN in 2 region is RARV = 0;

In other cases, the two different consequents may not be mutually exclusive.
This is an ambiguity, not a contradiction. For example, the same set of
symptoms may be characteristics of different diseases; in this case, the
doctor would want to know all possibilities, in order to decide what
treatment to employ, or the program may call up a different set of rules to
look for data that would perform tbe differential diagnosis or suggest
methods of differential diagnosis if insufficient data are present.

> i) Although the conditions are seeming different, but the same in physical
>
> R1: If x1 is A1 and x2 is A2, then y is Positive Large
> R2: If x1 is A1 and x3 is A3, then y is Negative Large
>
> Although "x2 is A2" and "x3 is A3" appear to be different conditions, they
> might imply the same situation in some cases, such as in a chemical reactor:
> "If temperature is high" may imply "conversion rate is high"

What in the world is wrong about that? Different conditions may indeed
require the same corrective action.

> e.g. "The sun is bright and the rain is heavy"

It seems unlikely that these conditions would occur together, but it is not
impossible, as I have experienced in Hawaii. In any event, if the two
conditions do not occur together, the rule would not fire.

> e.g. If x is A then y is B and z is C
> However, y is B and z is C cannot happen together
>
In this case, subsequent rules can take care of this. For example:

if y is B and z is C and u is U then y is 0
if y is B and z is C and u is not U then z is 0

where U is a condition to resolve the contradictions.

or, if there is no condition U to resolve the contradiction,
if y is B and z is C then message "Contradictory results for y and z, setting
both to 0" and y is 0 and z is 0;

It seems that you are ignoring these facts:

1. Fuzzy sets, fuzzy values and fuzzy logic can handle both contradictions
and ambiguities with ease.
2. Reasoning is not necessarily a single-step affair.
3. The distinction between contradictions and ambiguities is a
problem-dependent affair. Often both are present in the same problem.
Contradictions need to be resolved, by additional rules loking at the same
data more closely or requesting additional data from the user.

William Siler

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<HTML><FONT FACE=arial,helvetica><FONT SIZE=2>In a message dated 10/4/01 12:23:34 AM Central Daylight Time, ftrjin@mail.ipm.net writes:
<BR>
<BR>
<BR><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">This might be an issue of consistency of fuzzy rules. Roughly, two fuzzy rules
<BR>mare inconsistent in the following situations
<BR>
<BR>i) The same condition, completely different consequents
<BR>e.g.
<BR>R1: If x1 is A1 and x2 is A2, then y is Positive Large
<BR>R2: If x1 is A1 and x2 is A2, then y is Negative Large</FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></BLOCKQUOTE>
<BR>
<BR>I see no reason why different consequents mean necessarily that the rules are inconsistent. We have permitted different consequents for many years. A great advantage of discrete fuzzy sets is that they permit contradictions and ambiguities to be handled with great ease. The program should be able to detect contradictions to be identified and handled in subsequent program steps. For example, our program for classification of regions in an echocardiogram has these two rules:
<BR>
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0">IF (in Data xbar is Left and ybar is Center) THEN region is RV;
<BR>IF (in Data xbar is Left and ybar is Center) THEN region is RARV;
<BR>
<BR>This is, in fact, a contradiction, since the consequents are mutually exclusive. A subsequent rule to resolve this contradiction, bringing in additional data, rules out RARV if RA is present in the same frame:
<BR>
<BR>IF (in Data frame is &lt;FR&gt; and region is RARV and region is RA)
<BR>IF (in Data frame is &lt;FR&gt; and region is RV )
<BR>THEN in 2 region is RARV = 0;</FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0">
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0">
<BR>In other cases, the two different consequents may not be mutually exclusive. This is an ambiguity, not a contradiction. For example, the same set of symptoms may be characteristics of different diseases; in this case, the doctor would want to know all possibilities, in order to decide what treatment to employ, or the program may call up a different set of rules to look for data that would perform tbe differential diagnosis or suggest methods of differential diagnosis if insufficient data are present.
<BR>
<BR>i<BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">i) Although the conditions are seeming different, but the same in physical
<BR>meanings, and the consequents are totally different</BLOCKQUOTE>
<BR><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">R1: If x1 is A1 and x2 is A2, then y is Positive Large
<BR>R2: If x1 is A1 and x3 is A3, then y is Negative Large
<BR>
<BR>Although "x2 is A2" and "x3 is A3" appear to be different conditions, they
<BR>might imply the same situation in some cases, such as in a chemical reactor:
<BR>"If temperature is high" may imply "conversion rate is high"</FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></BLOCKQUOTE>
<BR>
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0">What in the world is wrong about that? Different conditions may indeed require the same corrective action.
<BR></FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0">
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0"><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">iii) Contradictory conditions
<BR>e.g. "The sun is bright and the rain is heavy"</FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></BLOCKQUOTE>
<BR>
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0">It seems unlikely that these conditions would occur together, but it is not impossible, as I have experienced in Hawaii. In any event, if the two conditions do not occur together, the rule would not fire.
<BR></FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0">
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0"><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">iv) Contradictory consequents
<BR>e.g. If x is A then y is B and z is C
<BR>However, y is B and z is C cannot happen together
<BR></FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></BLOCKQUOTE>
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0">In this case, subsequent rules can take care of this. For example:
<BR>
<BR>if y is B and z is C and u is U then y is 0
<BR>if y is B and z is C and u is not U then z is 0
<BR>
<BR>where U is a condition to resolve the contradictions.
<BR>
<BR>or, if there is no condition U to resolve the contradiction,
<BR>if y is B and z is C then message "Contradictory results for y and z, setting both to 0" and y is 0 and z is 0;
<BR>
<BR>It seems that you are ignoring these facts:
<BR>
<BR>1. Fuzzy sets, fuzzy values and fuzzy logic can handle both contradictions and ambiguities with ease.
<BR>2. Reasoning is not necessarily a single-step affair.
<BR>3. The distinction between contradictions and ambiguities is a problem-dependent affair. Often both are present in the same problem. Contradictions need to be resolved, by additional rules loking at the same data more closely or requesting additional data from the user.
<BR>
<BR>William Siler</FONT></HTML>

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