# Re: Consistency of fuzzy rules

From: Yaochu Jin (ftrjin@mail.ipm.net)
Date: Mon Oct 15 2001 - 19:06:00 MET DST

• Next message: Pim van den Broek: "Re: Consistency of fuzzy rules"

>> 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.

Sure, fuzzy rules allows ambiguities, but fully contradictory rules not only
confuse users, but also degrade the performance of the fuzzy system. Otherwise,
why do you need another rules to resolve the contradiction??? I don't think a
fuzzy system will be easily accepted if the user find it is full of

>> ii) 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.

I hope you have got me correctly. This is the same situation as in i), except
that the premises looks different.

R1: "If temperature is high, reduce the flow rate"
R2: "If conversion rate is high, increase the flow rate"

R1 and R2 looks o.k., but if "temperature is high" always imply "conversion
rate is high", then R1 and R2 are inconsistent.

>> 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.

This can happen not only in Hawaii. Nevertheless, I will remove this fuzzy rule
from my fuzzy system.

>> 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.

Contradictions should be avoided if you can and if you want to build an
understandable and compact fuzzy system.

>2. Reasoning is not necessarily a single-step affair.

Agree.

>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.

Fully agree. Contridictory fuzzy rules generated from data are usually caused by
incomplete or highly contaminated data. Once inconsistent fuzzy rules are
generated, it is necessary to look into the contradictory fuzzy rules and try to
resolve the contradiction. Therefore, a fuzzy rule system that is finally
presented to the user should be consistent and compact. Garbage rules should be
removed.

-------------------------------------------------------------------
Yaochu Jin
Future Technology Research
Honda R&D Europe (D) GmbH
Carl-Legien-Strasse 30
63073 Offenbach/Main
Germany

Tel: +49 69 89011735
Fax: +49 69 89011749
Email: yaochu.jin@hre-ftr.f.rd.honda.co.jp
Alias: yaochu.jin@ieee.org
http://www.soft-computing.de/

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