Need for Expert Systems Need for Degrees of . . . Need for “And”- . . . Archimedean t-Norms What If We Use Different Inverse Operations “And”-Operations in the Formulation of the . . . Main Definition Same Expert System Main Result Possible Application to . . . Mahdokht Afravi and Vladik Kreinovich Home Page Title Page Department of Computer Science University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ mafravi@miners.utep.edu vladik@utep.edu Page 1 of 27 Go Back Full Screen Close Quit
Need for Expert Systems Need for Degrees of . . . 1. Need for Expert Systems Need for “And”- . . . • We often rely on expert knowledge; e.g.: Archimedean t-Norms Inverse Operations • we ask medical experts to help cure patients, Formulation of the . . . • we ask human expert in piloting to pilot planes. Main Definition • Ideally, everyone should have access to the top experts: Main Result • top experts in medicine should cure all the patients, Possible Application to . . . Home Page • top pilots should pilot every plane, etc. Title Page • However, there are very few best experts. ◭◭ ◮◮ • So, it is not realistic to expect these top experts to satisfy all the demands. ◭ ◮ • It is therefore desirable to describe the knowledge of Page 2 of 27 the top experts inside a computer. Go Back • Then other experts can use this knowledge. Full Screen • This descriptions are known as expert systems. Close Quit
Need for Expert Systems Need for Degrees of . . . 2. Need for Degrees of Certainty Need for “And”- . . . • Experts are usually not 100% certain about their state- Archimedean t-Norms ments. For example: Inverse Operations Formulation of the . . . • a medical expert may indicate some visible signs of Main Definition a heart attack, but Main Result • but experts cannot tell with absolute certainty Possible Application to . . . whether a patient is experiencing a heart attack. Home Page • The expert system must store the experts’ degrees of Title Page certainty in different statements. ◭◭ ◮◮ • In the computer, “absolutely true” is usually repre- ◭ ◮ sented by 1, and “absolutely false” by 0. Page 3 of 27 • Thus, intermediate degrees of certainty are usually de- Go Back scribed by numbers between 0 and 1. Full Screen Close Quit
Need for Expert Systems Need for Degrees of . . . 3. Need for “And”-Operations Need for “And”- . . . • One of the main objectives of an expert system is to Archimedean t-Norms help decision maker make decisions. Inverse Operations Formulation of the . . . • Decisions are rarely based on a single expert statement. Main Definition • Usually, two or more statements are used to argue for Main Result the proper decision. Possible Application to . . . Home Page • For example, we want is, given the symptoms, come up with an appropriate cure. Title Page • However, medical rules rarely go from symptoms di- ◭◭ ◮◮ rectly to cure. Usually: ◭ ◮ • some rules describe a diagnosis based on the symp- Page 4 of 27 toms (and test results), and Go Back • other rules describe a cure based on the diagnosis. Full Screen Close Quit
Need for Expert Systems Need for Degrees of . . . 4. Need for “And”-Operations (cont-d) Need for “And”- . . . • So, to decide on an appropriate cure based on given Archimedean t-Norms symptoms, we must use at least two rules: Inverse Operations Formulation of the . . . • a rule describing the diagnosis, and Main Definition • a rule selecting a cure based on the diagnosis. Main Result • It is desirable, in addition to a recommendation r , to Possible Application to . . . also estimate our degree of certainty in r . Home Page • For a recommendation based on several statements: Title Page • we are certain in this recommendation ◭◭ ◮◮ • if we are certain in all the statements used in de- ◭ ◮ riving this recommendation. Page 5 of 27 • Thus, the degree to which we are confident is a given Go Back recommendation is the degree to which: Full Screen • the first statement holds and • the second statement holds, etc. Close Quit
Need for Expert Systems Need for Degrees of . . . 5. Need for “And”-Operations (cont-d) Need for “And”- . . . • So, we need to know the degrees to which each possible Archimedean t-Norms “and”-combination of these statement hold. Inverse Operations Formulation of the . . . • Ideally, we should elicit, from the experts, the degrees Main Definition to which each such combination holds. Main Result • However, this is not practically possible: for n state- Possible Application to . . . ments, we can have 2 n − ( n +1) possible combinations. Home Page • So even for a reasonable value n ≈ 100, we have an Title Page astronomical number of combinations. ◭◭ ◮◮ • We cannot elicit the degrees for all “and”-combinations ◭ ◮ directly from the experts. Page 6 of 27 • We must therefore estimate these degrees based on the Go Back known degrees of confidence in individual statements. Full Screen Close Quit
Need for Expert Systems Need for Degrees of . . . 6. “And”-Operations and t-Norms Need for “And”- . . . • In other words, we need to be able: Archimedean t-Norms Inverse Operations • given the expert’s degrees a = d ( A ) and b = d ( B ) Formulation of the . . . in two statements A and B , Main Definition • to come up with an estimate for the expert’s degree Main Result of confidence in the “and”-combination A & B . Possible Application to . . . • This estimate – depending on a and b – will be denoted Home Page by f & ( a, b ); it is known as an “and”-operation. Title Page • Usually, we assume that the same “and”-operation can ◭◭ ◮◮ be used for all possible pairs of statements ( A, B ). ◭ ◮ • Under this assumption, we get reasonable requirements Page 7 of 27 on the “and”-operation known as t-norms . Go Back • For example, A & B means the same as B & A . Full Screen • It is thus reasonable to require that f & ( a, b ) = f & ( b, a ). Close Quit
Need for Expert Systems Need for Degrees of . . . 7. t-Norms (cont-d) Need for “And”- . . . • Similarly, A & ( B & C ) means the same as Archimedean t-Norms ( A & B ) & C , so we should have Inverse Operations Formulation of the . . . f & ( a, f & ( b, c )) = f & ( f & ( a, b ) , c ) . Main Definition Main Result • In mathematical terms, this means that the “and”- operation should be associative. Possible Application to . . . Home Page • Also: Title Page • if we increase our degree of confidence in A and/or ◭◭ ◮◮ B , ◭ ◮ • this should either increase our degree of confidence in A & B . Page 8 of 27 • So, the “and”-operation should be monotonic : Go Back Full Screen if a ≤ a ′ and b ≤ b ′ , then f & ( a, b ) ≤ f & ( a ′ , b ′ ). Close Quit
Need for Expert Systems Need for Degrees of . . . 8. Archimedean t-Norms Need for “And”- . . . • If a = d ( A ) = 0, then increasing our degree of confi- Archimedean t-Norms dence in B does not change the estimate for A & B : Inverse Operations Formulation of the . . . b < b ′ , but f & ( a, b ) = f & ( a, b ′ ) = 0. Main Definition • However, if a > 0, then it’s reasonable to require that Main Result in b increases confidence in A & B : Possible Application to . . . Home Page if a > 0 and b < b ′ , then f & ( a, b ) < f & ( a, b ′ ). Title Page • t-norms that satisfy this additional requirement are ◭◭ ◮◮ known as Archimedean . ◭ ◮ • Not all t-norms are Archimedean: e.g., f & ( a, b ) = Page 9 of 27 min( a, b ) is not an Arhimedean t-norm. Go Back Full Screen Close Quit
Need for Expert Systems Need for Degrees of . . . 9. Archimedean t-Norms (cont-d) Need for “And”- . . . • However, it can be proven that every t-norm can be Archimedean t-Norms approximated, Inverse Operations Formulation of the . . . • with any given accuracy, Main Definition • by an Archimedean one. Main Result • In practice, the degrees are known with some accuracy Possible Application to . . . anyway. Home Page • Thus, without losing any generality, we can always as- Title Page sume that our t-norms are Archimedean. ◭◭ ◮◮ • A general Archimedean t-norm can be obtained from ◭ ◮ f & ( a, b ) = a · b by a re-scaling: Page 10 of 27 f & ( a, b ) = g − 1 ( g ( a ) · g ( b )) for some 1-1 cont. g : [0 , 1] → [0 , 1] . Go Back • When a ≤ b , then, for f & ( a, b ) = a · b , there exists a Full Screen unique c for which a = f & ( b, c ): namely, c = a/b . Close Quit
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