From Expert Words Directly to Numerical Simulations Group Theoretic Approach to Computing with Words in Information Intelligent Systems

In many real life situations e g when making an environmental decision it is important to be able to predict long term consequences of di erent decisions Very often these predictions must be done in the situation where the only available information consists of expert rules which are formulated by words from natural language One possible way to transform these expert words into numerical simulation lead ing to prediction is to use the fuzzy control methodology However there is a problem with using this methodology it invokes replacing each word by a membership function and this replacement drastically increases the required computer space and thus increases the compu tation time i e it de granulates the original compact description It is therefore desirable to get from the original words directly to numerical simulations thus avoiding this de granulation In seeking this direct transformation we will use the experience of modern physics where symmetry groups are a tool that enables to compress complicated di erential equations into compact form In our previous papers we have shown that the symmetry group approach can be used to nd optimal membership functions optimal t norms and t conorms and optimal defuzzi cation procedures In this paper we show that the same approach can also be used to combine these steps and produce an optimal direct transformation from words to numerical results From Expert Words to Numerical Simulations Necessity For many complex systems long term predictions are necessary In the century there have been many situations in which an environment related decision that seemed at rst to be very reasonable and successful turned out in the long run to have been a mistake Such decisions include the use of pesticides e g DDT the design of some river dams etc To avoid such mistakes we must be able to predict long term conse quences of each decision Numerical simulations are needed The ideal situation is when we have an analytical formula that would enable us to exactly predict the consequences of each decision However in reality such formulas are extremely rare In most cases we have to rely on numerical simulations instead Often expert words are the only information we have In some cases we know the di erential or di erence equations that describe the sys tem However in many other cases especially for environmental systems we do not know the exact equations Instead we have the informal expert knowledge This knowledge is usually formulated in terms of rules that use only words from natural language such as if x increases then y slightly decreases We must transform fuzzy expert words into crisp numerical simulations Thus to make meaningful decisions we must somehow trans form the fuzzy expert words that describe the system s dynamics into crisp equations that would enable us to run numerical computer simulations of the consequences of di erent possible decisions From Expert Words to Numerical Simulations How It Is Done Now For the desired translation we can use the experience of fuzzy control There is an area where the methodology of transforming expert rules like the one described above into numerical formulas has been already successfully developed the area of intelligent control based on fuzzy expert rules The corresponding fuzzy control methodology was rst developed by Mamdani in for the latest overview see e g see Klir and Yuan Nguyen and Walker and Nguyen and Sugeno So we can use this methodology to transform expert words into numerical simulations Fuzzy control methodology in brief Why explain This paper has three main objectives to explain the fuzzy control methodology and how it can be used for simulations to explain the problems with applying this methodology to simulations and to propose a better methodology Thus fuzzy control methodology is crucial for us and so we will brie y describe this methodology for those readers who are not familiar with it readers familiar with fuzzy control can skip this explanation In this explanation we will only describe the simplest basic version of fuzzy control Rules In the fuzzy control methodology we start with expert rules of the type If x is Ar and xn is Arn then u is Br Here x xn are inputs i e parameters whose values we measure in order to decide what control to apply e g the position and velocity of a spaceship u is the desired control e g the force applied to the spaceship r R is the rule number and Ari and Br are words from natural language that are used in r th rule like small medium large approximately etc To transform these rules into a precise control strategy u u x xn we do the following First stage First we describe the words Ari and Br in numerical terms In fuzzy control methodology we usually describe each such word by a mem bership function ri xi or correspondingly r u i e a function that describes for each xi to what extent the experts believe this very value xi to satisfy the corresponding property Ari e g to what extent the experts believe that xi is small These degrees of belief run from complete disbelief xi does not satisfy the property Ari to complete belief i e from false to true In the computer false is usually represented by and true by Therefore in most implementations the membership functions take values from to Second stage Next for each input x xn and for each possible value we describe to what extent i th rule holds i e to what extent it is true that x satis es the property Ar and x satis es the property Ar and xn satis es the property Arn and u satis es the property Br We have n statements Ar x Arn xn Br u and for each of these statements we know its degree of belief truth value We are interested in the degree of belief of their and combination disjunction Ar x Arn xn Br u If all the combined statements were known to be exactly true or exactly false then we would be able to use the known and operation for Boolean truth values Thus what we need is to generalize the traditional Boolean and operation that is well de ned for truth values from the set f g to the entire interval Many such generalizations have been proposed they are usually called and operations or t norms Two most widely used examples of t norms are a b min a b and a b a b In terms of a t norm the degree of belief that r th rule is applicable is equal to br r x rn xn r u Third stage To compute for given x xn and u the degree of belief that this u is a reasonable control for the given x xn we must estimate the degree of belief that one of the rules is applicable i e that either the rst rule is applicable or the second rule is applicable etc We know the degree of belief br that each rule is applicable so to combine them we need an extension of the Boolean or operation to the interval This extended or operation is usually called a t conorm The most widely used t conorms are a b max a b and a b a b a b So for each u we can estimate the desired degree of belief as u b bR Fourth stage After the previous step for every possible value u we get the degree of belief u that u is a reasonable control We need to use the membership function u to choose a single value u that corresponds to the given x xn The transformation from the fuzzy membership function u to a single crisp value u is called a defuzzi cation In fuzzy control one of the most widely used defuzzi cation procedures is the following centroid defuzzi cation