ABSTRACT
Fuzzy set theory and fuzzy logic are a highly suitable and applicable basis for developing knowledge-based systems in medicine for tasks such as the interpretation of sets of medical findings, syndrome differentiation in Eastern medicine, diagnosis of diseases in Western medicine, mixed diagnosis of integrated Western and Eastern medicine, the optimal selection of medical treatments integrating Western and Eastern medicine, and for real-time monitoring of patient data. This was verified by trials with the following systems that were developed by our group in Vietnam: a fuzzy Expert System for Syndromes Differentiation in Oriental Traditional Medicine, an Expert System for Lung Diseases using fuzzy logic, Case Based Reasoning for Medical Diagnosis using fuzzy set theory, a diagnostic system combining disease diagnosis of Western Medicine with syndrome differentiation of Oriental Traditional Medicine, a fuzzy system for classification of Western and Eastern medicaments and finally, a fuzzy system for diagnosis and treatment of integrated Western and Eastern Medicine.

INTRODUCTION
In recent years, computational intelligence has been used to solve many complex problems by developing intelligent systems. Fuzzy logic has proved to be a powerful tool for decision making systems, such as expert and pattern classification systems. Fuzzy set theory has been used in some medical expert systems for example.
In traditional rule-based approaches, knowledge is encoded in the form of antecedent consequent structure. When new data are encountered, it is matched to the antecedents clause of each rule, and those rules where antecedents match a data exactly are fired, establishing the consequent clauses. This process continues until the desired conclusion
is reached, or no new rule can be fired. In the past decade, fuzzy logic has proved to be useful for intelligent systems in medicine. Some examples of using fuzzy logic to develop fuzzy intelligent systems are fuzzy systems in microprocessors, fuzzy control of N. H. Phuong, V. Kreino_ich /International Journal of Medical Informatics 62 (2001) 165–173 166 the subway system in the Japanese city of Sendai, fuzzy washing machines, fuzzy cam-eras and camcorders, which map image data to lens settings and fuzzy voice commands: ‘up’, ‘land’, ‘hover’ to control an unmanned helicopter. In this paper, we discuss how fuzzy set theory and fuzzy logic can be used to develop knowledge-based systems in medicine. Some notions of fuzzy logic in both a narrow and broad sense are introduced in Section 2. Section 3 describes the formalism of a fuzzy rule-based system in medicine. An example of applying fuzzy logic in knowledge based systems in medicine is given in Section 4. Some conclusions are made in Section 5.

2. NOTIONS OF FUZZY LOGIC
In order to show how fuzzy sets theory and fuzzy logic are a suitable tool in representing and handling medical concepts, let us answer three questions: what is logic, what is fuzziness, and what meaning has the term ‘fuzzy logic’ then, we will discuss the application of fuzzy logic in medicine.
Logic studies the notions(s) of consequence: it deals with propositions, sets of propositions and the relationship of consequence between the propositions. The task of formal logic is to represent all this by means of well-defined logical calculi. Various calculi differ in their definitions of sentences and notion(s) of consequence (propositional logics, predicate logics, modal prepositional/predicate logics, many-valued prepositional/predicate logics). Often, a logical calculus has two notions of consequence: syntactical (based on a notion of proof) and semantical (based on a notion of truth); then, there are the natural questions of soundness (does probability imply truth?) and completeness (does truth imply probability?).
Medical fuzziness is impreciseness: a fuzzy proposition may be true to some degree. The word ‘crisp’ is used as meaning ‘non-fuzzy’. Standard examples of fuzzy propositions use linguistic variables such as age with possible values young, medium, old or similar. The sentence ‘the patient is young’ is true in some degree the lower the age of the patient, the truer the sentence. Truth of a fuzzy proposition is a matter of degree.
‘Fuzzy logic’ in medicine: the term ‘fuzzy logic’ has two different meanings—wide and narrow.
Let recall the preface made by Zadeh. In a narrow sense, fuzzy logic, FLn, is a logical system that aims at a formalization of approximate reasoning. In this sense, FLn is an extension of multivalued logic. However, the agenda of FLn is quite different from that of traditional multivalued logics. In particular, such key concepts in FLn as a concept of a linguistic variable, canonical form, fuzzy if –then rule, fuzzy quantification, the extension principle, the compositional rule of inference and interpolative reasoning, among others, are not addressed in traditional systems. This is the reason why FLn has a much wider range of applications than traditional systems. In its widest sense, fuzzy logic, FLw, is fuzzily synonymous with fuzzy set theory, FST, which is the theory of classes with unsharp boundaries. FST is much broader than FLn and includes the latter as one of its branches.
Based on Zadeh’s opinions on ‘fuzzy logic’, we may conclude two things. First, in the broad sense, everything dealing with fuzziness may be called ‘fuzzy logic’. Second, in the narrow sense, formal calculus of many-valued logic is the base of fuzzy logic.
Now, let us deal with ‘fuzzy logic’ in medicine in a broad sense. In medicine, especially in oriental medicine, most medical concepts are fuzzy. The imprecise nature of medical concepts and their relationships re-quires the use of ‘fuzzy logic’. It defines inexact medical entities as fuzzy sets and provides a linguistic approach with an excellent approximation to texts. ‘fuzzy logic’ offers reasoning methods capable of drawing approximate inferences. for example, in oriental medicine for back pain that is not caused by a disease, acupuncture is often very efficient. rules of oriental medicine include words like ‘severe pain’ that are difficult to formalize and to measure. however, traditionally, mathematics uses crisp (well-defined) properties p(x), i.e. properties that are either true or false.
Each such property defines a set: {x x has a property P}. In 1965, Zadeh proposed a theory that explains how to formalize ‘fuzzy’ (non-crisp) properties: a crisp property P can be described by a characteristic function : x  {0,1}. A fuzzy property can be described as a function  : x [0,1]. The value _(x) indicates the degree to which x has the property (e.g. to which x has pain). An example of representing a medical concept ‘high fever’ as a fuzzy set is illustrated in Fig. 1. In Fig. 1, (a) if x is greater than 39°C, then membership function (x) of the medical concept ‘High Fever’ is 1 (x has surely ‘high fever’), (b) if x is less than 38.5°C, then membership function (x) of medical concept.

‘High Fever’ is 0 (x has surely not ‘high fever’), and © if x is in the interval (38.5– 39 °C), then x has a property ‘high fever’ with some degree in [0,1]. We will show reasoning with medical fuzzy sets. In the knowledge base of rule-based systems, fuzzy properties are often connected by logical words like ‘and’, ‘or’, ‘not’ .In traditional set theory, these operations correspond to , V, ’. So, we need to extend these operations to fuzzy sets.



3. RULE-BASED FUZZY SYSTEMS IN MEDICINE
In rule-based fuzzy systems in medicine, experts often formulate their statement in terms of rules of the type: If x is A,and y is B,then z is C For example: If the back pain is severe, and the patient is old, then apply acupuncture to a certain point for a long time. Here: x is patient’s pain, A is ‘severe’; y is patient’s age, B is ‘old’ z describes treatment time, C is ‘long time’. Now, we describe the formalization of such If –Then rules in the rule base of expert systems. For each rule:

We can compute the degree to which its conditions are applicable as

Then, for each possible z, we can compute the degree to which this rule holds:

If we have several rules, r1,…,rn, then the degree, _(z), to which one of them is applicable for a given effect, z, is:

Finally, we find the ‘most probable’ value, z, and use it, e.g. we take z for which

Remark: Most probable would mean the value for which the membership function takes the largest possible value; centroid defuzzification is that which is the ‘average’ or for which the least-square error is the smallest.

4. APPLICATION OF FUZZY LOGIC IN DEVELOPING RULE BASED SYSTEMS IN MEDICINE
Based on a study of the famous medical expert systems such as MYCIN and CA-DIAG, in this section, we will illustrate the application of the above formalism for incorporating negative knowledge into fuzzy knowledge-based systems by using an ordered Abelian group. We used this formal-ism to design expert systems for lung disease diagnosis, for syndrome differentiation in Eastern medicine and for diagnosis combining Western and Eastern medicine indiagnosis and several other medical applications. Due to limited space, therefore, in this paper, we will show only the performance of the diagnostic system for lung diseases diagnosis using fuzzy logic developed with Hung H. Dang, Nandipuram R. Prasad as an example.