Fuzzy Semantic Analysis and Formal Specification of Conceptual Knowledge

NORTH- H O ~

Fuzzy Semantic Analysis and Formal Specification of Conceptual Knowledge

DAN E. TAMIR Department of Computer Science, Florida Institute of Technology, Melbourne, Florida 32901

and

ABRAHAM KANDEL Department of Computer Science and Engineering, College of Engineering, University of South Florida, 4202 East Fowler Avenue, ENG 118, Tampa, Florida, 33620-5399

ABSTRACT

Conceptual knowledge can be specified using one of the methods of formal specification of the semantics of a computer program: axiomatic semantics, denotational semantics, or operational semantics. For example, axiomatic semantics can be used to specify the conceptual knowledge of a medical doctor in an expert system for medical diagnosis. The problem is, however, that the knowledge of the expert is not always crisp and well defined. In such cases, a mean for specifying fuzzy conceptual knowledge is required. This paper proposes a method for the specifications of fuzzy conceptual knowledge. To this end, the concepts of fuzzy axiomatic semantics and fuzzy denotational semantics are developed. Fuzzy semantics is a generalization of classical semantics.

1. I N T R O D U C T I O N

One of the goals of research in the area of automatic knowledge representation is to find effective methods for the representation, storage, and retrieval of human knowledge. There is a strong relationship between language and knowledge. In fact, the main means of knowledge representation

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182 D.E. TAMIR AND A. KANDEL

and communication used by human beings is natural and formal languages. For centuries, scientists from several disciplines, including philosophy,

linguistics, and cognitive sciences, have studied various aspects of human knowledge. An important subject of research is the relationship between form and meaning in natural and formal languages. That is, the syntax and semantics of languages. In most cases, the methods of automatic knowledge representation emulate human knowledge representation schemes. There- fore, research in knowledge representation is based largely upon the results of extended research in linguistics, cognition, logic, and philosophy.

It is common to describe formal and natural languages in terms of three components: syntax, semantics, and pragmatics. Generally speaking, the syntax of a language determines the form of the expressions of that language. The semantics is the interpretation of the expressions, and the pragmatics is a set of environment-dependent restrictions that govern the interpretation of the syntax. Alternatively, syntax, semantics, and pragmatics can be viewed as a set of constraints imposed on a stream of symbols (code) to govern the encoding and the decoding of information. Human beings use these constraints efficiently to perform complicated tasks, such as generating and understanding natural and formal languages. Automat- ing the process of natural language processing is difficult, however, because the tasks of generating and understanding natural languages are generally NP-complete [3]. Moreover, it seems that the most difficult problems are those related to the semantics of natural languages.

There are two widely accepted approaches to the study of the semantics of natural languages: the logical/philosophical approach [5] and the linguistic/cognitive approach [31, 36]. The logical/philosophical approach applies the methods of semantic analysis of formal languages to a natural language or to subsets of natural languages [14]. For example, Frege [10], Russell [28], Tarski [35], and Carnap [5] investigated the logical approach and developed formal semantics for subsets of natural languages. Montague [21] proposed a denotational semantics to a subset of English. The cognitive approach connects the entities of a natural language to their image in the human mind [1], [30]. Because this technique relies on an analysis of the way humans process linguistic information, it is a highly subjective method. Nevertheless, Hintikka [14] showed that these two different approaches are not contradictory, and in many aspects can be unified.

In the context of computer science, the research on semantics of formal and natural languages has several application areas, such as software engineering, database management, natural language processing, and knowledge engineering. Generally, semantic analysis is done off-line and manually. Automating the process of semantic analysis is an important and relatively undeveloped subject of research. The following sections describe

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some of the computer science applications of automatic and manual semantic analysis.

The main three methods for semantic analysis of programming languages are the operational, axiomatic, and denotational semantics. These methods can be used for verification and validation of computer programs [2, 26 I. Hence they are a part of the core theory of software engineering, and some computer-aided software engineering tools (CASE tools) incorporate automatic semantic analysis in the verification and validation phase [43].

In the area of database management systems, semantic analysis is used for the specification of the conceptual schema of a database, that is, the "real-world" model captured by the database management system [19]. In addition to the semantic analysis methods used for programming language analysis, the discipline of database management systems uses knowledge representation methods, simulation, and modeling, for the construction of the conceptual schema. In this context, automatic semantic analysis is generally implemented through either logic programming or simulation and modeling. For example, Chang [6] applied semantic networks in the process of specifying the conceptual schema of a database. Ngu [23] developed a specification language that is used for modeling transaction schema in a financial organization.

Semantic analysis is an integral part of natural language processing. Generally, the semantic analysis is done manually at the stage of system design and automatically at the stage of processing. The main techniques for semantic analysis used in natural language processing are the logic-based axiomatic and denotational semantic [13]. Generally, these techniques are applied through a meaning representation language that represents the interpretation (meaning) of a small, well defined, and restricted domain [16].

In the context of knowledge engineering, semantic analysis is used for specification, validation, and verification of a knowledge base. For example, Roman [27] proposed a method for formal specification of geographical data processing requirements. Diederich and Milton [9] examined the problem of specifying the metadata for scientific data and knowledge bases. Zeruel [43] presented a knowledge-based system for modeling software system specifications.

A particularly difficult problem in the definition of semantics of formal and natural languages is dealing with uncertainty and ambiguity. This problem is common to both the logical and the cognitive approaches to semantic analysis. Human knowledge is complicated, vague, and contains fuzzy concepts. Automatic knowledge representation systems are often equipped with some means of resolving uncertainty [41 . These techniques include probability [18, 25], Dempster-Shafer evidence theory [20], degrees of certainty [29], and fuzzy techniques [38, 39].


Careful examination shows that most of the existing formalisms use only syntactic methods to resolve problems of ambiguity and uncertainty. For example, Goodman and Nguyen [12] examined different models for uncertainty in knowledge-based systems, but in the models presented, uncertainty is treated only at the syntatic level.

Fuzzy semantics can be used as a tool for the definition of concepts in an environment of uncertainty and incompleteness. Moreover, fuzzy semantics can be used to unify the logical and cognitive approaches to semantics of natural languages. Zadeh [40, 41] initiated research into fuzzy semantics and related some of the ideas of fuzzy semantics to the concept of linguistic variables. However, this direction of research has not been completely explored yet either by Zadeh or by other researchers.

In view of the similarities between knowledge-based and database management systems, Tamir and Kandel [33] proposed a multilayer reference model for knowledge-based systems. This model is analogous to the open system reference model used for standardizing data communication networks. One of the layers in the proposed model is referred to as the conceptual schema. It contains formal specifications of the conceptual knowledge stored in the knowledge base.

This paper is concerned with the process of formal specifications of conceptual knowledge. The objective is to develop specification methods ca- pable of representing the ambiguity vagueness and uncertainty inherent in conceptual knowledge. This is done through formal specifications of the semantics of conceptual knowledge.

This research contributes to the fields of artificial intelligence/knowledge engineering in three ways. First, fuzzy semantics can be used as a part of a knowledge-based verification process. Second, formal specification of a knowledge in a knowledge-based system can be used to standardize knowledge representation systems. Finally, the formal specification of knowledge is essential for cooperative expert system configurations.

The remainder of this paper is organized as follows. After the definitions of syntax and semantics of formal languages given in Section 2, Section 3 provides the formulation of crisp and fuzzy semantics relations. Section 4 presents two methods for the formal specification of conceptual knowledge. The methods are the fuzzy axiomatic specification of conceptual knowledge and the fuzzy denotational specification of conceptual knowledge. An example concerning the specifications of some of the components of a monetary system, using fuzzy denotational semantics is provided in Section 5.

C O N C E P T U A L K N O W L E D G E 185

2. SYNTAX AND SEMANTICS OF FORMAL SYSTEMS

The following definitions assume first order logic, ZF set theory (Zemelo [42] and Fraenkel [11], and fuzzy set theory as formulated in Tamir, Cao, Kandel, and Mott [34]. By this formulation, a fuzzy set is a subclass of the class of all the functions with range [0, 11.

DEFINITION 1. Let z be a set. The Kleene closure of z, the Cartesian product of the z with itself n times (r~ > 0), is denoted by z*. Thus, z* is also a set.

DEFINITION 2. Let E be a finite set of symbols. A language L t is defined to be any subset of E*. The subscript E is omitted from L if there is no potential ambiguity in this omission.

The hierarchy of languages as proposed by Chomsky [7] is assumed along with the commonly accepted definitions of the grammar G of a language L(GL) [15]. The term "grammar" is defined as follows:

DEFINITION 3. A grammar GL is a structure of the form {E, N, s, P) , where (1) E is a finite set of symbols (referred to as terminal symbols), (2) N is a finite set of nonterminal symbol (referred to as primitives), such that N N E = 0, (3) s E ~ , and (4) P is a set of partial functions, where the function Pi E P is the mapping P~ : (N x E)* --* (N x E)*. P is referred to as the set of production rules and the language L C E* is formed from P.

DEFINITION 4. A formal language is the pair {L, G}, where L is a language and G is a grammar.

DEFINITION 5. A fuzzy grammar is the structure {E, N, s, P) as defined in Definition 3, where one or more Of the components E, N, or P is a fuzzy set [17].

DEFINITION 6. Let [s] denote the power set of a set s. A formal system F, is the structure {L, Ax, In}, where L is a formal language, Ax c_ L, and In is an effective relation from [L] to L.

The set As is referred to as the set of axioms of the formal system. The relation In constitutes a set of inference rules in the system. Generally, the language L, the g rammar GL, the set Ax, and the relation In are referred to as the syntactic entities.

Montague [22] defines semantics as follows: "Semantics is concerned with

186 D . E . TAMIR AND A. KANDEL

relations between expressions and objects." This definition serves as the start ing point for semantic analysis and the construction of fuzzy semantics presented in this paper. Let E be a set of expressions and let O be a set of objects. By Montague's definition and by the definition of a relation, a semantic of E in O is any set S satisfying S C (E x O). The following definition formalizes the idea of a semantic relation.

DEFINITION 7. Let C be a set of concepts. A semantic relation (interpretation) is defined to be a relation from a language L to the set C. Thus SL,C, the semantics of L in C, is any set satisfying S C_ (L x C).

This definition of semantic relation is basically the same as Montague's definition. It includes, however, several important differences. First, instead of using a set of objects O, a set of concepts C is used. The set C may contain "real" objects, abstract objects, and attributes. For example, C can contain the abstract concept of real numbers. The use of the name C, instead of O as in Montague's definition, is merely a technical difference, which serves to stress that the goal is to formalize specifications of concepts. Second, instead of a general set of expressions E, sets of expressions that establish a language as defined in the foregoing text are used. Finally, from the way tha t Montague uses the definitions of semantics in his work, it seems that although he defines semantics as a relation from E to O, he actually means that semantics is a function from E to O. In the context of specification of concepts, the use of a semantic relation should be preferred over a semantic function, because a semantic relation enables the association of an element in the domain with more than one element in the codomain. Hence, it is a less restrictive form of semantic analysis tha t can enable modeling of ambiguity, uncertainty, and fuzzy semantics. A model of a formal system is defined as the follows:

DEFINITION 8. Let F~ = {L, As, In) be a formal system and let S L C (L × C) be an interpretation of L. A model M of the formal system F~ is an interpretation of Fs, where it is assumed that the axioms of F~ hold, tha t is, it is assumed that the axioms are valid.

In addition to the specification of a model, the semantic analysis of a formal system may also include analysis of the validity of inferences made by the system and the inference capability of the system under a given a model. These concepts are usually referred to as the soundness and completeness of the system. A system is sound if and only if the inference

C O N C E P T U A L KNOWLEDGE 187

rules of that system map a set of valid statements to a valid statement and is complete if any valid statement about the model can be deduced by the system [24].

3. CRISP AND FUZZY FORMULATIONS OF THE SEMANTIC RE- LATION

A semantic relation is a relation from a language L to a set of concepts C. Thus, the formulation of the semantic relation consists of the specification or construction of three sets: a set of expressions E that constitutes a language L~ a set of concepts C, and a set of pairs S, where SL C_ (L x C). This section assumes that the syntax is given, that is, that the set E is already defined. In order to set the semantic relation, the sets C and S must be specified or constructed.

Set theory allows for several basic methods of set construction, including enumerating the items of a set, using the axiom of unions or using the theorem of intersection. Two important methods of set construction used extensively in semantic analysis are the separation and the replacement theorems. The separation theorem constructs a set X by dividing a set Y into two sets X G Y and X - Y, where X is a set of items that satisfy a logical proposition, and the symbol " - " denotes set subtraction. The theorem of replacement asserts that if the domain of a mapping F from a set A to set B(F : (A ~ B)) is a set, then so is the codomain (F(A) C B).

Semantics analysis includes the construction of relations from a language L to a set C, where C is a set of human concepts. Stoy [32J distinguishes between two basic methods of logical semantics: axiomatic and denotational semantics. The main difference between the two lies in the way the sets C and S are constructed. Although both methods of semantic analysis can use any set construction axiom or theorem, axiomatic semantics tends to use the separation theorem for the construction of C and S, whereas denotational semantics tends to use the replacement theorem.

The different formalisms of semantics can now be described through the way the set S is defined. Let L be a language. Axiomatic semantics constructs the sets C and SL as follows:

1. A set of objects D, the set of objects denoted by the primitives of the syntax (nonterminal symbols), is enumerated.

2. The entire set C is constructed using logical propositions. 3. Propositions are used to construct S, which is a relation from L to C.

The denotational semantics constructs C and S in the following way:

1. Set-construction axioms and theorems are used to construct the set

188 D. E. TAMIR AND A. KANDEL

D C C (the denotation of the primitives of the syntax). 2. The axiom of replacement is used to set mappings from L to C. In the

classical case, the denotation of a complicated syntactic item of the language is a function of the denotation of its immediate constituents.

3. The set C is the range of the denotational functions. The semantic relation S is a by-product of this process.

Both methods of logical semantics exclude the case where the set of concepts C is ill-defined and contains vague, uncertain, or ambiguous components. Natural languages and hmnan knowledge, however, 'do not com- ply with this exclusion. They present ambiguity and contain vague and ill-defined concepts. One solution to this problem, adapted by many researchers, is to restrict semantic analysis to well-defined subsets of natural languages [21]. Another approach is to add syntactic entities to the language in order to enable the modeling of uncertainty. Nevertheless, it is possible to develop special formalisms of semantics such as fuzzy semantics, which allows incomplete specification of the sets D, C. The formalism of fuzzy semantics is a generalization of the classical methods of semantic analysis.

The difference between classical semantics and fuzzy semantics is that the relation SL C (L × C) is allowed to be a fuzzy relation. This approach can be used to define fuzzy axiomatic semantics, to define fuzzy denotational semantics, and to generalize Zadeh's definition of fuzzy semantics [40].

DEFINITION 9. Let C be a set (either crisp or fuzzy) of concepts, A fuzzy semantic relation is a fuzzy relation from a set of expressions consti- tuting a language L (possibly a fuzzy language) to the set C. Following the line of logic used to define axiomatic semantics and denotational semantics, fuzzy axiomatic semantics and fuzzy denotational semantics are developed.

3. I. FUZZY AXIOMATIC SEMANTICS

Let L be a language and let GL be the grammar (syntax) of L. Let C be a set (either crisp or fuzzy) of concepts. The fuzzy axiomatic semantics is formalized as follows:

1. A fuzzy set (D C_ C) of the denotations of the primitives of the syntax of L is constructed.

2. Fuzzy propositions are used to construct a fuzzy set C' from D, where C' C_C.

3. Fuzzy propositions are used to construct the fuzzy relation S, which is a fuzzy subset of L × C'.


3.2. F U Z Z Y DENOTATIONAL SEMANTICS

Let E C_ L be the set of expressions of a language L. Let C be a set of concepts. The fuzzy denotational semantics is formalized as follows:

1. A fuzzy set (D) of the denotations of the primitives of L is constructed.

2. A set of fuzzy denotational functions (F) from E to C is defined. A denotational function f (e) E F (where e E E) depends only on the immediate constitutes of e. In this way, the fuzzy set C ~ C_ C is defined as the fuzzy union of the denotational functions, i.e., C ~ =

U(f E F)&(ee E) f ( e ). 3. A set of fuzzy relations S from E to C t is defined.

Zadeh [40] equates meaning with a fuzzy subset of a universe of discourse U. The universe of discourse tha t Zadeh uses is the counterpart of the set of concepts C proposed here. The universe U as used by Zadeh is actually a set of at tr ibutes. For example, in the proposition "X is Tall," U can be a set of real numbers denoting.heights in centimeters, (e.g., IT0, 250]). Zadeh does not distinguish between fuzzy denotational and axiomatic semantics, but it seems that the semantics he proposes is a fuzzy version of denotational semantics.

The set of concepts C presented here is more general than the universe of discourse in the sense that it may contain concepts corresponding to "real" objects such as humans, at tr ibutes such as heights, and abstract concepts such as complex number. Thus, the "fuzzification" of semantics presented here is more general. Furthermore, the set C t, the set S of relations set over the expressions of L, and the elements of the set C ' are fuzzy sets. Apar t from the difference in the universe of discourse, the definition of denotational fuzzy semantics coincides with Zadeh's definition and may serve as a base for fuzzy denotational semantics. Moreover, because the fuzzy relation S can be established with fuzzy logic, the approach developed here can be used to define fuzzy axiomatic semantics and to generalize Zadeh's approach.

4. DENOTATION OF THE PRIMITIVES

Both the fuzzy and crisp formalisms of semantics make use of the set D, the set of the denotations of the primitives of the syntax. An essential feature of fuzzy semantics is that D is a fuzzy set. The specification of the denotation of the primitives of the syntax is basically a dictionary. Hence, the following notation is defined.

190 D . E . TAMIR AND A. KANDEL

DEFINITION 10. Let P be the set of primitives of the syntax and let D C C be the set of the denotations of the primitives (where C is a set of concepts), a dictionary T is defined to be a relation from P to D such tha t T c__ ( p x D).

Classical semantic methods does not allow ambiguity and assume tha t a primitive denotes a single object. The dictionary in a classical semantics is therefore a one-to-one mapping from P onto D. Due to the ambiguity inherent in natural languages, however, a term may have more than one denotation. It is therefore natural to allow the dictionary to be a fuzzy relation from P to a fuzzy set D. In this case, T is defined to be a fuzzy subset of (P x D).

A fuzzy specification of the denotations of the primitives of a language is an acyclic dictionary. This dictionary allows multiple definitions for the same term, yet demands a grade of membership or compatibili ty for each definition. This idea is actually implemented in some writ ten dictionaries, such as the dictionary by Dale [8], which includes a subjective measure of the compatibili ty of different definitions of identical terms.

5. SPECIFICATION OF CONCEPTS

The concepts developed so far are sufficient for establishing the process of formal specification of a set of concepts. Generally, the process of formal specification of a set of concepts C takes the following form:

1. Selection of a formal system to represent the concepts. 2. Formulation of the syntax of that system. 3. Selection of a method for the definition of the semantics of the formal

system:

• Specification of the interpretation using the method selected in step 2.

• Specification of D, the set of the denotation of the primitives. • Specification of the set C C C, the set of values that composite

terms in the language may assmne.

4. Setting the relations between terms in L and items in C t. 5. Specification of the model and analysis of the soundness and com-

pleteness of the system.

Step 3 in the specification procedure calls for the selection of the semantic analysis method. The previous section has described two new methods for setting the semantic relation: the fuzzy axiomatic and the fuzzy denotational methods. Each of these methods may be used in the process

CONC EP TUAL KNOWLEDGE 191

of semantic analysis of the system. The first question that the knowledge engineer has to approach, however, is whether fuzzy semantics is needed for a specific set of concepts. If the concepts of the underlying system are well defined, then the classical methods of semantic analysis can be applied to the concept-specification process. On the other hand, if the concepts are vague or if complete specification of the concepts is not practical, then it may be more appropriate to use a fuzzy scheme of semantic analysis. Examples in which schemes of incomplete reasoning seem to be a more practical method of semantic analysis are given in Yorick [37].

EXAMPLE. Specification of the Concepts of a Monetary System. The following is an example of the specifications of some of the components of a monetary system using fuzzy semantics. The language for definition is first order logic along with the ZF set theory. It is assumed that the syntactic entities of the formal system are given. The concern here is with the semantic entities of the specifications.

Suppose that the concept of a monetary system is to be explained to a group of people for the first time. For the sake of simplicity, it is assumed that the members of this group speak English, know logic and set theory, and are familiar with the concepts of fuzzy sets. In addition, the familiar- ity of the group with some basic concepts, such as "metals," "gold", and "paper" is assumed. This group, however, still uses a primitive system of exchanging goods for goods and is not familiar with a monetary system. Furthermore, the monetary system in this example is restricted to U.S. currency and contains coins but no other forms of' money, such as bills or bank notes.

Hence, a formal system, a "monetary system," is assumed. This system contains a "monetary language," a set of axioms, and a set of inference rules. In this paper, only the semantic analysis of the monetary language is given. The syntax of the monetary language is not defined here, yet it can generate a subset of English with sentences of the form: "Mary bought a car," John is selling a house," "The price of the house is $66,000," or "The value of the car is $4000." The formal language for the purpose of specification is first order logic and set theory, the metalanguage is English, and the specification is based on semantic analysis methods using fuzzy denotational semantics. The specification is done in two stages. In the first stage, the primitives and their denotation are specified. In the second stage, a set of denotational functions is defined.

A possible formal specification using fuzzy denotational semantics is as follows:

A. Definition of the primitives and their denotation:

192 D . E . T A M I R A N D A. K A N D E L

1. The pr imit ives are "coins," "goods," and "persons" 1 where:

• "Coins" is a name for a fuzzy set of s t a m p e d metals . Some of its member s are

agora meta l ring p ru t a

1.0 ' 0.0 ' 0 . 0 ' 0.5 '

{penny , nickel ,dime,qoarter ,s i lver-dollar } , . . . , } .

• "Goods" is a name for a fuzzy set of real and abs t rac t entities. Some of its member s are

shoe sheep knowledge } 2

0.8 ' 0.9 ' 0.8 ' ' " ' "

• "Persons" is a name for the set of human beings.

B. Definit ion of the denota t iona l functions: 1. Belong is a funct ion from a set of the form { (X × persons)} to the set

{True, False} (Belong : ( (X × persons) --~ (True, False))). I t is defined in the following way: Let X be a set, K E persons and X K C_ X. T h e n

(V(K E persons) ,V(xi c X) )

(Belong (xi, k) = True) if(xi E Xk)'~ (Belong (xi, k) False) o therwise ] " (1)

2. Money is a fuzzy funct ion f rom the set coins to the set R ~ = {$} × R, where R is the set of real numbers . T h a t is, money: Coins --* R ~, I t is defined in the following way:

• Money(penny) = $0.01 • Money(n icke l )= $0.05 • Money(d ime) = $0.1 • Money(quar te r ) = $0.25

In addit ion, an opera t ion of money addi t ion (denoted as a • b is defined in conjunct ion with the funct ion Money() : Let m o n e y ( m 1 ) =

1,,Coins,,, "goods," and "persons" are syntactic entities (names of sets). The semantic entities (the sets) are referred to as coins, goods, and persons.

2The degree of membership of items such as shoes in the set "goods" is subjective and has been determined arbitrarily in this example.

CONCEPTUAL KNOWLEDGE 193

$r l , money(m2) = $r2, where ml , m2 are two coins and r l , r2 are two real numbers. Then money (ml • m2) = $(rl + r2).

3. Value is a fuzzy function from the set {goods × persons} to the set money. Thus, value: (goods × persons) --~ money. For example, for some k C persons,

valuek = { shoe, $30 sheep, $220 0 . ~ ' 0.75 ' (2)

Sears_building, $10, 000,000k } 0.8 . . . . . (3)

4. Price is a fuzzy function from the set {goods × persons} to the set money price: (goods × persons) --* money. For example, for some j E persons,

shoe, S20 sheep, $200 } Pricej = [ 0.99 ' 1.0 , . . . . (4)

Note that value is a fuzzy function that denotes the "value" of an item g in goods to a person i, and price is the "price" that a person 1 would ask for an item g in goods.

5. Let j , k E persons, let Gk denote the goods that belong to person k, and let Mk be the money that belongs to k. The set of possessions of k, Pk, is defined to be Pk -=- Gk CJ Mk. Assume that gj C Gj (g are goods that belong to j) . Buy is a fuzzy function from the set {Pk × {gj}} to the set Pk so that buy(Pk,gj) = (GkAgj) -pricey(g). 3

6. Let j , k be members of persons and let Pj be the set of possessions of j . Let gj E G (g belongs to j) . Then sell is a fuzzy function from the set {Pj x {gj}} to the set Pk such that sell(Pj,gj) = (Gj v a l u e k , ( g ) ) -

The denotational functions defined in this example formally specify some of the components of a monetary system. Furthermore, they can be used in a process of semantic analysis of a subset of English that includes sentences describing the use of a monetary system. One advantage of using fuzzy semantics is that it appeals to human cognition and, therefore, may be used to unify the logical and the cognitive approaches to semantic analysis. Some researchers have proposed ways of unifying these two approaches to semantics [14, 40]. Another a t tempt to unify the two semantic analysis methods is presented here: The notation of fuzzy semantics enables unification of the logical and cognitive approaches.

3 ,,_, denotes set subtraction.


6. CONCLUSIONS

Formal specifications of a knowledge base is a part of the process of de- signing an artificial intelligence system such as expert systems. The specifications document is used in the stages of system development, testing, verification, validation, and maintenance. The specification document includes formal specifications of the following subjects: (1) the data structures used for the knowledge representation, (2) the semantics and pragmatics of the knowledge representation structures, and (3) the specifications of the conceptual knowledge. This paper is concered with the formal specifications of the conceptual knowledge. The other parts of the specifications document are described in Tamir and Kandel [33].

Automating the process of compiling the specification document can be a part of a computer-aided knowledge engineering tool similar to computer- aided software development tools.

As a step toward automating the process of formal specification of knowledge, this paper defines the terms related to the syntax and semantics of a language. It presents two basic schemes of semantic analysis--axiomatic semantics and denotational semantics--and develops the notion of fuzzy denotational semantics and fuzzy axiomatic semantics as a generalization of the classical semantic schemes. These formulations are used to propose axiomatic specification of conceptual knowledge and denotational specification of conceptual knowledge. The approach is exemplified through the specification of some components in the concept of a monetary system. The specification is done via denotational fuzzy semantics.

This work was partially supported by the Florida High Technology and Industry Council grant UPN 85100316, and by USF DSR research grant #2108-934R0

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Received 1 January 1994; revised 28 June 1994

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Fuzzy Semantic Analysis and Formal Specification of Conceptual Knowledge