PRUF - A Language for the Representation of Meaning in Natural Languages

PRUF -A LANGUAGE FOR THE REPRESENTATION OF MEANING IN NATURAL LANGUAGES

L.A. Zadeh*

Computer Science Division University of California Berkeley, CA 94720

PRUF--an acronym for Possibilistic Relational Qniversal !uzzy--is a designation for a novel type of synthetic language which is intended to serve as a target language for the representation of meaning of expressions in a natural language.

PRUF is based on the premise that the uncer­ tainty in the interpretation of a proposition in a natural language is possibilistic rather than prob­ abilistic in nature. This implies that a proposi­ tion, p, of the form "X is F" where X is the name of an object and F is a fuzzy subset of a universe of discourse U, translates in PRUF into an expres­ sion, P, which defines a procedure whose domain is the set of possibly fuzzy relations in a database. Acting on these relations, P yields a possibility distribution (or a set of possibility distributions) which characterizes the information, I(p), conveyed by p.

The concept of a possibility distribution is defined as follows (Zadeh 1977). Let X be a vari­ able taking values in U, and let F be a fuzzy sub­ set of U which is characterized by a membership function ).JF: U+ [0,1], with l1F(u), uEU, represent­ ing the grade of membership of u in F. Then the proposition "X is F" induces a possibility distri­ bution, rrx. which is given by the possibility asso­ ciation equation

Thus, (1) implies that the possibility that X may take the value u is given by

and, more generally, if A is a nonfuzzy subset of

U,

For example, the proposition "X is small," where small is a fuzzy set defined by

implies that the possibility that X is 3 is 0.8, and that X is either 4 or 5 is 0.6.

As a simple illustration, the propos:!.tion "John is young" leads to the possibility associa­ tion equation ITAge(John) = young, where Age(John) is an implied attribute of John and young is a fuzzy subset of the interval [0,100].

A proposition such as "John resides near Boston," translates in PRUF into the expression

RESIDES [subject= John;

Currently on leave with the Computer Science Department, IBM Research Laboratory, San Jose, CA 95143. Research supported by the National Science

* Foundation Grant MCS-76-06693.

where NEAR is a fuzzy relation which associates with each pair of cities (cityl,city2) the degree to which they are near one another; RESIDES is a relation which is particularized by (a) assigning the value John to the attribute subject; and (b) assigning the possibility distribution

citylNEAR[city2 =Boston] to the attribute location, in which NEAR[city2 =Boston] denotes a particulari­ zation of the relation NEAR with city2 set equal to Boston, and citylNEAR[city2 =Boston] denotes its projection on the domain of the attribute city!. Typically, an expression in PRUF contains names of relations, names of attributes, particularized pos­ sibility distributions which are associated -with attributes, and labels of tests. In appearance, the simpler expressions in PRUF resemble those of the query language SQUARE and may involve an arbi­ trary number of levels of nesting.

PRUF has a variety of facilities for modeling the logical organization of natural languages. Among these are: attribute modification, truth qualification, probability qualification, possibil­ ity qualification, quantification, particulariza­ tion and composition. Its expressive power is con­ siderably greater than that- of semantic networks or first order predicate calculus, and by employing the concept of a possibility distribution PRUF pro­ vides an effective framework for dealing with the imprecision of natural languages in a systematic fashion.

The logic underlying PRUF is a fuzzy logic, FL, in which the truth-values are linguistic, i.e., are expressed as true, quite true, very true, more or less true, etc., with the base logic for FL being Lukasiewicz's LAlephl logic in which the truth­ values are points in the unit interval (Bellman & Zadeh 1976). In addition to the use of linguistic truth-values, PRUF allows the use of linguistic quantifiers, e.g.,many, few, several, most, almost all, etc.; linguistic probabilities expressed as likely, very likely, more or less likely, etc.; and linguistic possibilities expressed as possible, quite possible, slightly possible, almost impossi­ ble, etc.

PRUF has only two basic rules of inference: (a) The projection rule, which asserts that from an n-ary possibility distribution IT(x 1 , ... ,Xn) one can infer by projection a marginal possibility distri­ bution IT(Xi , ... ,Xik) where (il·····ik) is a subse­ quence of t~e index sequence (1, ... ,n); and(b) that from IT(Xl, ... ,Xn) and IT(xi 1 , ..· ,Xik) one can infer the intersection of IT(x1 , ... ,Xn) with the cylindrical extension of IT(xi 1 , ... ,Xik)· In combination, these two rules lead to the compositional rule of inference which includes the classical modus ponens as a special case.