The concept of a generalized assignment statement and its application to commonsense reasoning

The concept of a generalized assignment statement provides a basis for a unified approach to knowledge representation and inference. The basic idea underlying this approach is that a proposition, p, in a natural language may be viewed as an elastic constraint on a variable, X. More specifically, the meaning of p may be represented as p → X isr C in which → should be read as “translates into” and the right-hand member is a representation of the meaning of p in the form of a generalized assignment statement. In this statement, X plays the role of a constrained variable; C is the constraint on X; and r in the copula isr is a variable whose values define the role of C in relation to X. The prinipal values of r are d, standing for disjunctive; c, standing for conjunctive; p, standing for probabilistic; g, standing for granular; and h, standing for hybrid. Since in most cases the value of r is d, it is convenient to abbreviate isd to is. With this understanding, X is C should be interpreted as X isd C.In general, X, C, and r are implicit rather than explicit in p. Viewed in this perspective, the problem of meaning representation is that of explicitating X, C and r, given p. In general, X and C are defined procedurally. Thus the principal steps in representing the meaning of p are: (1) construction of an explanatory database, ED; (2) construction of a procedure which acts on ED and yields X; and construction of a procedure which acts on ED and yields C.If p1, …, pu are propositions in a knowledge-base, inference from p1, …, pu starts with representing the meaning of each p1 in the form of a generalized assignment statement, Xi isri Ci, i = 1, … n. Then the constraints are combined through the application of a collection of rules of inference, resulting in a computed constraint, C, on a specified variable Y as a function of the Ci. In general, the determination of C requires the solution of a non-linear program.Although the approach to knowledge representation and inference based on the concept of a generalized assignment statement does not follow the traditional lines, it is in fact quite natural and easy to understand. Several illustrative examples show how a proposition can be expressed as a generalized assignment statement, and how an answer to a question may be obtained through the solution of a nonlinear program.A particularly important application of the concept of a generalized assignment statement relates to the representation and inference from commonsense knowledge. More specifically, a distinguishing characteristic of commonsense knowledge is that the facts and rules which comprise such knowledges, e.g., birds can fly and tomatoes are red when they are ripe, are, for the most part, preponderantly rather than universally true. For this reason, the concepts of typicality, normality, and default play an essential role in most of the existing approaches to commonsense reasoning and knowledge representation.In an alternative approach based on the concept of a generalized assignment statement, a comparable role is played by the concept of usuality. As its name suggests, the concept of usuality relates to what is usual or, more precisely, to events of high probability. A basic concept which derives from usuality is that of a usual value of a variable. Thus, in the proposition a cup of coffee usually costs about fifty cents, about fifty cents is a usual value of the variable Cost (Cup Coffee)). In general, a usual value is imprecise and non-unique.Viewed in the perspective of usuality, commonsense knowledge may be regarded as a collection of usuality-qualified generalized assignment statements of the form usually (X is F) and usually (X is F if Y is R), in which F is a usual value of the variable X, R is a constraint on the conditioning variable Y, and usually is a fuzzy quantifier which is representable as a fuzzy proportion in the inverval [0,1].Based on this view of commonsense knowledge, various inference rules for reasoning with usuality-qualified facts and rules are developed. In particular, it is shown that from usually (X is A) and usually (X is B), it follows that (2usually-1) (X is A ∩ B), where A ∩ B is the conjunction of A and B, and (2usually-1) is a fuzzy number which is less specific that usually. Rules of this type are of direct relevance to property inheritance in the presence of exceptions, to the combination of uncertain evidence in expert systems and, more generally, to inference under uncertainty.