The theory which is outlined in this lecture, call it RCC for short, is a system of reasoning and computation which is not in the traditional spirit. In large measure, RCC is oriented toward reasoning and computation in an environment of uncertainty, imprecision and partiality of truth. The centerpiece of RCC is the concept of a restriction--a basic concept which is deceptively simple. Informally, a restriction is an answer to a question of the form: What is the value of a variable, X? More concretely, a restriction, R(X), is a limitation on the values which X can take. A restriction is precisiated if R(X) is mathematically well defined; otherwise it is unprecisiated. Generally, restrictions which are described in a natural language are unprecisiated. A restriction is precisiable if it lends itself to precisiation. A restriction is singular if R(X) is a singleton; otherwise it is nonsingular. Nonsingularity implies uncertainty. Examples. Robert is staying at a hotel in Berkeley. He asks the concierge, "How long will it take me to drive to SF Airport?" Possible answers: one hour; one hour plus minus fifteen minutes; about one hour; usually about one hour, etc. Each of these answers is a restriction on the variable, Driving time. The first two answers are precisiated restrictions. The last two answers are unprecisiated. Another example. Consider the proposition, p: Most Swedes are tall. What is the truth-value of p? Possible answers: True; 0.8; about 0.8; high; likely high, etc. In this case, the first two answers are precisiated restrictions; the rest are unprecisiated. The concept of a restriction is closely related to the concept of information granule--a concept which was introduced in my 1979 paper on information granularity.

The concept of a restriction is considerably more general than the concept of an interval, set, fuzzy set and probability distribution. In one form or another, much of human cognition involves restrictions, particularly in the realms of everyday reasoning and decision-making. Humans have a remarkable capability to reason and, to some degree, compute with restrictions. What is needed is a theory which formalizes this capability. RCC may be viewed as a step in this direction. What should be noted is that existing approaches to reasoning and computation, other than RCC, do not have the capability of reasoning and computation with restrictions which are described in a natural language. The canonical form of a restriction is an expression of the form X isr R, where X is the restricted variable, R is the restricting relation and r is an indexical variable which defines the way in which R restricts X.

There are two principal issues which are addressed in RCC. First, how can a semantic entity, e.g. a proposition, be represented as a restriction? Second, how can restrictions be reasoned and computed with? In RCC, for computation with restrictions what is employed is the extension principle. The extension principle is a collection of computational rules which address the following problem. Assume that Y=f(X). Given a restriction on X and/or a restriction on f, what is the restriction on Y, R(Y), which is induced by R(X) and R(f)? Basically, the extension principle involves propagation of restrictions. In essence, in RCC the objects of reasoning and computation are not values of variables, but restrictions on values of variables. A key to understanding of RCC is that in RCC the focus of attention is shifted from reasoning and computation with values of variables to reasoning and computation with restrictions on values of variables. This shift has wide-ranging ramifications. Representation and computation with restrictions are illustrated with examples.