Outline of a restriction-centered theory of reasoning and computation in an environment of uncertainty and imprecision

The theory which is outlined in this lecture, call it RRC for short, is a departure from traditional approaches to reasoning and computation. A principal advance is an enhanced capability for reasoning and computation in an environment of uncertainty and imprecision. The point of departure in RRC is a basic premise—in the real world such environment is the norm rather than exception. A concept which has a position of centrality in RRC is that of a restriction. Informally, a restriction is an answer to the question: What is the value of a variable X? More concretely, a restriction, R(X), on a variable, X, is a limitation on the values which X can take—a limitation which is induced by what is known or perceived about X. A restriction is singular if the answer to the question is a singleton; otherwise it is nonsingular. Generally, nonsingularity implies uncertainty. A restriction is precisiated if the limitation is mathematically well defined; otherwise it is unprecisiated. Generally, restrictions which are described in a natural language are unprecisiated. There are many kinds of restrictions ranging from very simple to very complex. Examples. 3≤X≤6; X is normally distributed with mean m and variance σ 2 ; X is small; it is very likely that X is small; it is very unlikely that there will be a significant increase in the price of oil in the near future. 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. In RRC there are two principal issues—representation and computation. Representation involves representing a semantic entity, e.g., a proposition, as a restriction. 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 1 Department of EECS, University of California, Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510642-1712; E-Mail: [email protected]. Research supported in part by ONR N00014-02-1-0294, Omron Grant, Tekes Grant, Azerbaijan Ministry of Communications and Information Technology Grant, Azerbaijan University Grant and the BISC Program of UC Berkeley.