Basically, in computing with words and perceptions, or CWP for short, the objects of computation are words, propositions and perceptions described in a natural language. A very simple example is: Usually Robert returns from work at about 6 pm. What is the probability that Robert is home at about 6:l5 pm? Another elementary example is: A box contains about 20 balls of various sizes. Most are large. There are many more large balls than small balls. How many are neither large nor small?
In science, there is a deep-seated tradition of striving for progression from perceptions to measurements, and from the use of words to the use of numbers. Why and when, then, should the use of CWP be considered?
There are two principal rationales. The first, briefly stated, is: When precision is desired but the needed information is not available, in which case the use of CWP is a necessity rather than an option. And second, when precision is not needed, in which case the tolerance for imprecision may be exploited to achieve tractability, robustness, simplicity and low solution cost.
Another important point is that humans have a remarkable capability to perform a wide variety of physical and mental tasks without any measurements and any computations, e.g., parking a car, driving in city traffic, playing golf and summarizing a book. In performing such tasks, humans employ perception--rather than measurements--of distance, direction, speed, count, likelihood, intent and other attributes.
Reflecting the bounded ability of sensory organs and, ultimately, the brain, to resolve detail, perceptions are intrinsically imprecise. More concretely, perceptions are f-granular in the sense that (a) the perceived values of attributes are fuzzy; and (b) the perceived values of attributes are granular, with a granule being a clump of values drawn together by indistinguishability, similarity, proximity or functionality.
F-granularity of perceptions is the reason why in the enormous literature on perceptions—in fields ranging from linguistics and logic to psychology and neuroscience—one cannot find a theory in which perceptions are objects of computation, as they are in CWP. What should be stressed is that in dealing with perceptions, the point of departure in CWP is not a perception per se, but its description in a natural language. This is a key idea which makes it possible to reduce computation with perceptions to computation with propositions drawn from a natural language.
A related key idea in CWP is that the meaning of a proposition, p, in a natural language may be represented as a generalized constraint of the form X isr R, where X is a constrained variable which, in general, is implicit in p; R is the constraining relation which, like X,is in general implicit in p; and r is an indexing variable whose value identifies the way in which R constrains X. The principal types of constraints are: equality (r is =); possibilistic (r is blank); veristic (r is v); probabilistic (r is p); random set (r is rs); fuzzy graph (r is fg); usuality (r is u); and Pawlak set (r is ps). In this system of classification of constraints, the standard constraint, X belongs to C, where C is a crisp set, is possibilistic. Representation of p as a generalized constraint is the point of departure in what is called precisiated natural language (PNL).
PNL associates with a natural language, NL, a precisiation language, GCL (Generalized Constraint Language), which consists of generalized constraints and their combinations and qualifications. A simple example of an element of GCL is: (X is A) and (Y isu B). A proposition, p, in NL is precisiable if it is translatable into GCL. In effect, PNL is a sublanguage of NL which consists of propositions which are precisiable through translation into GCL. More concretely, PNL is associated with two dictionaries (a) from NL to GCL, and (b) from GCL to what is referred to as the protoform language (PFL). An element of PFL is an abstracted version of an element of GCL. The translates of p into GCL and PFL are denoted as GC(p) and PF(p ), respectively.
In addition, PNL is associated with a deduction database, DDB, which consists of rules of deduction expressed in PFL. An example of such a rule is the intersection/product syllogism: if Q A's are B's and R (A and B)'s are C's, then QR A's are (B and C)'s, where Q and R are fuzzy quantifiers, e.g., most, many, few; A, B and C are fuzzy sets, and QR is the product of Q and R in fuzzy arithmetic.
The principal function of PNL is to serve as a system for computation and reasoning with perceptions. A related function is that of serving as a definition language. In this capacity, PNL may be used to (a) define new concepts, e.g., the usual value of a random variable; and (b) redefine existing concepts, e.g., the concept of statistical independence. The need for redefinition arises because standard bivalent -classic-based definitions may lead to counterintuitive conclusions.
Computing with words and perceptions provides a basis for an important generalization of probability theory, namely, perception-based probability theory (PTp). The point of departure in PTp is the assumption that subjective probabilities are, basically, perceptions of likelihood. A key consequence of this assumption is that subjective probabilities are f-granular rather than numerical, as they are assumed to be in standard bivalent-logic-based probability theory, PT.
A basic concept in PTp is that of f-granular probability distribution, P*(X), of a random variable, X, with X taking values in a space U. Symbolically, P*(X) is expressed as P*(X)= P1\A1+…+Pn\An, in which the Ai are granules of X, and Pi is a perception of probability of the event X is Ai. For example, if X is the length of an object and its granules are labeled short, medium and long, then P*(X) may be expressed as P*(X) = low \ short+high \ medium+low \ long, where low and high are granular probabilities. Basically, PTp adds to PT the capability to operate on perception-based information—a capability which plays an especially important role in decision analysis. More specifically, in most realistic settings in decision analysis, a decision involves a ranking of f-granular probability distributions. Furthermore, employment of the principle of maximization of expected utility leads to a ranking of fuzzy numbers.
In the final analysis, the importance of CWP derives from the fact that it opens the door to adding to any measurement-based theory, T, the capability to operate on perception-based information. Conceptually, computationally and mathematically, the perception-based theory, Tp, is significantly more complex than T. In this instance, as in many others, complexity is the price that has to be paid to reduce the gap between theory and reality.
* Professor in the Graduate School and Director, Berkeley initiative in Soft Computing (BISC), Computer Science Division and the Electronics Research Laboratory, Department of EECS, University of California, Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712; E-Mail: email@example.com . Research supported in part by ONR N00014-00-1-0621, ONR Contract N00014-99-C-0298, NASA Contract NCC2-1006, NASA Grant NAC2-117, ONR Grant N00014-96-1-0556, ONR Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the BISC Program of UC Berkeley.