Graduation and Granulation--Keys to Computation with Information Described in Natural Language
Lotfi A. Zadeh
Graduation and granulation play essential roles in human cognition. Both are concomitants of the bounded ability of human sensory organs, and ultimately the brain, to resolve detail and store information. Graduation relates to unsharpness of boundaries or, equivalently, fuzziness. Granulation involves clumping, with a granule being a clump of attribute values drawn together by indistinguishability, similarity, proximity or functionality. Graduation and granulation underlie the concept of a linguistic variable, a concept which plays a pivotal role in fuzzy logic and its applications.
A natural language is a system for describing perceptions which are intrinsically imprecise. This imprecision is a major obstacle to computation with information described in natural language or NL-computation for short. What is NL-computation? For example, a box contains about twenty balls of various sizes. Most are large. What is the number of small balls? What is the probability that a ball drawn at random is neither small nor large? Another example: usually the temperature is not very low, and usually the temperature is not very high. What is the average temperature?
As we move further into the age of machine intelligence and mechanized decision-making, NL-computation will grow in visibility and importance. NL-computation cannot be dealt with through the use of machinery of natural language processing.
Our approach to NL-computation centers on what is referred to as generalized-constraint-based computation, or GC-computation for short. A generalized constraint is expressed as X isr R, where X is the constrained variable, R is a constraining relation and r is an indexical variable which defines the way in which R constrains X. The principal constraints are possibilistic, veristic, probabilistic, usuality, random set, fuzzy graph and group. Generalized constraints may be combined, qualified, propagated and counter propagated, generating what is called the Generalized Constraint Language, GCL. The key underlying idea is that information conveyed by a proposition may be represented as a generalized constraint, that is, as an element of GCL.
NL-computation involves two modules: (a) Precisiation module; and (b) Computation module. The meaning of an element of a natural language, NL, is precisiated through translation into GCL and is expressed as a generalized constraint. An object of precisiation, p, is referred to as precisiend, and the result of precisiation, p*, is called a precisiand. Usually, a precisiend is a proposition or a concept. A precisiend may have many precisiands. Definition is a form of precisiation. A precisiand may be viewed as a model of meaning. The degree to which the intension (attribute-based meaning) of p* approximates to that of p is referred to as cointension. A precisiand, p*, is cointensive if its cointension with p is high, that is, if p* is a good model of meaning of p.
The Computation module serves to deduce an answer to a query, q. The first step is precisiation of q, with precisiated query, q*, expressed as a function of n variables u1, ..., un. The second step involves precisiation of query-relevant information, leading to a precisiand which is expressed as a generalized constraint on u1, ..., un. The third step involves an application of the extension principle, which has the effect of propagating the generalized constraint on u1, ..., un to a generalized constraint on the precisiated query, q*. Finally, the constrained q* is interpreted as the answer to the query and is retranslated into natural language.
The generalized-constraint-based computational approach to NL-computation opens the door to a wide-ranging enlargement of the role of natural languages in scientific theories. Particularly important application areas are decision-making with information described in natural language, economics, risk assessment, qualitative systems analysis, search, question-answering and theories of evidence.