# 2008 Research Summary

## Granular Computing--The Concept of Generalized Constraint-Based Computation

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Lotfi A. Zadeh

Basically, granular computing is a mode of computing in which the objects of computation are granular variables. Let X be a variable which takes values in a universe of discourse, U. Informally, a granule is a clump of elements of U which are drawn together by indistinguishability, similarity or proximity. For example, an interval is a granule; so is a fuzzy interval; so is a gaussian distribution; and so is a cluster of elements of U. A granular variable is a variable which takes granules as values. If G is value of X, then G is referred to as a granular value of X. If G is a singleton, then G is a singular value of X. A linguistic variable is a granular variable whose values are labeled with words drawn from a natural language. For example, if X is temperature, then 101.3 is a singular value of temperature, while "high" is a granular (linguistic) value of temperature.

A granular value of X may be interpreted as a representation of one's state of imprecise knowledge about the true value of X. In this sense, granular computing may be viewed as a system of concepts and techniques for computing with variables whose values are not known precisely.

A concept which serves to precisiate the concept of a granule is that of a generalized constraint. The concept of a generalized constraint is the centerpiece of granular computing.

A generalized constraint is an expression of the form X isr R, where X is the constrained variable, R is the constraining relation, and r is an indexical variable which serves to identify the modality of the constraint. The principal modalities are: possibilistic (r=blank); veristic (r=v); probabilistic (r=p); usuality (r=u); random set (r=rs); fuzzy graph (r=fg); bimodal (r=bm); and group (r=g). The primary constraints are possibilistic, veristic and probabilistic. The standard constraints are bivalent possibilistic, bivalent veristic and probabilistic. Standard constraints have a position of centrality in existing scientific theories.

A generalized constraint, GC(X), is open if X is a free variable, and is closed (grounded) if X is instantiated. A proposition is a closed generalized constraint. For example, "Lily is young," is a closed possibilistic constraint in which X=age(Lily); r=blank; and R=young is a fuzzy set. Unless indicated to the contrary, a generalized constraint is assumed to be closed.

A generalized constraint may be generated by combining, projecting, qualifying, propagating and counterpropagating other generalized constraints. The set of all generalized constraints together with the rules governing combination, projection, qualification, propagation and counterpropagation constitute the Generalized Constraint Language (GCL).

The point of departure in NL-Computation is: (a) an input dataset which consists of a collection of propositions described in a natural language; and (b) a query, q, described in a natural language. To compute an answer to the query, the given propositions are precisiated through translation into the Generalized Constraint Language (GCL). The translates which express the meanings of given propositions are generalized constraints. Once the input dataset is expressed as a system of generalized constraints, granular computing is employed to compute the answer to the query.

A natural language may be viewed as a system for describing perceptions. In turn, NL-Computation is reduced to granular computing through translation/precisiation into the Generalized Constraint Language (GCL).

As we move further into the age of machine intelligence and automated reasoning, the need for an enhancement of our ability to deal with imprecision, uncertainty and partiality of truth is certain to grow in visibility and importance. It is this need that motivated the genesis of granular computing and is driving its progress. In coming years, granular computing and NL-Computation are likely to become a part of the mainstream of computation and machine intelligence.