Formalization of Generalized Constraint Language: A Crucial Prelude to Computing With Words

The generalized constraint language (GCL), introduced by Zadeh, serves as a basis for computing with words (CW). It provides an agenda to express the imprecise and fuzzy information embedded in natural language and allows reasoning with perceptions. Despite its fundamental role, the definition of GCL has remained informal since its introduction by Zadeh, and to our knowledge, no attempt has been made to formulate a rigorous theoretical framework for GCL. Such formalization is necessary for further theoretical and practical advancement of CW for two important reasons. First, it provides the underlying infrastructure for the development of useful inference patterns based on sound theories. Second, it determines the scope of GCL and hence facilitates the translation of natural language expressions into GCL. This paper is an attempt to step in this direction by providing a formal syntax together with a compositional semantics for GCL. A soundness theorem is defined, and Zadeh's deduction rules are proved to be valid in the defined semantics. Furthermore, a discussion is provided on how the proposed language may be used in practice.