A Fuzzy-Algorithmic Approach to the Definition of Complex or Imprecise Concepts
Lotfi A. Zadeh · Birkhäuser Basel eBooks · 1976
It may be argued, rather persuasively, that most of the concepts encountered in various domains of human knowledge are, in reality, much too complex to admit of simple or precise definition. This is true, for example, of the concepts of recession and utility in economics; schizophrenia and arthritis in medicine; stability and adaptivity in system theory; sparseness and stiffness in numerical analysis; grammaticality and meaning in linguistics; performance measurement and correctness in computer science; truth and causality in philosophy; intelligence and creativity in psychology; and obscenity and insanity in law. The approach described in this paper provides a framework for the definition of such concepts through the use of fuzzy algorithms which have the structure of a branching questionnaire. The starting point is a relational representation of the definiendum as a composite question whose constituent questions are either attributional or classificational in nature. The constituent questions as well as the answers to them are allowed to be fuzzy, e.g., the response to: “How large is x?” might be not very large , and the response to “Is x large?” might be quite true. By putting the relational representation into an algebraic form, one can derive a fuzzy relation which defines the meaning of the definiendum. This fuzzy relation, then, provides a basis for an interpolation of the relational representation. To transform a relational representation into an efficient branching questionnaire, the tableau of the relation is subjected to a process of compactification which identifies the conditionally redundant questions. From a maximally compact representation, various efficient realizations which have the structure of a branching questionnaire, with each realization corresponding to a prescribed order of asking the constituent questions, can readily be determined. Then, given the cost of constituent questions as well as the conditional probabilities of answers to them, one can compute the average cost of deducing the answer to the composite question. In this way, a relational representation of a concept leads to an efficient branching questionnaire which may serve as its operational definition.