BISC: Zadeh/Causality is Undefinable

From: Masoud Nikravesh (nikravesh@eecs.berkeley.edu)
Date: Sun Sep 30 2001 - 11:52:06 MET DST

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    Causality is Undefinable - Toward a Theory of Hierarchical Definability
    Professor Lotfi A. Zadeh
      
    Attempts to formulate mathematically precise definitions of basic concepts such
    as causality, randomness, and probability have a long history. The concept of
    generalized definability that is outlined in the following suggests that such
    definitions may not exist. Furthermore, it suggests that existing definitions of
    many basic concepts, among them those of stability, statistical independence and
    Pareto-optimality, may need to be redefined.
    In essence, definability is concerned with whether and how a concept, X, can be
    defined in a way that lends itself to mathematical analysis and computation. In
    mathematics, definability of mathematical concepts is taken for granted. But as
    we move farther into the age of machine intelligence and automated reasoning,
    the issue of definability is certain to grow in importance and visibility,
    raising basic questions that are not easy to resolve.
    To be more specific, let X be the concept of, say, a summary, and assume that I
    am instructing a machine to generate a summary of a given article or a book. To
    execute my instruction, the machine must be provided with a definition of what
    is meant by a summary. It is somewhat paradoxical that we have summarization
    programs that can summarize, albeit in a narrowly prescribed sense, without
    being able to formulate a general definition of summarization. The same applies
    to the concepts of causality, randomness, and probability. Indeed, it may be
    argued that these and many other basic concepts cannot be defined within the
    conceptual framework of classical logic and set theory.
    The point of departure in our approach to definability is the assumption that
    definability has a hierarchical structure. Furthermore, it is understood that a
    definition must be unambiguous, precise, operational, general, and co-extensive
    with the concept it defines.
    The hierarchy involves five different types of definability. The lowest level is
    that of c-definability, with c standing for crisp. Thus, informally, a concept,
    X, is c-definable if it is a crisp concept, e.g., a prime number, a linear
    system, or a Gaussian distribution. The domain of X is the space of instances to
    which X applies.
    The next level is that of f-definability, with f standing for fuzzy. Thus, X is
    a fuzzy concept if its denotation, F, is a fuzzy set in its universe of
    discourse. A fuzzy concept is associated with a membership function that assigns
    to each point, u, in the universe of discourse of X, the degree to which u is a
    member of F. Alternatively, it may be defined algorithmically in terms of other
    fuzzy concepts. Examples of fuzzy concepts are small number, strong evidence,
    and similarity. It should be noted that many concepts associated with fuzzy sets
    are crisp concepts. An example is the concept of a convex fuzzy set. Most fuzzy
    concepts are context-dependent.
    The next level is that of f.g-definability, with g standing for granular, and
    f.g denoting the conjunction of fuzzy and granular. Informally, in the case of a
    concept which is f.g-granular, the values of attributes are granulated, with a
    granule being a clump of values that are drawn together by indistinguishability,
    similarity, proximity, or functionality. f.g-granularity reflects the bounded
    ability of the human mind to resolve detail and store information. An example of
    an f.g-granular concept that is traditionally defined as a crisp concept, is
    that of statistical independence. This is a case of misdefinition--a definition
    that is applied to instances for which the concept is not defined, e.g., fuzzy
    events. In particular, a common misdefinition is to treat a concept as if it
    were c-definable, whereas in fact it is not.
    The next level is that of PNL-definability, where PNL stands for Precisiated
    Natural Language. Basically, PNL consists of propositions drawn from a natural
    language that can be precisiated through translation into what is called
    precisiation language. An example of a proposition in PNL is: It is very
    unlikely that there will be a significant increase in the price of oil in the
    near future.
    In the case of PNL, the precisiation language is the Generalized Constraint
    Language (GCL). A generic generalized constraint is represented by Z isr R,
    where Z is the constrained variable, R is the constraining relation and r is a
    discrete-valued indexing variable whose values define the ways in which R
    constrains Z. The principal types of constraints are: possibilistic (r = blank);
    veristic (r = v); probabilistic (r = p); random set (r = rs); usuality (r = u);
    fuzzy graph (r = fg); and Pawlak set (r = ps). The rationale for constructing a
    large variety of constraints is that conventional crisp constraints are
    incapable of representing the meaning of propositions expressed in a natural
    language--most of which are intrinsically imprecise--in a form that lends itself
    to computation.
    The elements of GCL are composite generalized constraints that are formed from
    generic generalized constraints by combination, modification, and qualification.
    An example of a generalized constraint in GCL is ((Z isp R) and (Z,Y) is S) is
    unlikely.
    By construction, the Generalized Constraint Language is maximally expressive.
    What this implies is that PNL is the largest subset of a natural language that
    admits precisiation. Informally, this implication serves as a basis for the
    conclusion that if a concept, X, cannot be defined in terms of PNL, then, in
    effect, it is undefinable or, synonymously, amorphic.
    In this perspective, the highest level of definability hierarchy, which is the
    level above PNL-definability, is that of undefinability or amorphicity. A
    canonical example of an amorphic concept is that of causality. More
    specifically, is it not possible to construct a general definition of causality
    such that given any two events A and B and the question, "Did A cause B?" the
    question could be answered based on the definition. Equivalently, given any
    definition of causality, it will always be possible to construct examples to
    which the definition would not apply or yield counterintuitive results.
    In dealing with an amorphic concept, X, what is possible--and what we generally
    do--is to restrict the domain of applicability of X to instances for which X is
    definable. For example, in the case of the concept of a summary, which is an
    amorphic concept, we could restrict the length, type, and other attributes of
    what we want to summarize. In this sense, an amorphic concept may be partially
    definable or, p-definable, for short. The concept of p-definability applies to
    all levels of the definability hierarchy.
    The theory of hierarchical definability is not a theory in the traditional
    spirit. The definitions are informal and conclusions are not theorems.
    Nonetheless, it serves a significant purpose by raising significant questions
    about a basic issue--the issue of definability of concepts that lie at the
    center of scientific theories.
    Lotfi A. Zadeh is Professor in the Graduate School and director, Berkeley
    initiative in Soft Computing (BISC), Computer Science Division and the
    Electronics Research Laboratory, Department of EECs, Univeristy of California,
    Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712;E-Mail:
    zadeh@cs.berkeley.edu. Research supported in part by ONR Contract
    N00014-99-C-0298, NASAContract 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

    -- 
    Dr. Masoud Nikravesh
    Chairs: BISC-SIG-FLINT,ES, RT
    Berkeley Initiative in Soft Computing (BISC) 
    Computer Science Division- Department of EECS
    University of California, Berkeley, CA 94720
    Phone: (510) 643-4522; Fax: (510) 642-5775 
    Email: Nikravesh@cs.berkeley.edu
    URL: http://www.cs.berkeley.edu/~nikraves/
    

    Staff Scientist Lawrence Berkeley National Lab, Imaging and Collaborative Computing Group Masoud@media.lbl URL: http://vision.lbl.gov/ -------------------------------------------------------------------- If you ever want to remove yourself from this mailing list, you can send mail to <Majordomo@EECS.Berkeley.EDU> with the following command in the body of your email message: unsubscribe bisc-group or from another account, unsubscribe bisc-group <your_email_adress>

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