Number 27                                                June 15,1992

                      The Huntington Technical Brief 
                          By David Brubaker Ph.D.
     
                                  Hedges
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     INTRODUCTION - Adjectives and adjective phrases called hedges
     are used in fuzzy system design as a means of generating new
     membership functions derived from already defined functions. For
     example, the function for the fuzzy value "hot" can be skewed
     toward higher temperatures with the hedge "very", resulting in a
     membership function for a derived fuzzy value "very hot". If
     correctly defined, "very" can also be applied to "cold",
     resulting in "very cold", a shift in the opposite direction on
     the temperature scale.
     
        Hedges can also be used to "fuzzify", to transform a crisp
     value or variable (or even operator) into a fuzzy equivalent.
     
        HEDGING MEMBERSHIP FUNCTIONS - A large part of fuzzy system
     design is the definition of linguistic values for fuzzy
     variables, representing them as membership functions. For
     example, the relative distance to an elevator's stop point might
     be described by the fuzzy values "near", "medium", and "far".
     Often, in order to achieve greater precision, further refinement
     of a variable's values is needed, examples being the values
     "very_near" and "very_very_near".
     
        While these values can be defined independent of each other,
     they can also use a base fuzzy value, in this case "near",
     modified by one or more hedges. For example, the hedge "very"
     can be applied once to "near" to achieve "very near", and twice
     to achieve "very very near".
     
        As a membership function changes, any hedged values based on
     that function will also change. The appropriateness of the
     change in the hedged values depends on the choice of the hedge
     function.
     
        HEDGES AS FUZZIFIERS - Selected hedge functions can also be
     used to transform crisp values into fuzzy values, and are called
     "fuzzifiers" - not to be confused with the use of the term
     describing a component that transforms a crisp value into a
     truth value in a fuzzy set. Given a crisp value x, three hedged
     versions of x might be "nearly x", "approximately x", and
     "roughly x", each of which is a fuzzy value derived from x.
     
        HEDGES ON OPERATORS - The dominant fuzzy relational operator
     is "is", most often used to determine the degree to which a
     crisp input falls within a fuzzy set. As such, the operator "is"
     is crisp, relating a crisp input or output value to a crisp 
     degree of membership value.
     
        A more general realization would allow relational operators
     themselves to be fuzzy, either directly, as would be the case
     for the operators "approximately_is" or "roughly_is", or by
     hedging a crisp operator, as with "approximately is" and
     "roughly is".
     
        In providing for both hedged values and operators, we achieve
     representational redundancy.  For example the phrases "as
     track_angle is roughly 180_deg" and "as track_angle roughly is
     180_deg" achieve the same end, and in fact are synonymous for 
     appropriate choice of the hedge function "roughly". In the first
     expression, "roughly" hedges the crisp value 180_deg, and in the
     second, "roughly" hedges the crisp operator "is". In both cases
     a fuzzy range is created around 180_deg.
     
        SUMMARY - We have briefly discussed the use of hedges in
     fuzzy systems. In addition to the more common practice of
     hedging fuzzy values, we have also looked at hedging crisp
     values to make them fuzzy, and at hedging the crisp relational
     operator "is". All such applications of hedges add richness to
     both the power and representation of fuzzy logic systems.
     
     
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     The Huntington Technical Brief is published monthly as part of
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