FUZZY EXPRESSIONS

"When you have half an apple, you have both an apple and you don't have an apple. The half apple prevents an all-or-nothing description. The half apple is a fuzzy apple" (Bart Kosko).

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Fuzzy information

With MENTAL it is possible to represent fuzzy or fuzzy information (fuzzy) through the so-called (in this domain) "linguistic variables", whose semantic level is close to natural language.

A linguistic variable is composed of:

• A name (e.g. speed).
  
  
• A domain of what are often called "semantic definitions", which is a set of qualitative attributes. For example,
  
  ( attributes(speed) = {null low medium high extreme} )
  
  
• A set of possible values (e.g. 0 to 200 km/hour).
  
  ( values(speed) = {[0...200]*(Km÷hour)} )
  
  
• An interpretation. It is a function that assigns to each possible value a degree of approximation (between 0 and 1) to each qualitative attribute. For example:
  
  
  speed = 40
  degree = 0 (null)
  grade = 0.8 (low)
  grade = 0.4 (medium)
  grade = 0.1 (high)
  grade = 0 (extreme)
  
  Formally expressed:
  
  ( degrees(40) = {0*null 0.8*low 0.4*medium 0.1*high 0*extreme} )
  
  The interpretation is usually specified by a function over ranges of values that has the following structure:
  
  ⟨( (degrees(v) = (v∈{v1...v2} → {{f1*A1 ... fn*An} v∈{v3...v4} → ¡{f1*A1 ... fn*An} ... )! )⟩
  
  Where the qualitative attributes of the linguistic variable v are A1 ... An.

The specification of partial or incomplete

The fuzzy is usually associated with the qualitative affected by a factor or degree of approximation (between 0 and 1). But the factor or degree can, in general, affect any expression. Indeed, everything is a matter of degree.

MENTAL provides linguistic resources to express the concept of partial or incompleteness. For this purpose, the multiplicative factor of an expression is simply used:

• The partial membership of a certain element a to a group x. Partial membership means that the element is not fully included in the group (sequence or set). For example,
  
  (x = {0.8*a b c})
  
  indicates that element a belongs 80% to the set x, while elements b and c belong entirely to x.
  
  Components could be selected. For example, if
  
  (x = {0.7*a b c 0.6*a d 0.4*a})
  
  one has:
  x⇓(((>0.5)*a) // ev. 0.7*a 0.6*a
  
  i.e., elements a whose multiplier factor is greater than 0.5 are selected.
  
  
• The definition of incomplete groups. For example,
  0.7*{a b c}
  
  This expression is obviously different from
  {0.7*a 0.7*b 07*c}.
  
  
• The definition of incomplete groups formed in turn by incomplete elements. For example,
  0.7*{a 0.5*b c}.