general.hpp File Reference

Plans in general for fuzzy sets in Maxima/Lisp. More...

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Detailed Description

Plans in general for fuzzy sets in Maxima/Lisp.

Basic notions
  • As abbreviation for a fuzzy set we could use "fys".
  • A basic problem is that "fuzzy sets" are related to some base set; and this base set in general is infinite (like the natural or the real numbers).
  • Perhaps this base set is just left unspecified in general, and must be stated for each concrete application.
  • Then a fys A is a map A: B -> [0,1] (where the right-hand side denotes the interval from 0 to 1), where B is the base set.
  • So as a maxima object a fys would just be a function f(x), where all f(b) for b in B can be evaluated and yield a real number from 0 to 1.
  • We should allow as "number" here precise terms (for integers and rational numbers) as well as floating-point numbers (ordinary, or "big floats").
  • Should we use for example "fysr" for fuzzy-subsets of RR, "fysz" for fuzzy subsets of ZZ, "fysn", "fysn0" for fuzzy subsets of NN resp. NN0, and "fysq" for fuzzy subsets of QQ?
  • We need then the standard fuzzy set-operations, "fysunion", "fysintersection" and "fyscomplement", for the standard operation (max, min, difference with 1) as well as for arbitrary operations.
  • Or should we use "union_fys" etc. ? However, since it is not union, but only something similar, we should use "fysunion(A,B)" etc.
More general notions
  • What about more general types of "fuzzyfication"?
  • In a different context, one should also allow arbitrary "truth values", for example elements of a boolean algebra or a hyting lattice, that is, "subsets" of B of the form A: B -> P, where P is some quasi-ordered set.
  • Perhaps calling such "sets" "P-sets" ?
  • The basic P-set operations would be determined by operations from P^2 to P resp. P to P.
  • What are general axioms here?

Definition in file general.hpp.