ファジィマルチ集合の論理演算

• 論理演算は集合の『本質』
• Fuzzy logic
  • 色々なものが定義されている(以下がfuzzy_logic()でオプション選択できる定義一覧)

"Zadeh"
Zadeh's logic with T = \min and S = \max. Note that the minimum t-norm, also known as the Gödel t-norm, is the pointwise largest t-norm, and that the maximum t-conorm is the smallest t-conorm.
"drastic"
the drastic logic with t-norm T(x, y) = y if x = 1, x if y = 1, and 0 otherwise, and complementary t-conorm S(x, y) = y if x = 0, x if y = 0, and 1 otherwise. Note that the drastic t-norm and t-conorm are the smallest t-norm and largest t-conorm, respectively.
"product"
the family with the product t-norm T(x, y) = xy and dual t-conorm S(x, y) = x + y - xy.
"Lukasiewicz"
the Lukasiewicz logic with t-norm T(x, y) = \max(0, x + y - 1) and dual t-conorm S(x, y) = \min(x + y, 1).
"Fodor"
the family with Fodor's nilpotent minimum t-norm given by T(x, y) = \min(x, y) if x + y > 1, and 0 otherwise, and the dual t-conorm given by S(x, y) = \max(x, y) if x + y < 1, and 1 otherwise.
"Frank"
the family of Frank t-norms T_p, p &#8805; 0, which gives the Zadeh, product and Lukasiewicz t-norms for p = 0, 1, and Inf, respectively, and otherwise is given by T(x, y) = \log_p (1 + (p^x - 1) (p^y - 1) / (p - 1)).
"Hamacher"
the three-parameter family of Hamacher, with negation N_γ(x) = (1 - x) / (1 + γ x), t-norm T_α(x, y) = xy / (α + (1 - α)(x + y - xy)), and t-conorm S_β(x, y) = (x + y + (β - 1) xy) / (1 + β xy), where α &#8805; 0 and β, γ &#8805; -1. This gives a deMorgan triple (for which N(S(x, y)) = T(N(x), N(y)) iff α = (1 + β) / (1 + γ). The parameters can be specified as alpha, beta and gamma, respectively. If α is not given, it is taken as α = (1 + β) / (1 + γ). The default values for β and γ are 0, so that by default, the product family is obtained.
The following parametric families are obtained by combining the corresponding families of t-norms with the standard negation.

"Schweizer-Sklar"
the Schweizer-Sklar family T_p, -Inf <= p <= Inf, which gives the Zadeh (minimum), product and drastic t-norms for p = -Inf, 0, and Inf, respectively, and otherwise is given by T_p(x, y) = \max(0, (x^p + y^p - 1)^{1/p}).
"Yager"
the Yager family T_p, p &#8805; 0, which gives the drastic and minimum t-norms for p = 0 and Inf, respectively, and otherwise is given by T_p(x, y) = \max(0, 1 - ((1-x)^p + (1-y)^p)^{1/p}).
"Dombi"
the Dombi family T_p, p &#8805; 0, which gives the drastic and minimum t-norms for p = 0 and Inf, respectively, and otherwise is given by T_p(x, y) = 0 if x = 0 or y = 0, and T_p(x, y) = 1 / (1 + ((1/x - 1)^p + (1/y - 1)^p)^{1/p}) if both x > 0 and y > 0.
"Aczel-Alsina"
the family of t-norms T_p, p &#8805; 0, introduced by Aczél and Alsina, which gives the drastic and minimum t-norms for p = 0 and Inf, respectively, and otherwise is given by T_p(x, y) = \exp(-(|\log(x)|^p + |\log(y)|^p)^{1/p}).
"Sugeno-Weber"
the family of t-norms T_p, -1 <= p <= Inf, introduced by Weber with dual t-conorms introduced by Sugeno, which gives the drastic and product t-norms for p = -1 and Inf, respectively, and otherwise is given by T_p(x, y) = \max(0, (x + y - 1 + pxy) / (1 + p)).
"Dubois-Prade"
the family of t-norms T_p, 0 &#8804; p &#8804; 1, introduced by Dubois and Prade, which gives the minimum and product t-norms for p = 0 and 1, respectively, and otherwise is given by T_p(x, y) = xy / \max(x, y, p).
"Yu"
the family of t-norms T_p, p &#8805; -1, introduced by Yu, which gives the product and drastic t-norms for p = -1 and Inf, respectively, and otherwise is given by T(x, y) = \max(0, (1 + p) (x + y - 1) - p x y).
By default, the Zadeh logic is used.

• 基本４演算
  • N:negation
  • T:conjunction(t-norm)
  • D:disjunction(t-conorm)
  • I:residual implication