Basic Formal Properties of Triangular Norms and Conorms

• Autorzy: Adam Grabowski
• Języki publikacji: EN

Abstrakty

EN

In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2]. After defining corresponding Mizar mode using four attributes, we introduced the following t-norms: minimum t-norm minnorm (Def. 6), product t-norm prodnorm (Def. 8), Łukasiewicz t-norm Lukasiewicz_norm (Def. 10), drastic t-norm drastic_norm (Def. 11), nilpotent minimum nilmin_norm (Def. 12), Hamacher product Hamacher_norm (Def. 13), and corresponding t-conorms: maximum t-conorm maxnorm (Def. 7), probabilistic sum probsum_conorm (Def. 9), bounded sum BoundedSum_conorm (Def. 19), drastic t-conorm drastic_conorm (Def. 14), nilpotent maximum nilmax_conorm (Def. 18), Hamacher t-conorm Hamacher_conorm (Def. 17). Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm). This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].

Słowa kluczowe

EN

fuzzy set; triangular norm; triangular conorm; fuzzy logic

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 94D05: Fuzzy sets and logic (in connection with questions of Section 94)
• 03E72: Fuzzy set theory

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2017
• Tom: 25
• Numer: 2
• Strony: 93-100

Daty

wydano

2017-07-01

otrzymano

2017-06-27

online

2017-09-23

Twórcy

autor

Adam Grabowski

• , ,

Identyfikatory

DOI

10.1515/forma-2017-0009