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A new universal approximation result for fuzzy systems, which reflects CNF DNF duality
Author(s) -
Perfilieva Irina,
Kreinovich Vladik
Publication year - 2002
Publication title -
international journal of intelligent systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.291
H-Index - 87
eISSN - 1098-111X
pISSN - 0884-8173
DOI - 10.1002/int.10063
Subject(s) - relation (database) , mathematics , fuzzy logic , propositional calculus , duality (order theory) , conjunction (astronomy) , function (biology) , disjunctive normal form , conjunctive normal form , discrete mathematics , algebra over a field , computer science , pure mathematics , artificial intelligence , physics , data mining , astronomy , evolutionary biology , biology
There are two main fuzzy system methodologies for translating expert rules into a logical formula: In Mamdani's methodology, we get a DNF formula (disjunction of conjunctions), and in a methodology which uses logical implications, we get, in effect, a CNF formula (conjunction of disjunctions). For both methodologies, universal approximation results have been proven which produce, for each approximated function f ( x ), two different approximating relations R DNF ( x , y ) and R CNF ( x , y ). Since, in fuzzy logic, there is a known relation F CNF ( x ) ≤ F DNF ( x ) between CNF and DNF forms of a propositional formula F , it is reasonable to expect that we would be able to prove the existence of approximations for which a similar relation R CNF ( x , y ) ≤ R DNF ( x , y ) holds. Such existence is proved in our paper. © 2002 Wiley Periodicals, Inc.