Applications of Tied Implications to Approximate Reasoning and Fuzzy Control

Abstract eng:
The logic of tied implications deals with implications ⇒: P × L → L (on two lattices (P, ≤P , 1P ) and (L, ≤L )), tied by an integral commutative ordered monoid operation ⊗ on P , in the sense of the following identity: ((x ⊗ y) ⇒ c) = (x ⇒ (y ⇒ c)) . We demonstrate the usefulness of tied implications through some applications. We use the connectives of a tied algebra to interpret Generalized Modus Ponens (GMP) inference schemata, in the vein of both the Compositional Rule of Inference (CRI) of Zadeh, and the Consequent Dilation Rule (CDR), due to Magrez and Smets and developed by Morsi and Fahmy. We show that a multiple-rule, generalized modus ponens inference scheme is equivalent, as far as CRI or CDR are concerned, to a scheme that satisfies the "basic requirement for fuzzy reasoning", proposed by Fukami, Lehmke, Perfilieva, Tian and Turksen. We end by investigating the principles of fuzzy control in general with interpretations based on a particular case of CRI, called Generalized Conjunctive Rule (GCR), due to Hájek. We show that the basic requirement for fuzzy reasoning is satisfied by GCR in all single rule inference schemata, using the connectives of a tied algebra, and we indicate a special type of multiple-rule schemata in which this requirement is satisfied.

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