Left continuity of T-norms and completeness of fuzzy SLD-resolution

Rok, strany: 1997, 51 - 63

This is a continuation of a joint work with P. Vojtáš where the authors consider fuzzy logic programming based on generalization of P. Hájek's RPL and RQL logic. They proved soundness and completeness of fuzzy SLD-resolution under condition that function $et^·$ which interprets conjunction is continuous. Here we show that if we do not assume that $et^·$ is continuous, then the operator $T_P$ need not be continuous. From a previous result of L. Paulík, that during iteration $T_P\mathbin{\uparrow}ω$ every ground atom attains his maximal value, we prove directly that $T_P\mathbin{\uparrow}ω$ is the least fixpoint of $T_P$. But if we want $T_P\mathbin{\uparrow}ω$ to be a model of a definite fuzzy logic program $P$ and fuzzy SLD-resolution to be complete, using S. Gottwald's results concerning connection between fuzzy conjunction and fuzzy implication, we show that left continuity of $et^·$ is necessary. We also show that S. Gottwald's conditions $(Φ 1){-}(Φ 3)$ are equivalent to J. Pavelka's adjoint condition.

ISO 690:

Paulík, L. 1997. Left continuity of T-norms and completeness of fuzzy SLD-resolution. In Tatra Mountains Mathematical Publications, vol. 12, no.3, pp. 51-63. 1210-3195.

APA:

Paulík, L. (1997). Left continuity of T-norms and completeness of fuzzy SLD-resolution. Tatra Mountains Mathematical Publications, 12(3), 51-63. 1210-3195.