Robustness analysis of logic metrics on F(X)F(X)

• SRSR was chosen to measure the perturbation of fuzzy sets since it is consistent with fuzzy reasoning.
• A sufficient condition under which the measure of perturbation “HR=1−SRHR=1−SR” becomes a metric on F(X)F(X) was proposed.
• The distributions of isolated points in four logic metric spaces were investigated in detail.
• The four perturbation measurements HGHG, H0H0, HπHπ and HLHL were compared from the viewpoint of robustness analysis.

Based on the notion of logic similarity degree, logic metric spaces are constructed on the family of fuzzy subsets of the universe X by means of some regular implication operators. The structures of four specific logic metric spaces induced by four important logic implication operators are discussed and compared. It can be concluded that the logic metrics induced by Łukasiewicz implication and Goguen implication are more beneficial to the robustness analysis of fuzzy reasoning.