Upper and Lower Bounds of the α-Universal Triple I Method for Unified Interval Implications

The α-universal triple I (α-UTI) method is a recognized scheme in the field of fuzzy reasoning, which was proposed by our research group previously. The robustness of fuzzy reasoning determines the quality of reasoning algorithms to a large extent, which is quantified by calculating the disparity between the output of fuzzy reasoning with interference and the output without interference. Therefore, in this study, the interval robustness (embodied as the interval stability) of the α-UTI method is explored in the interval-valued fuzzy environment. To begin with, the stability of the α-UTI method is explored for the case of an individual rule, and the upper and lower bounds of its results are estimated, using four kinds of unified interval implications (including the R-interval implication, the S-interval implication, the QL-interval implication and the interval t-norm implication). Through analysis, it is found that the α-UTI method exhibits good interval stability for an individual rule. Moreover, the stability of the α-UTI method is revealed in the case of multiple rules, and the upper and lower bounds of its outcomes are estimated. The results show that the α-UTI method is stable for multiple rules when four kinds of unified interval implications are used, respectively. Lastly, the α-UTI reasoning chain method is presented, which contains a chain structure with multiple layers. The corresponding solutions and their interval perturbations are investigated. It is found that the α-UTI reasoning chain method is stable in the case of chain reasoning. Two application examples in affective computing are given to verify the stability of the α-UTI method. In summary, through theoretical proof and example verification, it is found that the α-UTI method has good interval robustness with four kinds of unified interval implications aiming at the situations of an individual rule, multi-rule and reasoning chain.