  It is worth noticing that a fuzzy conjunction and its corresponding fuzzy
implication can form a residual pair if and only if it is left-continuous. In
order to get a more general result related on residual implications that
induced by aggregation functions, we relax the definition of general overlap
functions, more precisely, removing its right-continuous, and then introduce a
new kind of aggregation functions, which called semi-overlap functions.
Subsequently, we study some of their related algebraic properties and
investigate their corresponding residual implications. Moreover, serval
scholars have provided kinds of methods for fuzzy modus ponens (FMP,for short)
problems so far, such as Zadeh's compositional rule of inference (CRI, for
short), Wang's triple I method (TIM, for short) and the quintuple implication
principle (QIP, for short). Compared with CRI and TIM methods, QIP method has
some advantages in solving FMP problems, in this paper, we further consider the
QIP method for FMP problems and prove that it satisfies the reducibility of
multiple-rules fuzzy reasoning. Finally, we propose a new classification
algorithm that based on semi-overlap functions and QIP method, which called
SO5I-FRC algorithm. Through the comparative tests, the average accuracy of
SO5I-FRC algorithm is higher than FARC-HD algorithm. The experimental results
indicate that semi-overlap functions and QIP method have certain advantages and
a wide range of applications in classification problems.
