Unified forms of fully implicational restriction methods for fuzzy reasoning

The goal of this paper is to extend the scope of previously proposed “fully implicational restriction methods” for fuzzy reasoning, and provide a unified form of these methods which can employ diverse implications. By means of the residual implications generated by left-continuous t-norms, we obtain the supremum of triple I (the abbreviation of triple implications) restriction solutions for fuzzy modus ponens and the supremum of multiple I restriction solutions for general fuzzy reasoning respectively. Furthermore, we obtain the infimum of triple I restriction solutions and the supremum of triple I* restriction solutions for fuzzy modus tolens. Finally, the optimal (i.e., the largest or smallest, depending on context) restriction solutions of both basic and general reasoning problems are derived for four specific residual implications.