Uniqueness of reconstruction for Yager's t-norm combination of probabilistic and possibilistic knowledge

Often, about the same real-life system, we have both measurementrelated probabilistic information expressed by a probability measure P (S) and expert-related possibilistic information expressed by a possibility measure M(S). To get the most adequate idea about the system, we must combine these two pieces of information. For this combination, R. Yager – borrowing an idea from fuzzy logic – proposed to use a t-norm f&(a, b) such as the product f&(a, b) = a · b, i.e., to consider a set function f(S) = f&(P (S),M(S)). A natural question is: can we uniquely reconstruct the two parts of knowledge from this function f(S)? In our previous paper, we showed that such a unique reconstruction is possible for the product t-norm; in this paper, we extend this result to a general class of t-norms. 1 Formulation of the Problem Need to combine probabilistic and possibilistic knowledge. In many practical situations, we have both probabilistic information about some objects – e.g., information coming from measurements with known probability of measurement errors – and possibilistic information – describing expert knowledge. In the probabilistic case, for every set S, we have a probability P (S) ∈ [0, 1] that the actual (unknown) state s of the object belongs to the set S. In the possibilistic case, for each set S, we know the possibility M(S) ∈ [0, 1] that s belongs to S. It is often desirable to combine these two numbers P (S) and M(S) into a single value f(S).