**3. Fusion within Possibility Theory**

To provide a basis for a discussion on fusion topology design, the importance of associativity, and the role of consistency and redundancy, the core principles of possibility theory (PosT) are recapitulated. For this, common fusion rules are also reported in detail.

The main motivation behind PosT is that probability theory (ProbT) is not able to model epistemic uncertainty adequately—such as imprecision or missing information. Probability theory models random phenomena quantitatively; PosT handles incomplete information qualitatively [5,43]. Zadeh [44] introduced PosT based on fuzzy sets in the context of natural language processing. He interpreted fuzzy membership functions as possibility distributions allowing uncertainties in the sense of imprecisions as well as a lack of confidence in statements [45].

Consequently, PosT is mathematically close to fuzzy set theory [46]. This proximity often allows mathematical operations defined in the context of fuzzy sets—such as similarity measures or t-norms—to be applied to possibility distributions. Since Zadeh's introduction of PosT, Dubois and Prade [4,6,47–49] and YAGER [50–53] have mainly contributed to the advancement of possibility theory. If not explicitly mentioned otherwise, a numerical, real-valued representation of possibility values is assumed (cf. Dubois et al. [6] for an overview of qualitative and numerical possibility scales).

A possibility distribution is defined as a mapping of mutually exclusive and exhaustive alternative events to a numerical representation. Let the set of all alternative events be described as the *frame of discernment X* and let *v* ∈ *X* be an imprecisely known element whose true value is unknown. Then, a possibility distribution is defined by

$$
\pi\_{\mathcal{V}} : X \to [0, 1]. \tag{1}
$$

Alternatives *x* ∈ *X* that are assigned higher values are deemed more plausible. Alternatives with *πv*(*x*) = 0 are considered impossible, and alternatives with *πv*(*x*) = 1 are fully plausible. Possibility theory is strongly guided by the *minimum specificity principle*, which states that any alternative *x* not known to be impossible should not be disregarded [45]. Extreme cases of knowledge about *v* are *total ignorance* and *complete knowledge*. In the first case, ∀*x* ∈ *X* : *πv*(*x*) = 1. In the case of complete knowledge, only one alternative is fully possible, and all others are impossible. A possibility distribution *πv*(*x*) is said to be *normal* if ∃*x* ∈ *X* : *πv*(*x*) = 1. The subset *A* ⊆ *X*, which ∀*x* ∈ *A* : *πv*(*x*) = 1 is referred to as

*core* of *πv*(*x*); if ∀*x* ∈ *A* : *πv*(*x*) ≥ 1, then *A* is referred to as *support*. In the following, the shortened notation *π*(*x*) = *πv*(*x*) is used.

Let multiple information sources **S** = {*S*1, ... , *Sn*} each provide an information item *Ii*, *i* ∈ {1, ... , *n*} in the form of a possibility distribution *π<sup>i</sup>* regarding the same imprecisely known element *v* ∈ *X*. A possibilistic fusion operator is then defined by *fu* : [0, 1] *<sup>n</sup>* → [0, 1] and the fused possibility distribution is obtained as *π*(fu)(*x*) = *fu*(*π*1(*x*),..., *πn*(*x*))). Multiple information sources allow the identification of even more impossible or hardly possible alternatives for the unknown *v* resulting in more precise, more specific, and thus more qualitative information. In this sense, the goal in possibilistic fusion is to reach a maximal specific outcome (the most certain outcome possible) although possibility theory follows the minimum specificity principle. It is important that none of the available information is disregarded or neglected—that is, that any information source is considered by the fusion process (see also the *fairness* property postulated for fusion operators [6]). This fairness constraint represents the minimum specificity principle stating that alternatives that are not known to be impossible are not to be ruled out [45].

Over time, multiple possibilistic fusion operators haven been proposed, verified, and brought to applications. We propose to categorise these operators as follows:

