Possibility Theory and Fuzzy Set Applications

Possibility theory and fuzzy set theory form a cohesive framework for modelling and reasoning under imprecision and vagueness. Whereas probability theory quantifies aleatory uncertainty via additive measures, possibility theory captures epistemic or fuzzy uncertainty by assigning to each event a degree of plausibility. Fuzzy sets extend classical sets by permitting partial membership, represented through membership functions that map elements to a continuum between zero (non-membership) and one (full membership). This flexibility allows practitioners to encode expert judgements, linguistic assessments and vague concepts in a mathematically rigorous way.

Applications span control engineering, decision support, environmental modelling and digital diagnostics. In control, fuzzy rules govern system behaviour via max-product or max-min operators, yielding robust responses even when precise models are unavailable. In decision-making, hybrid schemes integrate possibilistic and probabilistic data without forcing conversion between formalisms, thereby preserving the original information semantics. In environmental engineering, fuzzy approaches augment classical formulae with α-cuts to derive confidence intervals for key parameters under data imprecision. More recently, nonparametric inferential methods construct nested plausible regions for online model updating, treating the resulting structure as a joint fuzzy set and enabling near-real-time adaptation.

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Possibility Theory and Fuzzy Set Applications publication trend

The graph below shows the total number of articles in possibility theory and fuzzy set applications across all publications each year (not limited to Nature Index journals).

Technical terms

Fuzzy set: A set whose elements have graded membership values between 0 and 1, representing degrees of belonging.

Membership function: A mapping μ(x) in [0,1] that quantifies the degree to which element x belongs to a fuzzy set.

Possibility distribution: A function assigning to each event or value a degree of plausibility, used to model epistemic uncertainty.

α-cut: A crisp set obtained by retaining all elements whose membership in a fuzzy set meets or exceeds a threshold α, often used to derive interval approximations.

Intuitionistic fuzzy set: An extension of fuzzy sets characterised by a membership degree, a non-membership degree, and consequently a hesitation margin to represent incomplete knowledge.

References

1. Robust online updating of a digital twin with imprecise probability. Mechanical Systems and Signal Processing (2023).
2. Intuitionistic Fuzzy Sets for Spatial and Temporal Data Intervals. Information (2024).
3. A New Hybrid Possibilistic-Probabilistic Decision-Making Scheme for Classification. Entropy (2021).
4. Formalization of Fuzzy Control in Possibility Theory via Rule Extraction. IEEE Access (2019).
5. Estimation of Reservoir Storage Capacity Using the Gould-Dincer Formula with the Aid of Possibility Theory. Hydrology (2024).