Hyperfuzzy Petri Nets and n-Superhyperfuzzy Petri Nets: A Generalized Framework for Modeling Uncertainty

A fuzzy set assigns to each element in a universe a membership degree within the interval [0, 1], effectively capturing imprecision and vagueness. A hyperfuzzy set generalizes this notion by mapping each element to a nonempty subset of [0, 1], thereby representing uncertainty and variability through multiple possible membership degrees. An n-superhyperfuzzy set further extends this framework by assigning to each nonempty element in the nth power-set hierarchy a nonempty subset of [0, 1], thus enabling the modeling of hierarchical and recursively nested uncertainty. Petri nets are mathematical models designed to describe distributed systems involving conditions, events, and state transitions in concurrent processes. Fuzzy Petri nets enhance Petri nets by integrating fuzzy logic, allowing the representation of uncertainty and partial truth in transitions and states. In this paper, we define and explore the mathematical structure of Hyperfuzzy Petri Nets and n-SuperHyperfuzzy Petri Nets. These frameworks extend classical Petri nets by incorporating Hyperfuzzy Sets and n-SuperHyperfuzzy Sets, respectively. We hope these generalizations will contribute to the further advancement of research in fuzzy set theory and related fields.