Conjunctive and disjunctive combination rules in random permutation set theory: A layer-2 belief structure perspective

In uncertainty reasoning, conjunctive and disjunctive combination rules are the core tools for information updates. As an extension of evidential reasoning, random permutation set reasoning models uncertain information based on ordered focal sets. Existing combination rules in random permutation set theory, orthogonal sums, do not satisfy the commutativity of one of the original distributions and overly obey its order information. In this paper, the random permutation set theory is interpreted as an refined extension of Dempster-Shafer theory, and a layer-2 belief structure is proposed to describe the permutation event space. Compared with the traditional belief structure, the proposed structure can model both symbolic and numerical uncertainty. Based on the above, the conjunctive and disjunctive combination rules in Dempster-Shafer theory are extended to random permutation set theory. Through properties analysis and simulation demonstration, we demonstrate that the proposed methods can not only resolve the counter-intuitive results of orthogonal sums, but make full use of order information in distributions as well. In addition, we also extend the product space operations and discounting methods based on the proposed methods, and give a general framework of multi-source information fusion under the random permutation set theory.