Introduction to Algebra of Belief Functions on Three-element Frame of Discernment - A General Case

Authors: Milan Daniel
Publication year: 2012
Publication type: Konferenční příspěvek (mezinárodní konf.)
Periodical: Proceedings of 9th Workshop on Uncertainty Processing
Projects: Mathematical Fuzzy Logic in Computer Science
Bibliographic citation: DANIEL, Milan. Introduction to Algebra of Belief Functions on Three-element Frame of Discernment - A General Case. In: KROUPA, T., VEJNAROVÁ, J., eds. Proceedings of 9th Workshop on Uncertainty Processing. Prague: Faculty of Management, University of Economics, 2012, S. 46-57. ISBN 978-80-245-1885-5.
ID: 381403

Abstract

This contribution presents the second part of the introductive study of algebraic structure of belief functions (BFs) on 3-element frame of discernment. Algebraic method by Hájek & Valdés for BFs on 2-element frames is generalized to larger frame of discernment. Due to complexity of the algebraic structure, the study is divided into 2 parts, the present one is devoted to a case of general BFs. The definition of Dempster's semigroup (an algebraic structure) of BFs on 3-element frame is recalled from the first part of the study. Results related to Bayesian and quasi Bayesian BFs from the first part are also briefly recalled. Further substructures related to another subsets of general BFs are described and analyzed (including idempotents, simple complementary BFs, generalizations of subsemigroups of simple BFs) and subalgebras isomorphic to Dempster's semigroup on 2-element frame of discernment. Ideas and open problems for future research are presented.