Calculated Based On The Incidence Of Uncertainty Reasoning Theory Study

In the real world, only few problems can be described precisely, and most of them are imprecise or incomplete or uncertain. Obviously, orthodox reasoning methods, such as classic logic, cannot deal with these problems. In order to cater to the demand of objective problems, the study of uncertain reasoning is needed in the artificial intelligence areas.Among so many approaches to reasoning under uncertainty, probabilistic reasoning is accepted by most people as an important method to represent uncertain beliefs. The probabilistic methods that we are familiar with include classical probabilistic method, Subjective Bayes method, Bayesian networks, and so on. But they are all purely numeric mechanisms, which are not truth functional. Thus, it is inconvenient to use them to represent the knowledge in some cases.Logics have the strong ability to represent knowledge and infer new knowledge from old ones. So it is natural to combine probabilistic methods and logics, that is, to combine numeric approach with symbolic approach, to manage uncertain reasoning. Probabilistic logic was proposed by Boole~[42] and rediscovered by Nilsson~[12]There exist some severe problems in Nilsson's probabilistic logic, for example, high computational complexity and inferential vacuousness-inferences typically lead to large probability intervals. So Bundy introduced incidence calculus, which is a probabilistic logic developed from prepositional logic. The approach, which is different from Nilsson's probabilistic logic, associates the probabilities with formulae indirectly, and then some of its connectives are truth functional.During the study of incidence calculus theory, we firstly clarified original incidence calculus and the probabilistic reasoning mechanism on it. We then introduced the improvement of generalized incidence calculus theory (GICT) proposed by Liu~[18], which establishes incidence calculus theory on Lukasiewicz's three valued logic. Basing on Liu's work, Qi further developed the GICT. On the one hand, he established the theory on Kleene's three valued logic and proposed interval incidence calculus theory (IICT)~[21]. On the other hand, he proposed interval generalized incidence calculus theory (IGICT)~[22] by revising GICT.Basing on the theory discussed above, we analysed the probabilistic reasoning of IICT, and then proposed the relationship between the two interval incidence calculus theories, and the probabilistic reasoning mechanism according to the probabilistic reasoning of IICT. We proposed an axiomatic interpretation of IGICT in this paper, and proved that the pair of upper incidence function and lower incidence function is a interval structure and the equivalent relationship between this definition and that of IGICT defined by possible worlds, so we gave a good foundation for establishing the equivalent relationship between rough set and incidence calculus theory.