An introduction to DSmT

The management and combination of uncertain, imprecise, fuzzy and even paradoxical or high conflicting sources of information has always been, and still remains today, of primal importance for the development of reliable modern information systems involving artificial reasoning. The combination (fusion) of information arises in many fields of applications nowadays (especially in defense, medicine, finance, geo-science, economy, etc). When several sensors, observers or experts have to be combined together to solve a problem, or if one wants to update our current estimation of solutions for a given problem with some new information available, we need powerful and solid mathematical tools for the fusion, specially when the information one has to deal with is imprecise and uncertain. In this paper, we present a survey of our recent theory of plausible and paradoxical reasoning, known as Dezert-Smarandache Theory (DSmT) in the literature, developed for dealing with imprecise, uncertain and conflicting sources of information. Recent publications have shown the interest and the ability of DSmT to solve problems where other approaches fail, especially when conflict between sources becomes high. We focus this presentation rather on the foundations of DSmT, and on the main important rules of combination, than on browsing specific applications of DSmT available in literature. Several simple examples are given throughout the presentation to show the efficiency and the generality of DSmT.

The development of DSmT (Dezert-Smarandache Theory of plausible and paradoxical reasoning [8,31]) arises from the necessity to overcome the inherent limitations of DST (Dempster-Shafer Theory [24]) which are closely related with the acceptance of Shafer’s model for the fusion problem under consideration (i.e. the frame of discernment Θ is implicitly defined as a finite set of exhaustive and exclusive hypotheses θ i , i = 1, . . . , n since the masses of belief are defined only on the power set of Θ -see section 2.1 for details), the third middle excluded principle (i.e. the existence of the complement for any elements/propositions belonging to the power set of Θ), and the acceptance of Dempster’s rule of combination (involving normalization) as the framework for the combination of independent sources of evidence. Discussions on limitations of DST and presentation of some alternative rules to Dempster’s rule of combination can be found in [11,15,[17][18][19]21,23,31,38,46,49,50,[53][54][55][56] and therefore they will be not reported in details in this introduction. We argue that these three fundamental conditions of DST can be removed and another new mathematical approach for combination of evidence is possible. This is the purpose of DSmT.

The basis of DSmT is the refutation of the principle of the third excluded middle and Shafer’s model, since for a wide class of fusion problems the intrinsic nature of hypotheses can be only vague and imprecise in such a way that precise refinement is just impossible to obtain in reality so that the exclusive elements θ i cannot be properly identified and precisely separated. Many problems involving fuzzy continuous and relative concepts described in natural language and having no absolute interpretation like tallness/smallness, pleasure/pain, cold/hot, Sorites paradoxes, etc, enter in this category. DSmT starts with the notion of free DSm model, denoted M f (Θ), and considers Θ only as a frame of exhaustive elements θ i , i = 1, . . . , n which can potentially overlap. This model is free because no other assumption is done on the hypotheses, but the weak exhaustivity constraint which can always be satisfied according the closure principle explained in [31]. No other constraint is involved in the free DSm model. When the free DSm model holds, the classic commutative and associative classical DSm rule of combination, denoted DSmC, corresponding to the conjunctive consensus defined on the free Dedekind’s lattice is performed.

Depending on the intrinsic nature of the elements of the fusion problem under consideration, it can however happen that the free model does not fit the reality because some subsets of Θ can contain elements known to be truly exclusive but also truly non existing at all at a given time (specially when working on dynamic fusion problem where the frame Θ varies with time with the revision of the knowledge available). These integrity constraints are then explicitly and formally introduced into the free DSm model M f (Θ) in order to adapt it properly to fit as close as possible with the reality and permit to construct a hybrid DSm model M(Θ) on which the combination will be efficiently performed. Shafer’s model, denoted M 0 (Θ), corresponds to a very specific hybrid DSm model including all possible exclusivity constraints. DST has been developed for working only with M 0 (Θ) while DSmT has been developed for working with any kind of hybrid model (including Shafer’s model and the fr

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