The Combination of Paradoxical, Uncertain, and Imprecise Sources of Information based on DSmT and Neutro-Fuzzy Inference

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. In this chapter, 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 paradoxical sources of information. We focus our presentation here rather on the foundations of DSmT, and on the two important new 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 this new approach. The last part of this chapter concerns the presentation of the neutrosophic logic, the neutro-fuzzy inference and its connection with DSmT. Fuzzy logic and neutrosophic logic are useful tools in decision making after fusioning the information using the DSm hybrid rule of combination of masses.Comment: 20 page

Neutrality and Many-Valued Logics

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.Comment: 119 page

Uncertainty reasoning in expert systems

Intelligent control is a very successful way to transform the expert's knowledge of the type 'if the velocity is big and the distance from the object is small, hit the brakes and decelerate as fast as possible' into an actual control. To apply this transformation, one must choose appropriate methods for reasoning with uncertainty, i.e., one must: (1) choose the representation for words like 'small', 'big'; (2) choose operations corresponding to 'and' and 'or'; (3) choose a method that transforms the resulting uncertain control recommendations into a precise control strategy. The wrong choice can drastically affect the quality of the resulting control, so the problem of choosing the right procedure is very important. From a mathematical viewpoint these choice problems correspond to non-linear optimization and are therefore extremely difficult. In this project, a new mathematical formalism (based on group theory) is developed that allows us to solve the problem of optimal choice and thus: (1) explain why the existing choices are really the best (in some situations); (2) explain a rather mysterious fact that fuzzy control (i.e., control based on the experts' knowledge) is often better than the control by these same experts; and (3) give choice recommendations for the cases when traditional choices do not work

How far we are from the complete knowledge: Complexity of knowledge acquisition in Dempster-Shafer approach

When a knowledge base represents the experts' uncertainty, then it is reasonable to ask how far we are from the complete knowledge, that is, how many more questions do we have to ask (to these experts, to nature by means of experimenting, etc) in order to attain the complete knowledge. Of course, since we do not know what the real world is, we cannot get the precise number of questions from the very beginning: it is quite possible, for example, that we ask the right question first and thus guess the real state of the world after the first question. So we have to estimate this number and use this estimate as a natural measure of completeness for a given knowledge base. We give such estimates for Dempster-Shafer formalism. Namely, we show that this average number of questions can be obtained by solving a simple mathematical optimization problem. In principle this characteristic is not always sufficient to express the fact that sometimes we have more knowledge. For example, it has the same value if we have an event with two possible outcomes and nothing else is known, and if there is an additional knowledge that the probability of every outcome is 0.5. We'll show that from the practical viewpoint this is not a problem, because the difference between the necessary number of questions in both cases is practically negligible

On the foundations of Lévy finance: Equilibrium for a single-agent financial market with jumps

For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Lévy process. The agent is allowed to consume a lump at the terminal date; before, only flow consumption is allowed. The agent's utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium prices depend on the agent's risk-aversion through the felicity functions. It turns out that these prices will be the (stochastic) exponential of a Lévy process essentially only if this process is geometric Brownian motion.financial equilibrium, asset pricing, representative agent models, Lévy processes, nonstandard analysis

Advances and Applications of Dezert-Smarandache Theory (DSmT) for Information Fusion (Collected works), Vol. 2

This second volume dedicated to Dezert-Smarandache Theory (DSmT) in Information Fusion brings in new fusion quantitative rules (such as the PCR1-6, where PCR5 for two sources does the most mathematically exact redistribution of conflicting masses to the non-empty sets in the fusion literature), qualitative fusion rules, and the Belief Conditioning Rule (BCR) which is different from the classical conditioning rule used by the fusion community working with the Mathematical Theory of Evidence. Other fusion rules are constructed based on T-norm and T-conorm (hence using fuzzy logic and fuzzy set in information fusion), or more general fusion rules based on N-norm and N-conorm (hence using neutrosophic logic and neutrosophic set in information fusion), and an attempt to unify the fusion rules and fusion theories. The known fusion rules are extended from the power set to the hyper-power set and comparison between rules are made on many examples. One defines the degree of intersection of two sets, degree of union of two sets, and degree of inclusion of two sets which all help in improving the all existing fusion rules as well as the credibility, plausibility, and communality functions. The book chapters are written by Frederic Dambreville, Milan Daniel, Jean Dezert, Pascal Djiknavorian, Dominic Grenier, Xinhan Huang, Pavlina Dimitrova Konstantinova, Xinde Li, Arnaud Martin, Christophe Osswald, Andrew Schumann, Tzvetan Atanasov Semerdjiev, Florentin Smarandache, Albena Tchamova, and Min Wang

Great Expectations. Part II: Generalized Expected Utility as a Universal Decision Rule

Many different rules for decision making have been introduced in the literature. We show that a notion of generalized expected utility proposed in Part I of this paper is a universal decision rule, in the sense that it can represent essentially all other decision rules.Comment: Preliminary version appears in Proc. 18th International Joint Conference on AI (IJCAI), 2003, pp. 297-30

How far are we from the complete knowledge: complexity of knowledge acquisition in Dempster-Shafer approach

Abstract: When a knowledge base represents the experts ’ uncertainty, then it is reasonable to ask how far we are from the complete knowledge, that is, how many more questions do we have to ask (to these experts, to nature by means of experimenting, etc) in order to attain the complete knowledge. Of course, since we do not know what the real world is, we cannot get the precise number of questions from the very beginning: it is quite possible, for example, that we ask the right question first and thus guess the real state of the world after the first question. So we have to estimate this number and use this estimate as a natural measure of completeness for a given knowledge base. We give such estimates for Dempster-Shafer formalism. Namely, we show that this average number of questions can be obtained by solving a simple mathematical optimization problem. In principle this characteristic is not always sufficient to express the fact that sometimes we have more knowledge. For example, it has the same value if we have an event with two possible outcomes and nothing else is known, and if there is an additional knowledge that the probability of every outcome is 0.5. We’ll show that from the practical viewpoint this is not a problem, because the difference between the necessary number o