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Complementary neighborhood is a conception analogous to the neighborhood that we first introduced in a former paper. In this paper, we show that two different approximation operators may have a close relationship, namely, they can be defined almost in the same way except that one uses the notion of neighborhood and another uses the complementary neighborhood. We call such two approximation operators the twin approximation operators. We give some concrete examples of the twin approximation operators. Through detailed investigation on the relationship between the neighborhood and the complementary neighborhood, we further study the properties of given twin approximation operators and investigate the relationships between different twins. We also reveal the topological properties of those twin approximation operators. [ABSTRACT FROM AUTHOR]

Abstract: Two basic inference models of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). The Triple I method is a very important method to solve the problems of FMP and FMT. The aim of this paper is to extend the Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets. In the paper, we first investigate the algebra operators' properties on the lattice structure of intuitionistic fuzzy information and provide the unified form of residual implications which indicates the relationship between intuitionistic fuzzy implications and fuzzy implications. Then we present the intuitionistic fuzzy reasoning version of the Triple I principles based on the models of intuitionistic fuzzy modus ponens (IFMP) and intuitionistic fuzzy modus tollens (IFMT) and give the Triple I method of intuitionistic fuzzy reasoning for residual implications. Moreover, we discuss the reductivity of the Triple I methods for IFMP and IFMT. Finally, we propose α-Triple I method of intuitionistic fuzzy reasoning. [Copyright &y& Elsevier]

The conditional independencies from a joint probability distribution constitute a model which is closed under the semi-graphoid properties of independency. These models typically are exponentially large in size and cannot be feasibly enumerated. For describing a semi-graphoid model therefore, researchers have proposed a more concise representation. This representation is composed of a representative subset of the independencies involved, called a basis, and lets all other independencies be implicitly defined by the semi-graphoid properties. An algorithm is available for computing such a basis for a semi-graphoid independency model. In this paper, we identify some new properties of a basis in general which can be exploited for arriving at an even more concise representation of a semi-graphoid model. Based upon these properties, we present an enhanced algorithm for basis construction which never returns a larger basis for a given independency model than currently existing algorithms. [ABSTRACT FROM AUTHOR]

Embedding conglomerability as a rationality requirement in probability was among the aims of Walley's behavioural theory of coherent lower previsions. However, recent work has shown that this attempt has only been partly successful. If we focus in particular on the extension of given assessments to a rational and conglomerable model (in the least-committal way), we have that the procedure used in Walley's theory, the natural extension , provides only an approximation to the model that is actually sought for: the so-called conglomerable natural extension . In this paper we consider probabilistic assessments in the form of a coherent lower prevision P ̲ , which is another name for a lower expectation functional, and make an in-depth mathematical study of the problem of computing the conglomerable natural extension for this case: that is, where it is defined as the smallest coherent lower prevision F ̲ ≥ P ̲ that is conglomerable, in case it exists. Past work has shown that F ̲ can be approximated by an increasing sequence ( E ̲ n ) n ∈ N of coherent lower previsions. We solve an open problem by showing that this sequence can consist of infinitely many distinct elements. Moreover, we give sufficient conditions, of quite broad applicability, to make sure that the point-wise limit of the sequence is F ̲ in case P ̲ is the lower envelope of finitely many linear previsions. In addition, we study the question of the existence of F ̲ and its relationship with the notion of marginal extension. [ABSTRACT FROM AUTHOR]