Fully implicational methods for approximate reasoning based on interval-valued fuzzy sets

The aim of this paper is to discuss the approximate rea- soning problems with interval-valued fuzzy environments based on the fully implicational idea. First, this paper constructs a class of interval-valued fuzzy implications by means of a type of impli- cations and a parameter on the unit interval, then uses them to establish fully implicational reasoning methods for interval-valued fuzzy modus ponens （IFMP） and interval-valued fuzzy modus tel- lens （IFMT） problems. At the same time the reversibility properties of these methods are analyzed and the reversible conditions are given. It is shown that the existing unified forms of α-triple I （the abbreviation of triple implications） methods for FMP and FMT can be seen as the particular cases of our methods for IFMP and IFMT.

Theory of Approximate Reasoning in Two-Valued Predicate Logic Based on the Quasi-truth Degrees

Based on the theory of the quasi-truth degrees in two-valued predicate logic, some researches on approximate reasoning are studied in this paper. The relation of the pseudo-metric between first-order formulae and the quasi-truth degrees of first-order formulae is discussed, and it is proved that there is no isolated point in the logic metric space (F, ρ ). Thus the pseudo-metric between first-order formulae is well defined to develop the study about approximate reasoning in the logic metric space (F, ρ ). Then, three different types of approximate reasoning patterns are proposed, and their equivalence under some condition is proved. This work aims at filling in the blanks of approximate reasoning in quantitative predicate logic.

A unified approximate reasoning theory suitable for both propositional calculus system L and predicate calculus system K

The concepts of metric R0-algebra and Hilbert cube of type RO are introduced. A unified approximate reasoning theory in propositional caculus system ? and predicate calculus system (?) is established semantically as well as syntactically, and a unified complete theorem is obtained.

Intelligent reasoning and management decision making with grey rough influence diagrams

Purpose-Influence diagrams(IDs)have been widely applied as a form of knowledge expression and a decision analysis tool in the management and engineering fields.Relationship measurements and expectation values are computed depending on probability distributions in traditional IDs,however,most information systems in the real world are nondeterministic,and data in information tables can be interval valued,multiple valued and even incomplete.Consequently,conventional numeric models of IDs are not suitable for information processing with respect to imprecise data whose boundaries are uncertain.The paper aims to discuss these issues.Design/methodology/approach-The grey system theory and rough sets have proved to be effective tools in the data processing of uncertain information systems,approximate knowledge acquisition and representation are also the objectives in intelligent reasoning and decision analysis.Hence,this study proposes a new mathematical model by combining grey rough sets with IDs,and approximate measurements are used instead of probability distribution,an implicational relationship is utilized instead of an indiscernible relationship,and all of the features of the proposed approach contribute to deal with uncertain problems.Findings-The focus of this paper is to provide a more comprehensive framework for approximate knowledge representation and intelligent decision analysis in uncertain information systems and an example of decision support in product management systems with the new approach is illustrated.Originality/value-Collaboration of IDs and grey rough sets is first proposed,which provides a new mathematical and graphical tool for approximate reasoning and intelligent decision analysis within interval-valued information systems.

Randomization of classical inference patterns and its application

By means of randomization, the concept of D-randomized truth degree of formulas in two-valued propositional logic is introduced, and it is proved that the set of values of D-randomized truth degree of formulas has no isolated point in [0,1]. The concepts of D-logic pseudo-metric and D-logic metric space are also introduced and it is proved that there is no isolated point in the space. The new built D-randomized concepts are extensions of the corresponding concepts in quantified logic. Moreover, it is proved that the basic logic connectives are continuous operators in D-logic metric space. Lastly, three different types of approximate reasoning patterns are proposed.

Theory of truth degrees of propositions in the logic system Ln^＊

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lu- kasiewicz propositional logic system Ln^＊ is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0,1], and a general expres- sion of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.

Theory of (n) truth degrees of formulas in modal logic and a consistency theorem

The theory of （n） truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the （n） truth degree of a modality-free formula equals the truth degree of the formula in two-valued propositional logic. Variations of （n） truth degrees of formulas w.r.t. n in temporal logic is investigated. Moreover, the theory of （n） similarity degrees among modal formulas is proposed and the （n） modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic.

A fuzzy logic system based on Schweizer-Sklar t-norm

Based on the Schweizer-Sklar t-norm, a fuzzy logic system UL^＊ is established, and its soundness theorem and completeness theorem are proved. The following facts are pointed out＂ the well-known formal system SBL~ is a semantic extension of UL^＊; the fuzzy logic system IMTL△ is a special case of UL^＊ when two negations in UL^＊ coincide. Moreover, the connections between the system UL^＊ and some fuzzy logic formal systems are investigated. Finally, starting from the concepts of ＂the strength of an‘AND＇ operator＂ by R.R. Yager and ＂the strength of fuzzy rule interaction＂ by T. Whalen, the essential meaning of a parameter p in UL^＊ is explained and the use of fuzzy logic system UL^＊ in approximate reasoning is presented.

A variable precision rough set approach to knowledge discovery in land cover classification

This paper presents a granular computing approach to spatial classification and prediction of land cover classes using rough set variable precision methods.In particular,it presents an approach to characterizing large spatially clustered data sets to discover knowledge in multi-source supervised classification.The evidential structure of spatial classification is founded on the notions of equivalence relations of rough set theory.It allows expressing spatial concepts in terms of approximation space wherein a decision class can be approximated through the partition of boundary regions.The paper also identifies how approximate reasoning can be introduced by using variable precision rough sets in the context of land cover characterization.The rough set theory is applied to demonstrate an empirical application and the predictive performance is compared with popular baseline machine learning algorithms.A comparison shows that the predictive performance of the rough set rule induction is slightly higher than the decision tree and significantly outperforms the baseline models such as neural network,naïve Bayesian and support vector machine methods.