In this paper, we introduce the concept of relative pre-reducts to derive the relative reducts from a large dataset. The relative reduct is considered a consistency-based attribute reduction method that is commonly utilized to extract concise subsets of condition attributes. Nonetheless, calculating all relative reducts necessitates substantial time and memory to build a discernibility matrix. In this research, we demonstrate that all relative pre-reducts can be computed using a simplified matrix referred to as the partial discernibility matrix, which can be readily converted into relative reducts. We also suggest employing a data partitioning approach to generate the discernibility matrix. This method alleviates the issue of an increased number of results for each partition. The outcomes from this technique yield the relative pre-reducts proposed in this study. Since our enhancements to the computation of relative reducts are independent of other advancements, they can be implemented in conjunction with existing methods. Experimental findings indicate that utilizing relative pre-reducts for computing relative reducts is efficient for large datasets.

Traditional rough set models represent concepts as sets. Vector-based concept representations have become mainstream in recent years, particularly in word embedding models, but existing rough set models are not designed to accommodate such representations. To address this, we propose a model that represents concepts as vectors. In rough set theory, concepts are approximated by defining granules, or basic concepts, as the basic units of knowledge. Approximations are then constructed as unions of these granules, such that each approximation is entirely dependent on the inclusion or exclusion of individual granules. When approximations are restricted to decision classes, the inclusion of a granule in a decision class approximation can be determined by associating a vector with each of the granule and the decision class and then evaluating their inner product. This methodology remains valid even within probabilistic rough set models. Building on this foundation, we propose vector-based rough set models, wherein granules and decision classes are represented as vectors from the outset. The proposed model preserves the properties of probabilistic rough sets in the approximation of the decision classes. Although our previous considerations required the vectors representing the decision classes to be linearly independent, we demonstrate that this restriction is unnecessary.

Abstract User-based collaborative filtering (UBCF) is widely used in recommender systems (RSs) as one of the most successful approaches, but traditional UBCF cannot provide recommendations with satisfactory accuracy and diversity simultaneously. Covering-based collaborative filtering (CBCF) is a useful approach that we have proposed in our previous work, which greatly improves the traditional UBCF and could provide satisfactory recommendations to an active user which often has sufficient rating information. However, different from an active user, a new user in RSs often has special characteristics (e.g., fewer ratings or ratings concentrating on popular items), and the previous CBCF approach cannot provide satisfactory recommendations for a new user. In this paper, aiming to provide personalized recommendations for a new user, through a detailed analysis of the characteristics of new users, we reconstruct a decision class to improve the previous CBCF and utilize the covering reduction algorithm in covering-based rough sets to remove redundant candidate neighbors for a new user. Furthermore, unlike the previous CBCF, our improved CBCF could provide personalized recommendations without needing special additional information. Experimental results suggest that for the sparse datasets that often occur in real RSs, the improved CBCF significantly outperforms those of existing work and can provide personalized recommendations for a new user with satisfactory accuracy and diversity simultaneously.

Rough sets theory is used to handle uncertain and inconsistent information. While, Pawlak’s decision logic of rough sets is based on classical bivalence logic, this may cause a limitation for the various reasoning. In this study, we propose four-valued logics, as the deduction basis for the decision logic. To provide four-valued semantics to decision logic of rough set, we introduce a Ziarko’s variable precision rough set. As a deductive system, we adopt tableau calculi and define a consequence relation to construct deductive system based on four-valued semantics. Furthermore, weak-negation is introduced to compensate the deduction property of four-value logics. Finally, we discuss Henkin-type proof of the completeness theorem for the system.

Mining from big data is one of the hottest topics in current trends of computer science. Attribute reduction to compute relative reducts from a given dataset is a key technique to use rough set theory as a tool in data mining, however, attribute reduction from a given dataset with numerous objects and attributes is very difficult. In this paper, to achieve attribute reduction from data with numerous objects and attributes, we try to combine generalized dynamic reducts and a heuristic attributed reduction using reduced decision tables.

Epistemic logic was proposed by Hintikka to model human knowledge and beliefs. Since then, a lot of work has been done to improve epistemic logic for applications to several problems. One of the interesting problems is to relate the concept of ignorance to that of knowledge. In this paper, we introduce an epistemic logic LNI founded on nonignorance by extending the classical propositional logic with the operator Ni. We describe its axiomatization and Kripke semantics. We show some technical results including a completeness and discuss some applications and extensions.

Recently we face classification problems with many categorical features, as seen in genetic data and text data. In this paper, we discuss some ways to give weights on features in the framework of naïve Bayes classifier, that is, under independent assumption of features. Because no order exists in a categorical feature, we consider a histogram over possible values (bins) in the feature. Taking into the difference of number of samples falling in each bin, we propose two kinds of weights: 1) one is derived from the probability that the majority class takes the majority even in samples, and 2) another reflects the expected conditional entropy. With the latter entropy weight, it will be shown that more discriminative features gain higher weights and non-discriminative feature diminishes as the number of samples goes infinity. We reveal the properties of these two kinds of weights through artificial data and some real-life data.

An information table or a training/designing sample set is all that can be obtained to infer the underlying generation mechanism (distribution) of tuples or samples. However, how an information table is available in representation, in treatment, and in interpretation, can still be discussed. In this paper, these matters are discussed on the basis of “granularity.” First, an explanation is given to identify the reasons why different goals/treatments of information tables exist in some different research fields. In this stage, it will be emphasized that “granularity concept” plays an important role. Next, a framework of information tables is reformulated in terms of attribute sets and tuple sets. Here, a “Galois connection” helps to understand their relationship. Then, the use of “closed subsets” is proposed instead of given tuples, for efficiency and for interpretability. With a special type of closed subsets, the traditional logical DNF expression framework can be naturally extended to those with multivalues and continuous values. Last, several concepts on rough sets are reformulated using “variable granularity” connected to closed subsets. This paper determines how and in what points granularity can give flexibility in dealing with several problems. Through several concepts defined in this paper, some intuitions toward development of data exploration and data mining are given.

In multiple-attribute decision making (MADM) problems, one often needs to deal with decision information with uncertainty. During the last decade, Yang and Singh (1994) have proposed and developed an evidential reasoning (ER) approach to deal with such MADM problems. Essentially, this approach is based on an evaluation analysis model and Dempster's rule of combination in the Dempster-Shafer (D-S) theory of evidence. This paper reanalyzes the ER approach explicitly in terms of D-S theory and then proposes a general scheme of attribute aggregation in MADM under uncertainty. In the spirit of such a reanalysis, previous ER algorithms are reviewed and two other aggregation schemes are discussed. Theoretically, it is shown that new aggregation schemes also satisfy the synthesis axioms, which have been recently proposed by Yang and Xu (2002) for which any rational aggregation process should grant. A numerical example traditionally examined in published sources on the ER approach is used to illustrate the discussed techniques.

We propose a method of semantically nonmonotonic reasoning in prioritized knowledge bases based on incomplete information systems and granular reasoning. Our nonmonotonic reasoning is characterized as "reasoning with disregard of difference of detail." We provide granularized models and priority of each rule, and formulate the method of semantically non-monotonic reasoning based on the idea of ignored sets, rejected sets and possible tuples of atomic sentences that we propose.

In this paper, firstly, processes of classical inference are reviewed as granular reasoning from a point of view of reconstructing Kripke-style models with granularity. The essential point of the reconstruction is that some possible worlds are amalgamated to generate granules of worlds and vice versa. It is also called zoom reasoning systems. Then, the idea is applied for fuzzy reasoning processes by considering fuzzily granularized possible worlds. There linguistic truth values with linguistic hedges can be naturally introduced.

Implication of association rules is compared with material implication and conditional in standard conditional models for conditional logic. Then, association rules are shown to be translated into probability-based graded conditionals. Further cooccurrence as the basis for confidence of association rates is examined and then evidence-based confidence is newly introduced.

Functional dependencies, including weighted ones, are formulated in the same manner as other constraints in a relational database based on rough sets. A tuple that belongs to a relation is accompanied by a supplementary attribute value. When a functional dependency is imposed on a relation, a compatibility degree with the functional dependency is calculated for every tuple. Whether the tuple satisfies the functional dependency is determined by comparing the compatibility degree with a supplementary attribute value. Different inference rules from the conventional ones are obtained in augmentation for functional dependencies.

A previous way of generating the so-called non-transitive chain from fuzzy subsets using rough set theory (T. Murai et al., 1997, 1998) can only be applied to rational-valued fuzzy subsets. In this paper, an algorithm for generating a non-transitive chain is first described. Then, the algorithm is applied to real-valued fuzzy subsets adopting approximation in the usual sense.

The notion of degrees of significance of possible worlds under a given context are introduced and thereby a way of how an accessibility relation between possible worlds is generated. This is the first answer to the problem which was pointed out by Resconi et al. in uncertainty hierarchical metatheory.

This paper discusses how the semantic field can be used for making decisions under given data that consist of objects, properties and relations among objects and properties, using the example described in R. Wille's (1989) theory of context.

In the session entitled "New frontiers for logic and uncertainty by the semantic field," the new notion of semantic fields for modal logic is introduced, by which a degree of significance can be assigned to any possible world. Available information is the main source of the significance that shows the logical structure of the information itself. Thus, a guide is obtained to use information and also to discover and measure the uncertainty located in it. The worlds are useful for separating such information into its important parts. Any sentence is evaluated in worlds that are located at different levels of significance. An accessibility relation exists between the more significant and the less significant world. In this way, possible-worlds models of information are made possible using the semantic field. Uncertainty is given, normally, without any logic but with set theory. Thus, the semantic field is used to connect modal logic with set theory in order to obtain a better description of uncertainty.

In this paper, we describe a method by which several kinds of uncertainty, represented by probability, fuzzy sets and the Dempster-Shafer theory of evidence, can be formulated in the semantic field.

A new data structure for online data mining using binary trees is proposed. Two experiments using synthetic data show the effectiveness of the method of data mining based on the proposed data structure.

Proposes a faithful assignment function to construct W. Spohn's (1988) ordinal conditional functions (OCFs) that reflect ordinals in the current epistemic state represented by an OCF as long as possible. We also propose a composition schema for OCFs that construct a new OCF which satisfies postulates for updating and erasure in the AGM (Alchourron, Ga/spl uml/rdenfors & Makinson, 1985) paradigm. The faithful assignments for OCFs and the composition provide useful devices to represent iterated belief updates.

Belief change operations in logical settings and the possibilistic framework are reviewed. In logical settings, the Alchourron-Gardenfors-Makinson (AGM) paradigm (1985) and the Katsuno-Mendelzon framework (1992) are reviewed and interdefinability between update and erasure in the AGM paradigm is explained precisely. Moreover, a new identity that captures a relationship from contraction to erasure in the AGM paradigm is introduced and clear relationships among revision, contraction, update and erasure in the AGM paradigm are established. In the possibilistic framework, expansion and above four types of belief change are also formulated, and it is shown that two identities that capture the relationships between update and erasure in the AGM paradigm still remain valid between updating rule and the refined erasing rule.

A way of generating accessible worlds is described. For this purpose, the class of the so-called finite uniform Scott-Montague models is introduced and the properties are investigated as well as their relation with ambiguity and inconsistency. With the class of models, the approach is formulated from the point of view of possible-worlds restriction under available information.

A relational data model is extended under a modal logic approach so that it can deal with different kinds of imperfect information. In the extended model the imperfect information is uniformly represented in a set of possible worlds and an accessibility relation. This leads to the extended data model having the same operations to different kinds of imperfect information. Moreover, the model can support flexible queries. The approach gives fundamentals to integrate different kinds of databases handling different types of imperfect information.

A method of generating fuzzy k-partitions is proposed based on tolerance relations and maximal consistency. In particular, emphasis is put on the point that general cases of more than two clusters can be reduced to special cases of two clusters in both crisp and fuzzy partitions. Then a way of generating fuzzy subsets can be shown to be applied to the generation of fuzzy k-partitions.