Non-deterministic Information Systems Research Articles

Rough set‐based rule generation and Apriori‐based rule generation from table data sets: a survey and a combination

The authors have been coping with new computational methodologies such as rough sets, information incompleteness, data mining, granular computing, etc., and developed some software tools on association rules as well as new mathematical frameworks. They simply term this research Rough sets Non-deterministic Information Analysis (RNIA). They followed several novel types of research, especially Pawlak's rough sets, Lipski's incomplete information databases, Orlowska's non-deterministic information systems, Agrawal's Apriori algorithm. These are outstanding researches related to information incompleteness, data mining, and rule generation. They have been trying to combine such novel researches, and they have been trying to realise more intelligent rule generator handling data sets with information incompleteness. This study surveys the authors’ research highlights on rule generators, and considers a combination of them.

A probabilistic approach to rough set theory with modal logic perspective

We propose the notion of probabilistic information system (PIS) to capture situations where information regarding attributes of objects are not precise, but given in terms of probability. Notions of indistinguishability relations and corresponding approximation operators based on PISs are proposed and studied. It is shown that the deterministic information systems (DISs), incomplete information systems (IISs) and non-deterministic information systems (NISs) are all special instances of PISs. Moreover, the approximation operators defined on DIS (relative to indiscernibility), IISs and NISs (relative to similarity relations) are all originated from a single approximation operator defined on PISs. Further, a logic LPIS for PISs is proposed that can be used to reason about the proposed approximation operators. A sound and complete deductive system for the logic is given. Decidability of the logic is also proved.

Apriori-Based Rule Generation in Incomplete Information Databases and Non-Deterministic Information Systems

This paper discusses issues related to incomplete information databases and considers a logical framework for rule generation. In our approach, a rule is an implication satisfying specified constraints. The term incomplete information databases covers many types of inexact data, such as non-deterministic information, data with missing values, incomplete information or interval valued data. In the paper, we start by defining certain and possible rules based on non-deterministic information. We use their mathematical properties to solve computational problems related to rule generation. Then, we reconsider the NIS-Apriori algorithm which generates a given implication if and only if it is either a certain rule or a possible rule satisfying the constraints. In this sense, NIS-Apriori is logically sound and complete. In this paper, we pay a special attention to soundness and completeness of the considered algorithmic framework, which is not necessarily obvious when switching from exact to inexact data sets. Moreover, we analyze different types of non-deterministic information corresponding to different types of the underlying attributes, i.e., value sets for qualitative attributes and intervals for quantitative attributes, and we discuss various approaches to construction of descriptors related to particular attributes within the rules' premises. An improved implementation of NIS-Apriori and some demonstrations of an experimental application of our approach to data sets taken from the UCI machine learning repository are also presented. Last but not least, we show simplified proofs of some of our theoretical results.

Division Charts as Granules and Their Merging Algorithm for Rule Generation in Nondeterministic Data

We have been proposing a framework rough Nondeterministic information analysis, which considers granular computing concepts in tables with incomplete and nondeterministic information, as well as rule generation. We have recently defined an expression named division chart with respect to an implication and a subset of objects. Each division chart takes the role of the minimum granule for rule generation, and it takes the role of contingency table in statistics. In this paper, we at first define a division chart in deterministic information systems (DISs) and clarify the relation between a division chart and a corresponding implication. We also consider a merging algorithm for two division charts and extend the relation in DISs to nondeterministic information systems. The relation gives us the foundations of rule generation in tables with nondeterministic information.

Division charts for handling the consistency in granular computing

Non–deterministic information systems (NISs) have been proposed for handling tables with incomplete information. We have developed algorithms and software tools for calculating criterion values for rules, generating possible equivalence classes and so on for rough sets–based concepts in NISs. In the rough set theory, the concept of consistency can be characterised by inclusion relations between equivalence classes, and this property connects rough sets with granular computing. In the present paper, we propose diagrams that we call division charts for equivalence classes. Division charts present visual information about inclusion relations and consistency. We show how to use division charts to calculate criterion values for rules in NISs, and we also show how they can be used to calculate degrees of data dependencies.

Association rule-based decision making in table data

We have been coping with several aspects of rough sets in non–deterministic information systems (NISs) and tables with inexact data. We are simply calling this work rough non–deterministic information analysis (RNIA). As for decision making in RNIA, we at first obtain rules in tables, then we apply them to the current condition. Therefore, there may be a case that the current condition may not match with the condition part in the obtained rules. In order to recover this unmatched case, this paper newly considers direct question–answering in tables, and proposes association rule–based decision making in tables. The validity of the decision is defined by the criterion values of the association rule. This decision making recovers the complementary functionality in the application of obtained rules. Some examples of execution by the implemented software tool are also presented.

Stable rule extraction and decision making in rough non-deterministic information analysis

Rough Non-deterministic Information Analysis (RNIA) is a rough set-based data analysis framework for Non-deterministic Information Systems (NISs). RNIA-related algorithms and software tools developed so far for rule generation provide good characteri

Incomplete and Nondeterministic Information Systems: Object-Directed Semantics for Descriptor Languages

In the paper we discuss logical approaches to incomplete and/or nondeterministic data. As is well-known, complete and deterministic information systems induce indiscernibility relations and the lower and upper approximations regarded as operators obey the laws of S5 modal system. In the case of incomplete and/or nondeterministic systems, this modal approach yields a few more binary relations on the set of objects (e.g. NIL-structures and NIL-logics). Anyway, in both cases, there are relations which play a dominant role. The main idea of this study is to shift the focus from relations to objects - that is why we use the term object-directed. Following well-established traditions from modal logics, we would like to consider objects with empty or non-single attribute values as a special kind of worlds. In consequence, in the object-directed approach there would be two sorts of objects and (usually) one relation in contrast to the relation-directed approach where we have one sort of objects and a number of relations. In the first part of our paper we shall discuss a global kind of non-normality and show how rough approximations can be linked to weak modal systems. In the second part we shall consider a local kind of non-normality; this time we use a multi-valued modal system Q introduced by A. N. Prior. The key idea offered by the paper is to regard incomplete and/or nondeterministic information systems as a source of non-normal models for (modal) descriptor languages.

A grey-based rough approximation model for interval data processing

A new rough set model for interval data named grey-rough set is proposed in this paper. Information system in the real world are quite complicated. Most of information tables record not only categorical data but also numerical data including a range of interval data. The grey lattice operation in grey system theory is one of the operations for interval data that modifies endpoints non-arithmetically, and which is useful for interval data processing. The grey-rough approximation is based on an interval coincidence relation and an interval inclusion relation instead of an equivalence relation and an indiscernibility relation in Pawlak’s model. Numerical examples and four fields of practical examples, decision-making, information retrieval, knowledge discovery and kansei engineering are shown. The advantages of the proposal include: extending a treatable value compared with classical rough set for non-deterministic information systems, providing a maximum solution and minimum solution both in upper and lower approximations, and not only providing mathematical support to SQL but also functions for further extension in the future.

An Application of Discernibility Functions to Generating Minimal Rules in Non-Deterministic Information Systems

Minimal rule generation in Non-deterministic Information Systems (NISs), which follows rough sets based rule generation in Deterministic Information Systems (DISs), is presented. According to certain rules and possible rules in NISs, minimal certain rules and minimal possible rules are defined. Discernibility functions are also introduced into NISs for generating minimal certain rules. Like minimal rule generation in DISs, the condition part of a minimal certain rule is given as a solution of an introduced discernibility function. As for generating minimal possible rules, there may be lots of discernibility functions to be solved. So, an algorithm based on an order of attributes is proposed. A tool, which generates minimal certain rules and minimal possible rules, has also been implemented.

Special Issue on Rough Sets and Granular Computing

Rough sets and granular computing, known as new methodologies for computing technology, are now attracting great interest of researchers. This special issue presents 12 articles, and most of them were presented at the second Japanese workshop on Rough Sets held at Kyushu Institute of Technology in Tobata, Kitakyushu, Japan, on August 17-18, 2005. The first article studies the relation between rough set theory and formal concept analysis. These two frameworks are analyzed and connected by using the method of morphism. The second article introduces object-oriented paradigm into rough set theory, and object-oriented rough set models are proposed. Theoretical aspects of these new models are also examined. The third article considers relations between generalized rough sets, topologies and modal logics, and some topological properties of rough sets induced by equivalence relations are presented. The fourth article focuses on a family of polymodal systems, and theoretical aspects of these systems, like the completeness, are investigated. By means of combining polymodal logic concept and rough set theory, a new framework named multi-rough sets is established. The fifth article focuses on the information incompleteness in fuzzy relational models, and a generalized possibility-based fuzzy relational model is proposed. The sixth article presents a developed software EVALPSN (Extended Vector Annotated Logic Program with Strong Negation) and the application of this software to pipeline valve control. The seventh article presents the properties of attribute reduction in variable precision rough set models. Ten kinds of meaningful reducts are newly proposed, and hierarchical relations in these reducts are examined. The eighth article proposes attribute-value reduction for Kansei analysis using information granulation, and illustrative results for some databases in UCI Machine Learning Repository are presented. The ninth article investigates cluster analysis for data with errors tolerance. Two new clustering algorithms, which are based on the entropy regularized fuzzy c-means, are proposed. The tenth article applies binary decision trees to handwritten Japanese Kanji recognition. The consideration to the experimental results of real Kanji data is also presented. The eleventh article applies a rough sets based method to analysing the character of the screen-design in every web site. The obtained character gives us good knowledge to generate a new web site. The last article focuses on rule generation in non-deterministic information systems. For generating minimal certain rules, discernibility functions are introduced. A new algorithm is also proposed for handling every discernibility function. Finally, we would like to acknowledge all the authors for their efforts and contributions. We are very grateful to reviewers for their thorough and on-time reviews, too. We are also grateful to Prof. Toshio Fukuda and Prof. Kaoru Hirota, Editors-in-Chief of JACIII, for inviting us to serve as Guest Editors of this Journal, and to Mr. Uchino and Mr. Ohmori of Fuji Technology Press for their kind assistance in publication of this special issue.

AN APPLICATION OF ROUGH NON-DETERMINISTIC INFORMATION ANALYSIS TO CLASS EVALUATION DATA BY STUDENTS

Non-deterministic Information Systems (NISs) have been recognized to be the most important framework for handling information incompleteness in tables, and several theoretical work has been examined. We follow this robust framework, and we have been developing algorithms and tool programs, which can handle the rough sets based concepts in NISs, on computers. We are simply calling this work Rough Non-deterministic Information Analysis (RNIA). This paper briefly surveys RNIA, and applies RNIA to class evaluation data, which consists of evaluation on 16 questions by 60 students. In this data, Question 16 “(Q16) Rate this class in either 1, 2, 3, 4 or 5 grade”is the decision attribute, and data dependency and rules in the form of “Condition=> [Q16, 5]”are considered. The most characteristic point is that [Q16, 5] strongly depended upon question “(Q3) Was the teacher polite in every class?”. Question (Q3) is related to the emotional impression of a teacher, and the evaluation of a half-year class by every student seems to express every student's emotional impression of a teacher.

Non-deterministic information systems and their domains

In the theory of denotational semantics of programming languages Dedekind-complete, algebraic partial orders (domains) frequently have been considered since Scott's and Strachey's fundamental work in 1971 (Stoy, 1977). As Scott (1982) showed, these domains can be represented canonically by (deterministic) information systems. However, recently, more complicated constructions (such as power domains) have led to more general domains (Plotkin, 1976; Smyth and Plotkin, 1977; Smyth, 1983). We introduce non-deterministic information systems and establish the representation theorem similar to Scott (1982) for these more general domains. This result will be the basis for solving recursive domain equations.