Qingguo Li

Related family: A new method for attribute reduction of covering information systems

In terms of attribute reduction of covering based rough sets, the discernibility matrix is used as a conventional method to compute all attribute reducts. However, it is inapplicable to attribute reduction in certain circumstances. In this article, a new method, referred to as the related family, is introduced to compute all attribute reducts and relative attribute reducts for covering rough sets. Its core idea is to remove superfluous attributes while keeping the approximation space of covering information system unchanged. The related family method is more powerful than the discernibility matrix method, since the former can handle complicated cases that could not be handled by the latter. In addition, a simplified version of the related family and its corresponding heuristic algorithm are also presented.

Rough sets induced by ideals in lattices

This paper is to study the rough sets within the context of lattices. We study the special properties of the rough sets which can be constructed by means of the congruences determined by ideals of lattice. Also the properties of the generalized rough sets with respect to ideals of lattice are investigated. Finally we give an example of their application in formal concept analysis.

Algebraic properties of L-fuzzy finite automata

Fuzzy automata theory on lattice-ordered monoids was introduced by Li and Pedrycz. Dropping the distributive laws, fuzzy finite automata (L-FFAs for short) based on a more generalized structure L, named a po-monoid, are presented and investigated from the view of algebra in this paper. The notions of (strong) successor and source operators, fuzzy successor and source operators which are shown to be closure operators on certain conditions are introduced and discussed in detail. Using the weak primary submachines, a unique decomposition theorem of a fuzzy finite automaton based on a lattice-ordered monoid is obtained. Taking L as a quantale, fuzzy subsystems are proved to be the same as fuzzy submachines of an L-FFA. In particular, intrinsic connections between algebraic properties of L and properties of some operators of an L-FFA are discovered. It is shown that the join-preserving property of fuzzy successor and source operators can be fully characterized by the right and left distributive laws respectively, and the idempotence of successor operator can be characterized equivalently by the nonexistence of zero divisors when L is a lattice-ordered monoid.

Incremental approaches to knowledge reduction based on characteristic matrices

Knowledge reduction is complicated with the dynamic change of the object set in applications. In this paper, we propose incremental approaches to computing the type-1 and type-2 characteristic matrices of coverings with respect to variation of objects. Also we present two incremental algorithms of calculating the second and sixth lower and upper approximations of sets when adding and deleting more objects in dynamic covering approximation spaces. Subsequently, we employ experiments to validate that the incremental approaches are more effective and efficient to construct approximations of sets in dynamic covering information systems. Finally, we preform knowledge reduction of dynamic covering decision information systems by using the incremental approaches.

Fuzzy grammar theory based on lattices

In this paper, on the basis of breadth-first and depth-first ways, we establish a fundamental framework of fuzzy grammars based on lattices, which provides a necessary tool for the analysis of fuzzy automata. The relationship among finite automata with membership values in lattices (l-VFAs), lattice-valued regular grammars (l-RGs) and lattice-valued deterministic regular grammars (l-DRGs) is investigated. It is demonstrated that, based on each semantic way, l-VFAs and l-RGs are equivalent in the sense that they accept or generate the same classes of fuzzy languages. Furthermore, it is proved that l-VFAs,xa0l-valued deterministic finite automata, l-RGs and l-DRGs are equivalent based on depth-first way. For any l-RG,xa0the language based on breadth-first way coincides with the language based on depth-first way if and only if the truth-valued domain l is a distributive lattice.

Partial residuated structures and quantum structures

Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics—classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained.

Knowledge reduction of dynamic covering decision information systems caused by variations of attribute values

In practical situations, it is time-consuming to conduct knowledge reduction of dynamic covering decision information systems caused by variations of attribute values with the non-incremental approaches. In this paper, motivated by the need for knowledge reduction of dynamic covering decision information systems, we introduce incremental approaches to computing the type-1 and type-2 characteristic matrices for constructing the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces caused by revising attribute attributes. We also employ several examples to explain how to compute the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Then we propose the incremental algorithms for computing the second and sixth lower and upper approximations of sets and employ experimental results to illustrate the incremental algorithms are effective to calculate the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Finally, we give two examples to show how to conduct knowledge reduction of dynamic covering decision information systems caused by altering attribute values.

Formal query systems on contexts and a representation of algebraic lattices

Formal Concept Analysis (FCA) has proven to be a feasible framework for information retrieval (IR). In this paper, with the aim of exploring the mathematical properties of the FCA-based document retrieval paradigm, we propose the notions of conjunctive and disjunctive query on the formal contexts and investigate the derived query systems from the order-theoretical perspective. We show that the conjunctive query system is isomorphic to the Galois lattice of the underlying context, while the disjunctive query system can be join-densely generated from the Galois lattice of the underlying context up to isomorphism. Next, we introduce directed disjunctive query systems and study their connection with algebraic lattices. As a result, we show that the notion of directed disjunctive query provides an alternative approach to restructuring algebraic lattices.

Axiomatic approaches to rough approximation operators via ideal on a complete completely distributive lattice

In 2014, Zhou and Hu (Inf Sci 269:378–387, 2014) introduced a kind of rough sets on a complete completely distributive lattice (short for CCD lattice), which can be seen as a unified framework for the study of rough sets based on ordinary binary relations, rough fuzzy sets and interval-valued rough fuzzy sets. Han et al. (Soft Comput 20:1853–1861, 2016) introduced a new pair of rough approximation operators via ideal on a CCD lattice in 2016, which is more general and accurate than Zhou and Hu’s. In this paper, we further investigate its properties, and then the axiomatic approaches are studied. Through some of our axioms, the rough approximations via ideal on a complete atomic Boolean lattice can be viewed as special cases of rough approximation operators via ideal on a CCD lattice if the ideal is well given.

Rough approximations via ideal on a complete completely distributive lattice

The rough approximations on a complete completely distributive lattice L based on binary relation were introduced by Zhou and Hu (Inf Sci 269:378–387, 2014), where the binary relation was defined on the set of non-zero join-irreducible elements. This paper extends Zhou and Hu’s rough set model by defining new approximation operators via ideal. When I is the least ideal of L and R is a reflexive binary relation, these two approximations coincide. We present the essential properties of new approximations and also discuss how the new one relates to the old one. Finally, the topological and lattice structures of the approximations are given.