Modal sequents for normal modal logics

We present sequent calculi for normal modal logics where modal and propositional behaviours are separated, and we prove a cut elimination theorem for the basic system K, so as completeness theorems (in the new style) both for K itself and for its most popular enrichments. MSC: 03B45, 03F05.     

Classically complete modal relevant logics

A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.     

Topology vs generalized rough sets

This paper investigates the relationship between topology and generalized rough sets induced by binary relations. Some known results regarding the relation based rough sets are reviewed, and some new results are given. Particularly, the relationship between different topologies corresponding to the same rough set model is examined. These generalized rough sets are induced by inverse serial relations, reflexive relations and pre-order relations, respectively. We point that inverse serial relations are weakest relations which can induce topological spaces, and that different relation based generalized rough set models will induce different topological spaces. We proved that two known topologies corresponding to reflexive relation based rough set model given recently are different, and gave a condition under which the both are the same topology.     

Axiomatizations of team logics

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(～) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.     

单向奇异粗糙模糊集合的结构

用模糊集合与模糊等价关系对单向奇异粗集进行了研究,并给出了单向奇异粗糙模糊集合的数学结构及其并、交、补运算和性质.同时证明了单向奇异粗糙模糊集合对并、交、补运算构成完全可无限分配的软代数.     

Analyzing completeness of axiomatic functional systems for temporal × modal logics

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions (being injective, surjective, increasing, etc.) have been considered and axiomatic systems (called functional) which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prefinitely axiomatizable modal and intermediate logics

A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and in the lattice of normal extensions of K4.3. MSC: 03B45, 03B55.     

Frame definability in finitely valued modal logics

In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. [27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from [26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics.     

Probabilistic graded rough set and double relative quantitative decision-theoretic rough set

The probabilistic rough set (PRS) model ignores absolute quantitative information i.e., overlap between equivalence class and basic set. And graded rough set (GRS) model cannot reflect the distinctive degrees of information. In order to overcome these defects, this paper proposes the probabilistic graded rough set (PGRS), which is an extension of Pawlak's rough set and GRS. What is more, we propose double relative quantitative decision-theoretic rough set (Drq-DTRS) models, which essentially indicate the relative and absolute quantification.     

On some types of covering rough sets from topological points of view

The concept of coverings is one of the fundamental concepts in topological spaces and plays a big part in the study of topological problems. This motivates the research of covering rough sets from topological points of view. From topological points of view, we can get a good insight into the essence of covering rough sets and make our discussions concise and profound. In this paper, we first construct a type of topology called the topology induced by the covering on a covering approximation space. This notion is indeed in the core of this paper. Then we use it to define the concepts of neighborhoods, closures, connected spaces, and components. Drawing on these concepts, we define several pairs of approximation operators. We not only investigate the relationships among them, but also give clear explanations of the concepts discussed in this paper. For a given covering approximation space, we can use the topology induced by the covering to investigate the topological properties of the space such as separation, connectedness, etc. Finally, a diagram is presented to show that the collection of all the lower and upper approximations considered in this paper constructs a lattice in terms of the inclusion relation ⊆.     

A spatial modal logic with a location interpretation

A spatial modal logic (SML) is introduced as an extension of the modal logic S4 with the addition of certain spatial operators. A sound and complete Kripke semantics with a natural space (or location) interpretation is obtained for SML. The finite model property with respect to the semantics for SML and the cut‐elimination theorem for a modified subsystem of SML are also presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical programming in rough environment

Classical mathematics is usually crisp while most real-life problems are not; therefore, classical mathematics is usually not suitable for dealing with real-life problems. In this article, we present a systematic and focused study of the application of rough sets (Z. Pawlak, Rough sets, In. J. Comput. Informa. Sci. 11 (1982), pp. 341–356.) to a basic area of decision theory, namely ‘mathematical programming’. This new framework concerns mathematical programming in a rough environment and is called ‘rough programming’ (L. Baoding, Theory and Practice of Uncertain Programming, 1st ed., Physica-Verlag, Heidelberg, 2002; E.A. Youness, Characterizing solutions of rough programming problems, Eut. J. Oper. Res. 168 (2006), pp. 1019–1029). It implies the existence of the roughness in any part of the problem as a result of the leakage, uncertainty and vagueness in the available information. We classify rough programming problems into three classes according to the place of the roughness. In rough programming, wherever roughness exists, new concepts like rough feasibility and rough optimality come to the front of our interest. The study of convexity for rough programming problems plays a key role in understanding global optimality in a rough environment. For this, a theoretical framework of convexity in rough programming and conceptualization of the solution is created on the lines of their crisp counterparts.     

On intuitionistic modal and tense logics and their classical companion logics: Topological semantics and bisimulations

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics.