粗糙蕴涵

Rough implication operator is the emphasis and difficulty in the study of rough logic. Due to the shortage of rough implication in [3]～[5], we redefine rough set and rough implication operator by Stone algebra, and introduce new rough operators such as rough intersection, rough union, and rough complement. Moreover the characteristics of the proposed rough implication are investigated ,and we also point out that the proposed implication operation is superior to that of three-valued Lukasiewicz logic.     

A metaheuristic approach to the optimal definition of molecule-fixed axes in rovibrational Hamiltonians

We develop an algorithmic, metaheuristic approach to the definition of molecule-fixed axes orientation in molecules of arbitrary size. The goal is to simplify the treatment of overall rotation and rotation-vibration interaction in rovibrational Hamiltonians. Considering the kinetic elements of the rovibrational Hamiltonian, given by the G matrix, we select the optimal orientation of molecule-fixed axes minimizing specific G matrix elements. To such an end, we develop a global minimization method based in a hybrid Simulated Annealing+Powell's local minimization. The parameters of the method are calibrated using a set of non-rigid molecules: Acetaldehyde, glycolaldehyde, methyl formate and ethyl methyl ether. The results show that the principal axes of inertia do not give the simplest form to the pure rotational contribution. However, minimization of the G matrix rotational element does. Finally, we observe that in the cases considered it is not possible to nullify all the rotation-vibration coupling elements, since the torsional motions are coupled with the overall rotation. However, the treatment yields the optimal solution. The methodology proposed allows also to simplify simultaneously the pure rotational+rotation-vibration coupling elements in rovibrational Hamiltonians.     

Heuristic computation of the rovibrational G matrix in optimized molecule-fixed axes. Gmat 2.1

Gmat 2.1 is a program able to compute the rovibrational G matrix in different molecule-fixed axes extending the capabilities of Gmat 1.0. The present version is able to select optimal molecule-fixed axes minimizing the pure rotational kinetic elements, the rovibrational kinetic elements or both simultaneously. To such an end, it uses a hybrid minimization approach. Thus, it combines a global search heuristic based in simulated annealing with a gradient-free local minimization. As the previous version, the program handles the structural results of potential energy hypersurface mappings computed in computer clusters or computational Grid environments. However, since now more general molecule-fixed axes can be defined, a procedure is implemented to ensure the same minimum of the cost function is used in all the molecular structures. In addition, an algorithm for the unambiguous definition of the molecule-fixed axes orientation is used.

Program summary

Program title: Gmat 2.1Catalogue identifier: AECZ_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECZ_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 52 555No. of bytes in distributed program, including test data, etc.: 932 366Distribution format: tar.gzProgramming language: Standard ANSI C++Computer: AllOperating system: Linux, WindowsClassification: 16.2Catalogue identifier of previous version: AECZ_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 1183Does the new version supersede the previous version?: YesNature of problem: When building molecular rovibrational Hamiltonians, the kinetic terms depend on the molecule-fixed axes orientation. Thus, an appropriate orientation can significantly simplify the treatment of pure rotation and rovibrational coupling. The kinetic terms are collected in the rovibrational G matrix. Thus, selection of an appropriate molecule-fixed reference frame is equivalent to localize the axes that minimize specific G matrix elements. From this standpoint, three different kinds of molecule-fixed axes are of interest: first, axes minimizing pure rotational elements of the G matrix; second, axes minimizing all the rovibrational G matrix elements; third, axes minimizing simultaneously pure rotational + rovibrational coupling elements.Solution method: In order to carry out the optimal selection of molecule-fixed axes, we add a hybrid method of minimization to the capabilities included in the first version of the program [1]. Thus, we minimize specific elements of the rovibrational G matrix. To such an end, we apply a heuristic global optimization strategy, simulated annealing [2], followed by a Powell's local minimization [3]. We also include a procedure to ensure that the same minimum is used when several molecular configurations are considered. In addition, an unambiguous molecule-fixed axes ordering is implemented.Reasons for new version: The previous version of the program, Gmat 1.0, works in principal axes of inertia. Although this axes system is adequate for pure vibrational Hamiltonians, it is not always optimal for the construction of general rovibrational Hamiltonians. However, implementing the methods presented here, we can obtain molecule-fixed axes minimizing pure rotational or/and rovibrational interactions in the G matrix. In this form, we can derive the simplest analytical form of the rovibrational Hamiltonian.Summary of revisions: Some new methods have been introduced:

1.

A method to build the molecule-fixed axes rotation matrix from the Euler angles.

2.

Methods for rotating nuclear coordinates and their derivatives using the rotation matrix.

3.

A method for applying simulated annealing to the search of the global minimum of a cost function formed by rotational or rovibrational G matrix elements.

4.

A method implementing Powell.

Running time: The sample tests take a few seconds to execute.References:

[1]

M.E. Castro, A. Niño, C. Muñoz-Caro, Comput. Phys. Comm. 180 (2009) 1183.

[2]

S. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi, Science 220 (4598) (1983) 671.

[3]

W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes. The Art of Scientific Computing, Cambridge University Press, Cambridge, 2007.

     

Loop-free Gray code algorithm for the e-restricted growth functions

The subject of Gray codes algorithms for the set partitions of {1,2,…,n} had been covered in several works. The first Gray code for that set was introduced by Knuth (1975) [5], later, Ruskey presented a modified version of Knuth?s algorithm with distance two, Ehrlich (1973) [3] introduced a loop-free algorithm for the set of partitions of {1,2,…,n}, Ruskey and Savage (1994) [9] generalized Ehrlich?s results and give two Gray codes for the set of partitions of {1,2,…,n}, and recently, Mansour et al. (2008) [7] gave another Gray code and loop-free generating algorithm for that set by adopting plane tree techniques.In this paper, we introduce the set of e-restricted growth functions (a generalization of restricted growth functions) and extend the aforementioned results by giving a Gray code with distance one for this set; and as a particular case we obtain a new Gray code for set partitions in restricted growth function representation. Our Gray code satisfies some prefix properties and can be implemented by a loop-free generating algorithm using classical techniques; such algorithms can be used as a practical solution of some difficult problems. Finally, we give some enumerative results concerning the restricted growth functions of order d.     

WBR0-代数的Pt-模和Ps-模表示

三角模和蕴涵算子是逻辑代数中的一对相互关联的基本算子,而WBR0-代数是仅以蕴涵算子为基本算子定义于一般集合上的逻辑代数。对WBR0-代数进行了再研究。给出了WBR0-代数的Pt-模（偏序集上的t-模）的表现形式;在偏序集上通过 Pt-模结合蕴涵算子给出了WBR0-代数的表示形式;证明了WBR0-代数原始定义中的二元运算[⊕]是WBR0-代数的Ps-模（偏序集上的s-模）。结果对进一步探讨WBR0-代数及其相关的逻辑代数有积极的参考价值。     

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)-(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual non-distributive ones, in the scope of knowledge representation and approximation spaces is discussed.     

The complexity of constraint satisfaction problems for small relation algebras

Andréka and Maddux [Notre Dame J. Formal Logic 35 (4) 1994] classified the small relation algebras—those with at most 8 elements, or in other terms, at most 3 atomic relations. They showed that there are eighteen isomorphism types of small relation algebras, all representable. For each simple, small relation algebra they computed the spectrum of the algebra, namely the set of cardinalities of square representations of that relation algebra.In this paper we analyze the computational complexity of the problem of deciding the satisfiability of a finite set of constraints built on any small relation algebra. We give a complete classification of the complexities of the general constraint satisfaction problem for small relation algebras. For three of the small relation algebras the constraint satisfaction problem is NP-complete, for the other fifteen small relation algebras the constraint satisfaction problem has cubic (or lower) complexity.We also classify the complexity of the constraint satisfaction problem over fixed finite representations of any relation algebra. If the representation has size two or less then the complexity is cubic (or lower), but if the representation is square, finite and bigger than two then the complexity is NP-complete.     

Deterministic solutions to QSAT and Q3SAT by spiking neural P systems with pre-computed resources

In this paper we continue previous studies on the computational efficiency of spiking neural P systems, under the assumption that some pre-computed resources of exponential size are given in advance. Specifically, we give a deterministic solution for each of two well known PSPACE-complete problems: QSAT and Q3SAT. In the case of QSAT, the answer to any instance of the problem is computed in a time which is linear with respect to both the number n of Boolean variables and the number m of clauses that compose the instance. As for Q3SAT, the answer is computed in a time which is at most cubic in the number n of Boolean variables.     

粗糙集代数与BR0代数

讨论粗糙集代数与脓。代数的关系,以及由粗糙集代数构造BR0代数的方法。借助近似代数的原子及同余关系,证明了在适当选取蕴涵算子及余运算之后,粗糙集代数就成为BR0代数。     

粗糙集代数与蕴涵格

在粗糙集的代数刻画方面,一个重要方法是在偶序对〈R（X）,R（X）〉构成的集合中通过定义基本运算寻找刻画偶序对所成集合的代数结构。其中,最有影响的代数结构是正则双Stone代数和Nelson代数。本文从偶序对〈T（X）,R（X）〉构成的集合入手,通过定义蕴涵运算证明了偶序对〈R（X）,R（X）〉所成集合构成蕴涵格,讨论了粗蕴涵格与
正则双Stone代数的关系。本文的讨论可为粗糙逻辑和粗糙推理奠定基础。     

基于蕴涵算子的粗糙集代数与R0-代数

讨论了粗糙集代数与R0-代数的关系以及由粗糙集代数构造R0-代数的方法。粗糙集本身具有格结构,证明了在适当选取蕴涵算子之后,粗糙集代数就成为R0-代数。     

粗糙集代数与FI代数

讨论了粗糙集代数与FI代数的关系以及由粗糙集代数构造FI代数的方法。粗糙集本身具有格结构,证明了在适当选取蕴含算子之后,粗糙集代数就成为FI代数。     

Interval valued (\in,\in\!\vee\,q) -fuzzy filters of pseudo BL-algebras

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued -fuzzy filters of pseudo BL-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo BL-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo BL-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Connected sum of digital closed surfaces

During the past 20 years the research of digital surfaces has proceeded to find their properties in the digital space Zⁿ, such as a topological number, a simple k-point, the 3D-Jordan theorem, a k-separating set, a boundary detecting algorithm and so on. Actually, unlike surfaces in a continuous space, the features of digital surfaces have different characteristics. The aim of this paper is to introduce the notion of a digital closed k-surface in Zⁿ, n ? 3, with the general k-adjacency relations as a generalization of Malgouyres’ and Morgenthaler’s k-surfaces in Z³, to establish some minimal simple closed k-surfaces in Z³ and to find their digital topological properties in relation with the k-fundamental group and k-contractibility. Moreover, a connected sum of two digital closed surfaces is introduced and its digital topological properties are investigated.     

正则FBR_0-代数的弱t-模及其应用

吴望名教授建立的FI-代数(模糊蕴涵代数)是重要的基础逻辑代数,且通过弱化WBR0-代数建立的FBR0-代数与FI-代数有相同的代数结构。对FBR0-代数进行了较细致的研究。首先,证明了正则的FBR0-代数与RBR0-代数有相同的代数结构;其次,讨论了正则FBR0代数中弱t-模的基本性质;最后,给出了正则FBR0-代数的弱t-模表示形式。     

粗糙集代数与BL代数

讨论粗糙集代数与BL代数的关系以及由粗糙集代数构造BL代数的方法。粗糙集代数本身具有格结构,证明了在适当选取蕴涵算子之后,粗糙集代数就成为BL代数。     

集合代数是经典命题演算形式系统的语义解释

经典命题演算形式系统(CPC)中的公式只是一些形式符号,其意义是由具体的解释给出的.逻辑代数和集合代数都是布尔代数,都是CPC的解释.集合代数是CPC的集合语义,其中对联结词的解释就是集合运算;对形式公式的解释就是集合函数;对逻辑蕴涵.逻辑等价的解释就是集合包含和集合相等=.标准概率逻辑是在标准概率空间上建立的逻辑体系,命题表示随机事件,随机事件是集合,概率空间中的事件域是集合代数,概率逻辑就是CPC集合语义的实际应用.CPC完全适用于概率命题演算.     

Rough 3-valued algebras

Rough set theory is an important tool for dealing with granularity and vagueness in information systems. This paper studies a kind of rough set algebra. The collection of all the rough sets of an approximation space can be made into a 3-valued Lukasiewicz algebra. We call the algebra a rough 3-valued Lukasiewicz algebra. In this paper, we focus on the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras. Firstly, we examine whether the rough 3-valued Lukasiewicz algebra is an axled 3-valued Lukasiewicz algebra. Secondly, we present the condition under which the rough 3-valued Lukasiewicz algebra is also a 3-valued Post algebra. Then we investigate the 3-valued Post subalgebra problem of the rough 3-valued Lukasiewicz algebra. Finally, this paper studies the relationship between the rough 3-valued Lukasiewicz algebra and the Boolean algebra constructed by all the exact sets of the corresponding approximation space.     

FI代数上的一致拓扑空间

拓扑结构是逻辑代数中一个重要的研究内容。为描述Fuzzy蕴涵代数的拓扑结构,利用滤子诱导的同余关系在FI代数上构造一致结构和一致拓扑,证明了导出的一致拓扑空间是不连通的、零维的、局部紧的、完全正则的第一可数空间,是[T0]空间当且仅当诱导它的滤子为{1},且FI代数中的蕴涵运算关于导出的一致拓扑是连续的。此外,讨论了商空间的性质。这对从拓扑层面去揭示FI代数的内部结构具有促进作用。