Maximal contractions in Boolean algebras

In the present paper,the concepts of deductive element and maximal contraction are introduced in Boolean algebras,and corresponding theories of consistency and maximal contractions are studied.An algorithm principle is proposed to compute all maximal contractions for a consistent set with respect to its refutation in Boolean algebras.It is pointed out that the quotient algebra of the first-order language with respect to its provable equivalence relation constitutes a Boolean algebra,and hence the computation of R-contractions for closed formulas in first-order languages can be converted into the one in Boolean algebras proposed in this paper.Furthermore,the concept of basic element is introduced in Boolean algebras,which contributes to the definitions of clause and Horn clause transplanted from logic to a special type of Boolean algebras generated by basic elements.It is also pointed out that the computation of R-contractions for clauses in the classical propositional logic can be converted into the one in Boolean algebras generated by basic elements proposed in this paper.     

Refinement algebra with dual operator

Algebras of imperative programming languages have been successful in reasoning about programs. In general an algebra of programs is an algebraic structure with programs as elements and with program compositions (sequential composition, choice, skip) as algebra operations. Various versions of these algebras were introduced to model partial correctness, total correctness, refinement, demonic choice, and other aspects. We introduce here an algebra which can be used to model total correctness, refinement, demonic and angelic choice. The basic model of our algebra are monotonic Boolean transformers (monotonic functions from a Boolean algebra to itself).     

Torsion elements in effect algebras

We define the torsion element in effect algebras and use it to characterize MV-effect algebra and 0-homogeneous effect algebras in chain-complete effect algebras. As an application, we prove that every element of an orthocomplete homogeneous atomic effect algebra has a unique basic decomposition into a sum of a sharp element and unsharp multiples of atoms. Further, we characterize homogeneity by the set of all sharp elements in orthocomplete atomic effect algebras.     

Algebraic Properties of Cores of Generalized Neurofunctions

This paper considers generalized neural elements and identifies the conditions for implementation of functions of algebra of logic from these elements. We introduce the concept of a modified core of Boolean functions with respect to the system of characters of a group on which functions of algebra of logic are given. The criteria of belonging these functions to the class of generalized neurofunctions are provided. The algebraic structure of cores of Boolean neurofunctions is studied. On the basis of properties of tolerance matrices, a number of necessary conditions for implementation of Boolean functions by one generalized neural element are obtained. The obtained results allow to develop efficient methods of synthesis of integer-valued generalized neural elements with a large number of inputs that can be successfully applied in solving information compression and transmission problems, as well as discrete signal recognition problems.     

Fuzzy logic as a logic of the expressive strength of information

We develop alternative semantics for Łukasiewicz logic and for cancellative hoop logic according to the following idea. We formalize statements reflecting an inexact knowledge of certain (sharp) properties; we assume that all what can be known about a property is its expressive strength. To this end, we consider a Boolean algebra endowed with an automorphism group or, alternatively, with a measure. The Boolean algebra is meant to model a collection of properties; and the additional structure is used to identify pairs of properties which, although possibly distinct, are equally strong. Propositions are defined as subsets of the algebra containing with any element also those identified with it in this way. We show that then, the set of all propositions carries the structure of an MV-algebra or of a cancellative hoop.     

0-homogeneous effect algebras

In the present paper, we introduce a proper superclass of homogeneous effect algebras. We call this superclass as 0-homogeneous effect algebras. We prove that in every 0-homogeneous effect algebra, the set of all sharp elements forms a subalgebra. Every chain-complete 0-homogeneous effect algebra is homogeneous.     

To the problem of realizability of Boolean functions by circuits in a basis of unreliable functional elements

Maximal extensions of Post classes containing 0, 1, and x in the algebra of partially unreliable Boolean functions are described. Based on these extensions, criteria of expressibility of Boolean functions by circuits in a basis of partially unreliable elements are proved.     

The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states

We study remarkable sub-lattice effect algebras of Archimedean atomic lattice effect algebras E, namely their blocks M, centers C(E), compatibility centers B(E) and sets of all sharp elements S(E) of E. We show that in every such effect algebra E, every atomic block M and the set S(E) are bifull sub-lattice effect algebras of E. Consequently, if E is moreover sharply dominating then every atomic block M is again sharply dominating and the basic decompositions of elements (BDE of x) in E and in M coincide. Thus in the compatibility center B(E) of E, nonzero elements are dominated by central elements and their basic decompositions coincide with those in all atomic blocks and in E. Some further details which may be helpful under answers about the existence and properties of states are shown. Namely, we prove the existence of an (o)-continuous state on every sharply dominating Archimedean atomic lattice effect algebra E with B(E)\not = C(E).B(E)\not =C(E). Moreover, for compactly generated Archimedean lattice effect algebras the equivalence of (o)-continuity of states with their complete additivity is proved. Further, we prove “State smearing theorem” for these lattice effect algebras.     

Bounded algebra and current-mode digital circuits

This paper proposes two bounded arithmetic operations,which are easily realized with current signals.Based on these two operations,a bounded algebra system suitable for describing current-mode digital circuits is developed and its relationship with the Boolean algebra,which is suitable for representing voltagemode digital circuits,is investigated.Design procedure for current-mode circuits using the proposed algebra system is demonstrated on a number of common circuit elements which are used to realize arithmetic operations,such as adders and multipliers.     

Developing the Methods of Reliability Estimation for Structurally Complex Systems of Multistate Elements

For systems of multistate elements, the problem of developing Boolean reliability models was considered on the basis of multivalued logic, cortege algebra, algebra of groups of incompatible events, and classical logic-and-probabilistic method. Development of Boolean reliability models based on the multivalued logic was shown to be unadvisable. Numerical examples demonstrating applicability of the logic-and-probabilistic methods and their new capabilities were presented. Prospects of formulating a set of different problems that are tackled by the same logic-and-probabilistic methods with the aim of developing methods for calculation of the operational effectiveness of various levels of operability were emphasized. The possibility of constructing models for continual calculations on the basis of logic-and-probabilistic methods was demonstrated.     

Bounded commutative residuated ℓ-monoids with general comparability and states

Bounded commutative Rℓ-monoids are a generalization of MV-algebras as well as of BL-algebras. For such monoids the authors in [DvRa] introduced states, analogues of probability measures. We study Boolean elements and introduce the general comparability property. It entails that the monoids with the property are BL-algebras, and extremal states on Boolean elements can be uniquely extended to extremal states on the monoids. Moreover, the hull-kernel topology of maximal filters is totally disconnected.     

Sub-effect algebras and Boolean sub-effect algebras

 We show that Boolean effect algebras may have proper sub-effect algebras and conversely. Properties of lattice effect algebras with two blocks are shown. One condition of the completness of effect algebras is given. We also show that a lattice effect algebra associated to an orthomodular lattice can be embedded into a complete effect algebra iff the orthomodular lattice can be embedded into a complete orthomodular lattice.     

Extension of belief functions to infinite-valued events

We generalise belief functions to many-valued events which are represented by elements of Lindenbaum algebra of infinite-valued ?ukasiewicz propositional logic. Our approach is based on mass assignments used in the Dempster–Shafer theory of evidence. A generalised belief function is totally monotone and it has Choquet integral representation with respect to a unique belief measure on Boolean events.     

由可精确测量元控制的弱可换的伪效应代数

主要研究了由可精确测量元控制的弱可换的伪效应代数中可精确测量元。证明了可精确测量元控制的弱可换的伪效应代数中可精确测量元是弱可换的伪正交代数代数。讨论了弱可换的伪效应代数与BZ-偏序集之间的关系。讨论了弱可换的伪效应代数商代数中可精确测量元与正规Riesz理想之间的关系。     

BL*系统的一种扩张及相关性质

对BR₀代数,Boole 代数以及BL*系统,命题演算形式系统L进行了研究。首先讨论了BR₀代数与Boole 代数间的相互关系,随后在BL*系统中分别添加公理模式(A →B) →¬A ∨ B或(A →(B → C)) →(A ∧ B → C)得到BL*系统的两种扩张,并证明了BL*系统的这两种扩张与命题演算形式系统L之间是等价的。     

有限布尔代数上的线性自动机

自动机理论是计算机科学理论的重要组成部分。论文研究了布尔代数上的线性自动机,证明了任意一个线性有限自动机是函数布尔代数上的一个内动机。定出了有限布尔代数上的一类可逆线性内动机,给出并证明了有限布尔代数上内动机图型为下向森林的充分必要条件,给出了树型内动机中每一层节点数的计算公式,进而证明了有限布尔代数上的非可逆内动机图型为恰等叉支下向树的充分必要条件。     

异步时序电路反馈稳定性判据

本文提出了"含相开关代数"的概念,证明了它与布尔代数是同构的,从而使布尔代数可 用于设计时序电路.文中利用含相开关代数这一概念推导出异步时序电路反馈稳定性判据, 从而能方便地找出全部不稳定转换.     

优势关系下基于浓缩布尔矩阵的属性约简方法

在优势关系粗糙集方法(DRSA)的框架下,针对不协调的目标信息系统求属性约简。基于优势矩阵的方法是最常用的一类约简方法,但矩阵中不是所有的元素都有效。浓缩优势矩阵只保留对求约简有用的最小属性集,因而可以明显降低约简过程中的计算量。进一步地,浓缩布尔矩阵通过布尔代数的形式有效地弥补了优势矩阵生成效率低的缺点。文中将等价关系上的浓缩布尔矩阵属性约简方法扩展到优势关系上,针对优势矩阵提出了浓缩布尔矩阵的概念,建立了相应的高效约简方法,使效率得到明显提高。最后采用9组UCI数据进行实验,结果验证了所提方法的有效性。     

Region–based theory of discrete spaces: A proximity approach

We introduce Boolean proximity algebras as a generalization of Efremovič proximities which are suitable in reasoning about discrete regions. Following Stone’s representation theorem for Boolean algebras, it is shown that each such algebra is isomorphic to a substructure of a complete and atomic Boolean proximity algebra. Co-operation was supported by EC COST Action 274 “Theory and Applications of Relational Structures as Knowledge Instruments” (TARSKI), , and NATO Collaborative Linkage Grant PST.CLG 977641.     

On lower and upper intension order relations by different cover concepts

In this paper, the concept of intension is used to introduce two types of ordering relations based on information that generates a cover for the universal set. These types of ordering relations are distinct from the well-known ordering relation based on set inclusion. For these ordering relations, we consider the algebraic structures that arise in various types of covers. We show that in the case of a representative cover, the algebraic structure resulting from the lower intension inclusion is a double Stone algebra, while in the case of a reduced cover, it is a Boolean algebra. In addition, the algebraic structure resulting from the upper intension inclusion in the case of a representative cover is a Boolean algebra, and in the case of a reduced cover, the two Boolean algebraic structures from lower and upper intension inclusions are isomorphic.