格蕴涵代数中的极小素理想及α-理想

在格蕴涵代数中,证明了极小素理想与极小格素理想的等价性,继而给出了极小素理想与零化子的相互表示定理。提出了格蕴涵代数中的[α]-理想概念并给出其若干等价刻画,证明了极小素理想是[α]-理想。证明了全体素[α]-理想之集[Sα(L)]是一个紧的Stone空间,进一步给出[Sα(L)]分别是[T1、][T2]拓扑空间的充要条件。     

m序列频谱免疫度的研究

频谱免疫度是度量周期序列抵抗离散傅里叶频谱攻击的重要指标。周期序列的频谱免疫度越大,抵抗离散傅里叶频谱攻击的能力越强。通过搜索[m]序列的零化子,计算其频谱重量,提出猜想：[n]级[m]序列的零化子最低频谱重量是[n+1],而该[m]序列的补序列恰恰是其一个最低频重零化子。研究了[m]序列零化子及其补序列的性质,从理论上证明了该猜想。分析了[m]序列补序列的零化子性质,得出结论：[n]级[m]序列的频谱免疫度为[n];说明了[m]序列难以抵抗离散傅里叶频谱攻击。     

布尔函数的低次零化子研究

对布尔函数零化子的计数问题进行研究,在布尔函数系数矩阵的基础上给出线性独立零化子的一种新计数方式。提出布尔函数低次零化子概念,并在线性独立零化子新计数方式的基础上找到一种寻找布尔函数低次零化子的方法。对利用布尔函数低次零化子建立低错方程组实施攻击的思想进行了阐述。     

对序列密码算法的改进Cube攻击

Cube攻击是近来年兴起的攻击手段,针对序列密码算法提出改进的Cube攻击。将代数攻击与Cube相结合,把代数攻击作为Cube攻击的基础,以降低Cube攻击的复杂度。在前人研究的基础上对零化子与Cube攻击的结合方式进行扩展,将改进的攻击应用于Lili-128算法上,仅用三维Cube集合在数据复杂度[210]之内即可恢复88比特密钥。     

布尔函数零化子次数的一个错误证明的更正

指出了张文英等人关于零点集包含一个k维子空间的n元布尔函数一定存在[n-k]次零化子的证明,存在明显错误。利用线性空间笛卡儿积的思想,给出了关于上述结论的简洁、严密的逻辑证明。     

伪BL-代数的犹豫模糊滤子

提出了伪BL-代数的犹豫模糊滤子的概念，给出它的几个等价刻画，并研究了它的一些性质。进一步，引入了伪BL-代数的几种犹豫模糊滤子，如犹豫模糊Boolean滤子、犹豫模糊正规滤子、犹豫模糊超滤子和犹豫模糊固执滤子，讨论了它们的一些刻画，给出了其间的一些关系。通过研究伪BL-代数中犹豫模糊Boolean滤子与犹豫模糊正规滤子的关系，解决了伪BL-代数中是否每个Boolean滤子都是正规滤子这样一个开问题。     

正则FI代数的MP滤子与同构基本定理

正则FI-代数是仅基于蕴涵算子在一般集合上建立的逻辑代数,它是[BR0]-代数和BL-代数的基础逻辑代数。基于经典代数的思想和方法,讨论了正则FI-代数的MP滤子与同构基本定理。引入正则FI-代数中MP滤子的概念,并讨论了其基本性质,给出了正则FI-代数中包含任意子集的最小MP滤子的构造方法;讨论了正则FI-代数的MP滤子和同余关系之间的内在联系;给出了正则FI-代数的同构基本定理。     

IMTL-代数的零化算子和⊕理想及其相互关系

主要对IMTL-代数中的零化算子和⊕理想的性质及其相互关系进行了研究。首先,将零化算子的概念引入到IMTL-代数中,在IMTL-代数中分别讨论了单点零化算子,多点零化算子的性质;其次,将⊕理想的概念引入IMTL-代数中,并对⊕理想的基本性质进行了讨论;最后,给出了IMTL-代数上零化子和⊕理想之间的相互依赖的关系和相互决定的方法。     

一类二阶代数免疫函数的结构

对一阶代数免疫函数的结构进行了刻画,在此基础上讨论了一类二阶代数免疫布尔函数的构造问题,给出了两个[n]元一阶代数免疫函数通过级联得到[n]+1元二阶代数免疫函数的充分条件,并且给出了它们的若干结构和几个判断二阶代数免疫函数的判别准则。     

BL-代数的n-重模糊蕴涵滤子

引入BL-代数的n-重模糊蕴涵滤子概念,讨论了n-重模糊蕴涵滤子的性质,给出若干等价刻画,给出了验证n-重模糊蕴涵滤子的一个简单条件。用n-重模糊蕴涵滤子对n-重蕴涵BL-代数进行了刻画,证明了一个模糊滤子是一个n-重模糊蕴涵滤子当且仅当对应的商代数是一个n-重蕴涵BL-代数。     

Sheaf representations of BL-algebras

In this paper we present a survey on sheaf representations of BL-algebras, based on our PhD thesis [36]. We define sheaf spaces and sheaf representations of BL-algebras, we study completely regular and compact sheaf spaces of BL-algebras, compact representations of BL-algebras and we develop a Gelfand theory for BL-algebras. Thus, we prove that the category of nontrivial BL-algebras is equivalent to the category of compact local sheaf spaces of BL-algebras. The last section of our paper is a contribution to the representation theory of BL-algebras by (weak) Boolean products. We characterize the (weak) Boolean products of BL-chains, the weak Boolean products of local BL-algebras and the weak Boolean products of perfect BL-algebras.     

Commuting functions with annihilator elements

Commuting is an important property in any two-step information aggregation process where the results should not depend on the order in which the single steps are carried out. In case of aggregation functions with the neutral elements, Saminger et al. (IEEEs Transactions on Fuzzy Systems, 2007, 15, 1032–1045) already reduced characterization of commuting functions to resolving the functional equation of a unary function. In this paper, we will investigate the commuting functions with the annihilator elements, especially, the nullnorms. A characterization of the commuting aggregation functions with the annihilator elements will be obtained in this case.     

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.     

Generating the variety of BL-algebras

This paper collects some results from [AFM] and from [AM]. Our purpose is to illustrate some interesting classes of algebras which generate the whole variety of BL-algebras. In particular, we prove that such variety is generated by its finite members and by the class of finite ordinal sums of Lukasiewicz t-norm algebras. Finally, we characterize the BL-chains which generate the whole variety of BL-algebras.     

零化多项式与仿射空间关系研究

为把流密码的代数攻击问题转化为求解布尔函数的低次数零化多项式问题,讨论布尔函数的性质,介绍{0,1}上矩阵的特殊结构,研究两者间的关系,在此基础上探讨n元布尔函数f的零化多项式次数与f的支撑点集之间的关系,实验结果表明,寻找布尔函数零化多项式等价于在布尔函数的零点集合中寻找最大的仿射空间。     

一类低频重零化子的推导及频谱分析

针对序列密码实现有效的离散傅里叶频谱攻击,前提条件是寻找到序列的低频重乘积关系或低频重零化子。利用周期序列的离散傅里叶变换,得到满足乘积关系序列的一个充要条件,并以此为基础,定义频谱循环差分,推导出一类低频重乘积关系和低频重零化子。同时研究了m序列的频谱性质,给出了m序列的频谱空间快速计算方法以及计算实例。     

Functions represented by BL-algebra formulas: characterization and approximate representation

The paper is a contribution to the study of functional systems of BL-algebras on [0,1] extended by constants from [0,1] and suprema and infima over (infinite) sets of different instances of the same formula. We prove that the class of functions which are represented by thus specified formulas coincides with the class of extensional functions. Moreover, the class of these functions coincides with the class of Lipschitz continuous functions in the case of BL-algebra whose t-norm has a continuous additive generator. We also consider an approximate representation of functions, represented by BL-formulas with a possibly infinite length, by discrete normal forms and suggest an algorithmic approach to its construction. This paper has been partially supported by grant IAA1187301 of the GA AV ČR.     

BL-algebras and effect algebras

Although the notions of a BL-algebra and of an effect algebra arose in rather different contextes, both types of algebras have certain structural properties in common. To clarify their mutual relation, we introduce weak effect algebras, which generalize effect algebras in that the order is no longer necessarily determined by the partial addition. A subclass of the weak effect algebras is shown to be identifiable with the BL-algebras. Moreover, weak D-posets are defined, being based on a partial difference rather than a partial addition. They are equivalent to weak effect algebras. Finally, it is seen to which subclasses of the weak effect algebras certain subclasses of the BL-algebras, namely the MV-, product, and Gödel algebras, correspond.     

LTS中的弱不变量与限制乘积

本文在文献[1,2]的框架下给出了弱不变量的定义,讨论了其性质及与不变量之间的关系。此外,本文还引入了迁移系统的限制乘积概念,并以此为工具研究了弱互模拟和弱不变量之间的相互转化。