A multi-level conceptual data reduction approach based on the Lukasiewicz implication

Starting from fuzzy binary data represented as tables in the fuzzy relational database, in this paper, we use fuzzy formal concept analysis to reduce the tables size to only keep the minimal rows in each table, without losing knowledge (i.e., association rules extracted from reduced databases are identical at given precision level). More specifically, we develop a fuzzy extension of a previously proposed algorithm for crisp data reduction without loss of knowledge. The fuzzy Galois connection based on the Lukasiewicz implication is mainly used in the definition of the closure operator according to a precision level, which makes data reduction sensitive to the variation of this precision level.     

Extension of domains of states

We deal with categorical aspects of the extensions of generalized probability measures. In particular, we study various domains of fuzzy sets, describe the relationships between σ-fields of crisp sets and generated Łukasiewicz tribes of measurable functions, and mention some probabilistic aspects. D-posets and sequential continuity play an important role. This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0071-06, Protocol for Scientific and Technological Collaboration between the Republic of Italy and Slovak Republic, Project No. 15: Algebraic and Logical Systems of Soft Computing, and VEGA 2/6088/26.     

不确定性系统的统一性

本文综述不确定性系统的统一性.首先揭示Fuzzy系统的概率论意义,指出Fuzzy,系统中常用的清晰化方法,即重心法是合理的且在平均平方意义下是最优的方法.基于不同的Fuzzy蕴涵算予,一给出几种典型的概率分布,如Zadeh分布、Mamdani分布、Lukasiewicz分布等,它们充当Fuzzy系统的"系统内核"作用.此外,根据Fuzzy系统概率分布的一些性质,论证了由Zadeh提出的构造Fuzzy系统的CRI算法是基本合理的且有效的.此外还刻画了均匀概率分布在.Fuzzy 系统中的特殊作用.随后揭示了随机系统的 Fuzzy 推理意义.首先,相对于不确定性系统,给出了随机系统的定义,它视为对一个不确定系统的逼近.然后指出,对于任意给定的一个随机系统,总能将它转化为一组 Fuzzy 推理规则,由此可构造一个Fuzzy 系统,并且证明了这样构造的 Fuzzyr 系统能逼近给定的随机系统到指定的精度.还讨论了Fuzzy 系统与随机系统转换中的还原性.最后概述了不确定性系统的统一性.     

Lukasiewicz命题逻辑系统中真度的等价定义及相关性质

对Lukasiewicz命题逻辑系统中的公式真度理论进行了研究.首先,给出了Lukasiewiczn值命题逻辑系统中一个更为直观的真度定义的等价形式;其次,利用真度定义的等价形式简化了连接Lukasiewiczn值命题逻辑系统和Lukasiewicz连续值命题逻辑系统中真度理论的极限定理的证明;第三,得到了真度性质:在Lukasiewicz逻辑系统中,把命题公式中的原子命题与该原子命题的否定互换,公式的真度不变;第四,讨论了真度与推理规则之间的关系,给出了Lukasiewicz命题逻辑系统中真度与MP规则的精确关系式以及关于真度并推理规则的结果.     

Fuzzifying拓扑空间中的强半分离性

在[Fuzzifying]拓扑空间中，运用连续值逻辑[LΝ1]（赋值格为[Lukasiewicz]单位区间）语义的方法，以强半开集、强半邻域、强半闭包和强半内部等概念为工具引入了[SPT0、][SPT1、][SPT2、][SPT3、][SPT4]强半分离公理，并且给出了五种强半分离公理各自的等价刻画，讨论了它们彼此之间的关系。这套分离公理本身是协调的，可以作为对已有分离公理一种很好的补充。     

Lukasiewicz命题逻辑中命题的Choquet积分真度理论

将已有的不确定性测度概念引入到了Lukasiewicz命题逻辑中的全体赋值之集上,然后利用McNaughton函数关于该不确定性测度的Choquet积分定义了命题的Choquet积分真度概念.证明了当赋值空间上的不确定性测度满足有限可加性时Choquet积分真度函数就具有良好性质,由此可诱导出命题集上的一个伪距离,进而可建立逻辑度量空间并展开程度化推理,特别是证明了当赋值空间上的不确定性测度取为Borel概率测度时Choquet积分真度函数就退化为概率计量逻辑中的Borel概率真度函数.本文是已有命题逻辑概率计量化工作的继续与深入,为表示逻辑命题间不确定性的非线性关系提供了一种推理框架.     

Digital watermarking in coding/decoding processes with fuzzy relation equations

By normalizing the values of its pixels with respect to the length of the used scale, a gray image can be interpreted as a fuzzy relation R which is divided in submatrices (possibly square) called blocks. Every block R_B is compressed to a block G_B, which in turn is decompressed to a block D_B (unsigned) ⩾R_B. Both G_B and D_B are obtained via fuzzy relation equations with continuous triangular norms in which fuzzy sets with Gaussian membership functions are used as coders. The blocks D_B are recomposed in order to give a fuzzy relation D. We use the Lukasiewicz t-norm and a watermark (matrix) is embedded in every G_B with the LSBM (Least Significant Bit Modification) algorithm by obtaining a block , decompressed to a block (signed). Both and are obtained by using the same fuzzy relation equations. The blocks are recomposed by obtaining the fuzzy relation (signed). By evaluating the quality of the reconstructed images via the PSNR (Peak Signal to Noise Ratio) with respect to the original image R, we show that the signed image is very similar to the unsigned image D for low values of the compression rate.     

连续值Luk系统中公式的Γ-真度及其性质

对Lukasiewicz系统中的真度理论进行了进一步的研究。将n值Lukasiewicz系统中公式相对于有限理论的ΣΓ-真度理论与连续值Lukasiewicz系统中公式的积分真度相结合,在连续值Lukasiewicz系统中引入了公式相对于有限理论的Γ-真度理论,讨论了其中的主要性质;在公式集F(S)上引入了任意两公式相对于有限理论的Γ-伪距离,从而拓宽了真度理论的应用范围。