n值S-MTL命题逻辑系统中的近似推理理论及三I算法的还原性

发布时间：2018-02-27 12:47

本文关键词： MTL命题逻辑系统真度伪距离三I算法还原性　出处：《兰州理工大学》2011年硕士论文　论文类型：学位论文

【摘要】：基于左连续三角模的MTL逻辑,也是基于正则蕴涵算子的逻辑,其中左连续三角模作为逻辑强合取算子的语义对应,与其伴随的正则蕴涵算子作为逻辑蕴涵算子的语义对应.MTL逻辑作为模糊逻辑,具有很多良好的性质,同时,基于三I原则的模糊推理算法,其统一形式也是基于正则蕴涵算子给出的,因此三I算法和MTL逻辑之间存在着天然的联系.三I原则和算法可以看做是模糊推理的一种数值实现,但是在这种数值实现和逻辑的形式化推理之间,还存在着一定的距离,如果能为三I原则和算法提供一种逻辑上的解释,那将会为模糊推理找到合适的逻辑基础. 为了消除形式化的逻辑推理和数值计算之间的割裂,本世纪初,王国俊教授基于均匀概率的思想在经典二值命题逻辑中引入了命题的真度概念,提出了计量逻辑理论,建立了一套近似推理模式之后,国内外同行展开了广泛的研究.相似的结论被推广到n值Lukasiewicz命题逻辑系统和n值R0命题逻辑系统中.但是所有以上的结论都是建立在均匀概率测度空间上,由于实际应用中往往会对某些命题有所侧重,所以针对非均匀分布情形进行研究会更适合于应用.本文在n值MTL命题逻辑系统的统一框架中,基于一般的概率测度,建立了真度的统一理论,给出了这种统一框架下公式真度的积分表示形式;证明了真度推理规则在所有的n值S-MTL命题逻辑系统中成立,定义了一种伪距离,为在n值MTL命题逻辑系统中建立近似推理理论给出了一种可能的框架.考虑到三I算法的统一形式也是基于正则蕴涵算子的,而还原性是判断蕴涵算子与模糊推理方法配合效果的一个重要指标,只有蕴涵算子与推理方法搭配适当,才能使模糊推理有一个好的效果.因此本文还对三I算法的还原性进行了讨论. 以下是本文所得到的主要结果: (1)提出了强正则蕴涵算子与S-MTL命题逻辑系统的概念,并且证明了Lukasiewicz蕴涵是最大的强正则蕴涵算子. (2)在n值MTL命题逻辑系统中基于一般的概率测度空间定义了公式的真度,给出了真度的积分表示形式,并在n值S-MTL命题逻辑系统中证明了这种基于一般概率测度的真度满足真度推理规则.基于这种真度建立了S-MTL命题逻辑系统中公式之间的相似度及伪距离理论,进而为n值SMTL系统中建立了一种统一的近似推理机制. (3)对模糊推理三I算法具备还原性的条件进行了研究.当与蕴涵算子相伴随的三角模为连续三角模时,给出并证明了FMP问题三I算法具有还原性的充要条件;当蕴涵算子为连续的正则蕴涵算子时,给出了FMT问题三I算法具有还原性的充要条件;最后,当正则蕴涵算子关于补运算满足对合律时,给出了FMT问题三I算法满足还原性的一个充分条件.
[Abstract]:The MTL logic based on the left continuous triangular module is also the logic based on the regular implication operator, in which the left continuous triangular module is used as the semantic correspondence of the strong combination operator of the logic. With its accompanying regular implication operator as the semantic correspondence of the logical implication operator, the MTL logic as fuzzy logic has many good properties. At the same time, the fuzzy reasoning algorithm based on the three I principle is proposed. The unified form is also based on the regular implication operator, so there is a natural relationship between the three I algorithm and the MTL logic, and the three I principle and algorithm can be regarded as a numerical realization of fuzzy reasoning. However, there is still a certain distance between the numerical realization and the formal reasoning of logic. If we can provide a logical explanation for the three-I principle and algorithm, we will find the appropriate logical basis for fuzzy reasoning. In order to eliminate the separation between formal logic reasoning and numerical calculation, Professor Wang Guojun introduced the concept of truth of proposition in classical binary propositional logic based on the idea of uniform probability at the beginning of this century, and put forward the theory of quantitative logic. After establishing a set of approximate reasoning models, Similar conclusions have been extended to n-valued Lukasiewicz propositional logic systems and n-valued R0 propositional logic systems. Because some propositions are often emphasized in practical applications, it is more suitable to study the case of non-uniform distribution. In this paper, in the unified framework of n-valued MTL propositional logic system, based on the general probability measure, The unified theory of truth degree is established, the integral representation of formula truth degree is given under this unified framework, it is proved that the truth degree reasoning rule holds in all n-valued S-MTL propositional logic systems, and a pseudo distance is defined. This paper presents a possible framework for establishing approximate reasoning theory in n-valued MTL propositional logic systems, considering that the unified form of the triple I algorithm is also based on regular implication operators. Reducibility is an important index to judge the effect of the combination of implication operator and fuzzy reasoning method. Therefore, the reducibility of the triple-I algorithm is also discussed in this paper. The following are the main results obtained in this paper:. 1) the concepts of strongly regular implication operator and S-MTL propositional logic system are proposed, and it is proved that Lukasiewicz implication is the largest strongly regular implication operator. (2) in the n-valued MTL propositional logic system, the true degree of the formula is defined based on the general probability measure space, and the integral representation of the truth degree is given. In the n-valued S-MTL propositional logic system, it is proved that the truth degree based on the general probability measure satisfies the truth degree inference rule. Based on this truth degree, the similarity degree and pseudo-distance theory among the formulas in S-MTL propositional logic system are established. Then a unified approximate reasoning mechanism is established for n-valued SMTL systems. In this paper, we study the condition that the triple-i algorithm of fuzzy reasoning has the reducibility. When the triangular module associated with the implication operator is a continuous triangular module, the sufficient and necessary conditions for the triple-I algorithm of the FMP problem to be reducible are given and proved. When the implication operator is a continuous regular implication operator, a sufficient and necessary condition for the reducibility of the three-I algorithm for the FMT problem is given, and finally, when the regular implication operator satisfies the involutive law about the complement operation, In this paper, a sufficient condition for the triple-I algorithm of FMT problem to satisfy the reducibility is given.
【学位授予单位】：兰州理工大学
【学位级别】：硕士
【学位授予年份】：2011
【分类号】：O141.1

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