模糊逻辑形式系统的若干完备性定理
发布时间：2019-03-09 13:36
【摘要】：近年来，模糊控制技术在应用方面取得举世瞩目的成功．然而，作为 其核心的模糊推理，正如文[6]所言，在数学基础上却并非无懈可击，至今都没有 归入严密的逻辑系统中．所以，以研究模糊推理的数学基础为核心的模糊逻辑， 作为一个全新的数学领域，引起了世界上许多著名学者的关注，并且取得一系列 重要的研究成果． 1997年，工国俊教授在文[4]构造了模糊命题演算的形式系统L~*，该系统的 否定算子和析取算子都是标准的模糊算子，蕴涵算子被称为R_0-组涵算子，分 别定义如下： 随后，王国俊教授又从事于该系统的语义方面和应用方面的研究并产生了一 系列重要的研究成果，，首先，文[5]建立了L~*系统的语义系统（即修正的Kleene 系统）中的广义重言式理论；接着，文[7]提出了旨在为新型模糊控制器的研制 提供一种可能的理论依据的模糊推理的全蕴涵三I算法；随后，文[8]给出了L~* 系统的语义紧致性定理，文[16]给出了基于L~*系统的区间值刚推理方法，等 等． 最近，裴道武博士用代数方法证明了L~*关于（？）-语义的完备性定理[3].随 后，本文作者和裴道武博士又独立地完成了(?)-语义的公理化问题[45,49].在 此基础上，本掳四蛐造了L”所对应的谓词演算系统KZ，井在第五章证明了 它关于 W一解释的完备性定理［15］．迄今为止，在众多的非 Pavelka型模糊逻辑 形式系统中，只有G砧el系统和KZ系统具有完备性．另外，在KZ中，析取算 子和否定算子都是标准的模糊算子，而G5del系统中的这两个算子为了保证完备 性只能取弱算子（基本上退化为二值逻辑中相应算子），这在很大程度上丧失了模 糊性．从这一角度看，在理论上基于R卜蕴涵算子的逻辑系统KZ优越于基于 G5del蕴涵算子的G5del系统，那么可以设想在实践中，基于Rr蕴涵算子的模 糊推理模型应该优越于基干G5clel蕴涵算子的模型． KZ的完备性定理有一个很有意思的椎论．为了说明这一点，得先介绍一下 三1算法．对于最基本的模糊推理规则，即已知p－－＋4，且给定；”求。“，这里 f个”是集合X上的模糊集，。，，h，”是集合Y’上的模糊集．R。型三1算法的解 由下式确定 h·丫6］中 4．4厂）： 〔1川二＊。马｛丫x）八尸山叫x）·利川》，这里，马二扛E川（／（x》’＜ RO卜（0），y，（y》｝口由辽 中4．4．12，当厂是正规FuZZy集时贝若一＝fx．则＊·二乙·宜． 这样，在KZ中可以用狭义三1算法准p—p”的情形）代替N 规则，从而KZ 完备性的意义是：三1算法语义上推得的重言式集和人Z语构上的定理集一致． 这是 Zadeh等人的 CRI算法所无法比拟的．它为三 1算法进入模糊控制的实践 领域奠定了坚实的理论基础． 在本文的第二章和第三章中，主要介绍c，q的完备性的证明［46卜它们 都是通过Henkin方法完成的．这一方法最初应用于经典的二值系统完备性的证 明，要在模糊系统中使用它，首先需要对它进行彻底的改造．这也是本文的主要 贡献之一． 2 当然，要使用Henkin方法，还需要发展L”的一系列性质．腊研究结果在 本赠一邱绍，主要包括．＊）发现并证明了该系统的广义演绎定理，它也应 该是c”的基本定理之一；仰通过一台计算机进行了长达半小时左右的运行， 发现该系统中析取连接词的不独立性，这也反映了Rr算子具有很强的表达能 力；间发现该系统中交推理规则可以跑添加公理而去掉，从而获得该系统的 等价系统；（4）发现并证明了该系统中一些重要的推理规则（比如命题1．3．3），它 们对证明完备性至关重要．
[Abstract]:In recent years, the application of fuzzy control technology has been successful in the world. however, by way of example, The core of the fuzzy inference, as described in[6], is not based on a mathematical basis No, no, no, no. in a close-knit logic system. So, The fuzzy logic of the core of the mathematical basis of the fuzzy inference is studied as a whole The new field of mathematics, cited It has attracted the attention of many famous scholars in the world, and has made a series of important research In 1997, Professor Guojun constructed a fuzzy proposition in[4]. the formal system of calculation L~* And the negative operator and the extraction operator of the system are both standard modes. The operator of the paste, the implication operator is called R _ 0-group culvert operator, and the sub-definition is as follows: Then, the king of the kingdom of the kingdom of Research and production of the semantic aspects and applications of the system A series of important research results have been born. First, the L ~ * system is established in the paper[5] the semantic system (i.e., the modified k generalized re-expression in the leene system The theory of Next, the purpose of this paper is to develop a new type of fuzzy controller. provide a kind of can A full-implicative trii algorithm for fuzzy inference with a theoretical basis [8] In the following, L ~ * is given in[8] The semantic compactness theorem of the system, which is based on the L ~ * system, is given in the paper[16]. The method of reasoning, etc. Recently, Dr. Pei-wu's algebraic formula The method proves that L ~ * On the Completeness Theorem of (?)-Semantics[3]. After that, the author and Dr. Pei-wu independently completed the (?)-semantic problem[45,49]. ]. On that basis, the captives made L. "The corresponding predicate calculus system" KZ, well in the fifth chapter, proves the completeness theorem of the W-I interpretation. [15]. So far, in many non-Pavelka type fuzzy logic form in that system, only the G-anvil el system and the KZ system are complete. in addition, In KZ, both the disjunctor and the negative operator are the standard fuzzy operators. and these two operators in the g5del system are The guarantee completeness can only take the weak operator (basically degenerate into the corresponding calculation in the binary logic) In this view, the logic system based on the R-b implication operator in theory is largely lost. KZ is superior to base on G5del implication operator G5del system, it can be envisaged that in practice, the model based on the rr implication operator The paste-based reasoning model should be superior to the model of the G5clel implication operator. It's interesting The theory of the intervertebral disc. To illustrate this, we should first introduce the three-1 algorithm. for the most basic fuzzy inference rules, known as p--+ 4, and given;
[Abstract]:In recent years, the application of fuzzy control technology has been successful in the world. however, by way of example, The core of the fuzzy inference, as described in[6], is not based on a mathematical basis No, no, no, no. in a close-knit logic system. So, The fuzzy logic of the core of the mathematical basis of the fuzzy inference is studied as a whole The new field of mathematics, cited It has attracted the attention of many famous scholars in the world, and has made a series of important research In 1997, Professor Guojun constructed a fuzzy proposition in[4]. the formal system of calculation L~* And the negative operator and the extraction operator of the system are both standard modes. The operator of the paste, the implication operator is called R _ 0-group culvert operator, and the sub-definition is as follows: Then, the king of the kingdom of the kingdom of Research and production of the semantic aspects and applications of the system A series of important research results have been born. First, the L ~ * system is established in the paper[5] the semantic system (i.e., the modified k generalized re-expression in the leene system The theory of Next, the purpose of this paper is to develop a new type of fuzzy controller. provide a kind of can A full-implicative trii algorithm for fuzzy inference with a theoretical basis [8] In the following, L ~ * is given in[8] The semantic compactness theorem of the system, which is based on the L ~ * system, is given in the paper[16]. The method of reasoning, etc. Recently, Dr. Pei-wu's algebraic formula The method proves that L ~ * On the Completeness Theorem of (?)-Semantics[3]. After that, the author and Dr. Pei-wu independently completed the (?)-semantic problem[45,49]. ]. On that basis, the captives made L. "The corresponding predicate calculus system" KZ, well in the fifth chapter, proves the completeness theorem of the W-I interpretation. [15]. So far, in many non-Pavelka type fuzzy logic form in that system, only the G-anvil el system and the KZ system are complete. in addition, In KZ, both the disjunctor and the negative operator are the standard fuzzy operators. and these two operators in the g5del system are The guarantee completeness can only take the weak operator (basically degenerate into the corresponding calculation in the binary logic) In this view, the logic system based on the R-b implication operator in theory is largely lost. KZ is superior to base on G5del implication operator G5del system, it can be envisaged that in practice, the model based on the rr implication operator The paste-based reasoning model should be superior to the model of the G5clel implication operator. It's interesting The theory of the intervertebral disc. To illustrate this, we should first introduce the three-1 algorithm. for the most basic fuzzy inference rules, known as p--+ 4, and given;
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