模糊邏輯形式系統(tǒng)的若干完備性定理
發(fā)布時(shí)間：2019-03-09 13:36
【摘要】：近年來，模糊控制技術(shù)在應(yīng)用方面取得舉世矚目的成功．然而，作為 其核心的模糊推理，正如文[6]所言，在數(shù)學(xué)基礎(chǔ)上卻并非無懈可擊，至今都沒有 歸入嚴(yán)密的邏輯系統(tǒng)中．所以，以研究模糊推理的數(shù)學(xué)基礎(chǔ)為核心的模糊邏輯， 作為一個(gè)全新的數(shù)學(xué)領(lǐng)域，引起了世界上許多著名學(xué)者的關(guān)注，并且取得一系列 重要的研究成果． 1997年，工國俊教授在文[4]構(gòu)造了模糊命題演算的形式系統(tǒng)L~*，該系統(tǒng)的 否定算子和析取算子都是標(biāo)準(zhǔn)的模糊算子，蘊(yùn)涵算子被稱為R_0-組涵算子，分 別定義如下： 隨后，王國俊教授又從事于該系統(tǒng)的語義方面和應(yīng)用方面的研究并產(chǎn)生了一 系列重要的研究成果，，首先，文[5]建立了L~*系統(tǒng)的語義系統(tǒng)（即修正的Kleene 系統(tǒng)）中的廣義重言式理論；接著，文[7]提出了旨在為新型模糊控制器的研制 提供一種可能的理論依據(jù)的模糊推理的全蘊(yùn)涵三I算法；隨后，文[8]給出了L~* 系統(tǒng)的語義緊致性定理，文[16]給出了基于L~*系統(tǒng)的區(qū)間值剛推理方法，等 等． 最近，裴道武博士用代數(shù)方法證明了L~*關(guān)于（？）-語義的完備性定理[3].隨 后，本文作者和裴道武博士又獨(dú)立地完成了(?)-語義的公理化問題[45,49].在 此基礎(chǔ)上，本擄四蛐造了L”所對(duì)應(yīng)的謂詞演算系統(tǒng)KZ，井在第五章證明了 它關(guān)于 W一解釋的完備性定理［15］．迄今為止，在眾多的非 Pavelka型模糊邏輯 形式系統(tǒng)中，只有G砧el系統(tǒng)和KZ系統(tǒng)具有完備性．另外，在KZ中，析取算 子和否定算子都是標(biāo)準(zhǔn)的模糊算子，而G5del系統(tǒng)中的這兩個(gè)算子為了保證完備 性只能取弱算子（基本上退化為二值邏輯中相應(yīng)算子），這在很大程度上喪失了模 糊性．從這一角度看，在理論上基于R卜蘊(yùn)涵算子的邏輯系統(tǒng)KZ優(yōu)越于基于 G5del蘊(yùn)涵算子的G5del系統(tǒng)，那么可以設(shè)想在實(shí)踐中，基于Rr蘊(yùn)涵算子的模 糊推理模型應(yīng)該優(yōu)越于基干G5clel蘊(yùn)涵算子的模型． KZ的完備性定理有一個(gè)很有意思的椎論．為了說明這一點(diǎn)，得先介紹一下 三1算法．對(duì)于最基本的模糊推理規(guī)則，即已知p－－＋4，且給定；”求�！�，這里 f個(gè)”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 由下式確定 h·丫6］中 4．4廠）： 〔1川二＊。馬｛丫x）八尸山叫x）·利川》，這里，馬二扛E川（／（x》’＜ RO卜（0），y，（y》｝口由遼 中4．4．12，當(dāng)廠是正規(guī)FuZZy集時(shí)貝若一＝fx．則＊·二乙·宜． 這樣，在KZ中可以用狹義三1算法準(zhǔn)p—p”的情形）代替N 規(guī)則，從而KZ 完備性的意義是：三1算法語義上推得的重言式集和人Z語構(gòu)上的定理集一致． 這是 Zadeh等人的 CRI算法所無法比擬的．它為三 1算法進(jìn)入模糊控制的實(shí)踐 領(lǐng)域奠定了堅(jiān)實(shí)的理論基礎(chǔ)． 在本文的第二章和第三章中，主要介紹c，q的完備性的證明［46卜它們 都是通過Henkin方法完成的．這一方法最初應(yīng)用于經(jīng)典的二值系統(tǒng)完備性的證 明，要在模糊系統(tǒng)中使用它，首先需要對(duì)它進(jìn)行徹底的改造．這也是本文的主要 貢獻(xiàn)之一． 2 當(dāng)然，要使用Henkin方法，還需要發(fā)展L”的一系列性質(zhì)．臘研究結(jié)果在 本贈(zèng)一邱紹，主要包括．＊）發(fā)現(xiàn)并證明了該系統(tǒng)的廣義演繹定理，它也應(yīng) 該是c”的基本定理之一；仰通過一臺(tái)計(jì)算機(jī)進(jìn)行了長達(dá)半小時(shí)左右的運(yùn)行， 發(fā)現(xiàn)該系統(tǒng)中析取連接詞的不獨(dú)立性，這也反映了Rr算子具有很強(qiáng)的表達(dá)能 力；間發(fā)現(xiàn)該系統(tǒng)中交推理規(guī)則可以跑添加公理而去掉，從而獲得該系統(tǒng)的 等價(jià)系統(tǒng)；（4）發(fā)現(xiàn)并證明了該系統(tǒng)中一些重要的推理規(guī)則（比如命題1．3．3），它 們對(duì)證明完備性至關(guān)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;
"Pl Disease. ", here f涓
本文編號(hào)：2437504
本文鏈接：http://sikaile.net/shekelunwen/ljx/2437504.html
最近更新
教材專著
