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Yanqing Zhang, Martin Fraser, R. Gagliano, A. Kandel (2000)
Granular neural networks for numerical-linguistic data fusion and knowledge discoveryIEEE transactions on neural networks, 11 3
E. Czogala, J. Leski (2001)
On equivalence of approximate reasoning results using different interpretations of fuzzy if-then rulesFuzzy Sets Syst., 117
Guojun Wang, Hao Wang (2001)
Non-fuzzy versions of fuzzy reasoning in classical logicsInf. Sci., 138
Guangwu Meng (1995)
On countably strong fuzzy compact sets in L -fuzzy topological spacesFuzzy Sets and Systems, 72
C. Pappis, N. Karacapilidis (1993)
A comparative assessment of measures of similarity of fuzzy valuesFuzzy Sets and Systems, 56
Meng Guangwu (1993)
Lowen's compactness in L -fuzzy topological spacesFuzzy Sets and Systems, 53
D. Hong, S. Hwang (1994)
A note on the value similarity of fuzzy systems variablesFuzzy Sets and Systems, 66
T. Sudkamp (1993)
Similarity, interpolation, and fuzzy rule construction, 58
Zhang Wen (2004)
Measures of similarity between vague sets
Guojun Wang, Y. Leung (2003)
Integrated semantics and logic metric spacesFuzzy Sets Syst., 136
Mark Last, Yaron Klein, A. Kandel (2001)
Knowledge discovery in time series databasesIEEE transactions on systems, man, and cybernetics. Part B, Cybernetics : a publication of the IEEE Systems, Man, and Cybernetics Society, 31 1
Shyi-Ming Chen, Ming-Shiow Yeh, P.-Y. Hsiao (1995)
A comparison of similarity measures of fuzzy valuesFuzzy Sets and Systems, 72
R. Yager (2004)
On some new classes of implication operators and their role in approximate reasoningInf. Sci., 167
Shyi-Ming Chen (1995)
Measures of similarity between vague setsFuzzy Sets Syst., 74
(2003)
Stratified construction of fuzzy propositional logic
R. Yager, V. Kreinovich (2003)
Universal approximation theorem for uninorm-based fuzzy systems modelingFuzzy Sets Syst., 140
Bai Shi-zhong (1997)
Q -convergence of ideals in fuzzy lattices and its applicationsFuzzy Sets and Systems, 92
W. Kickert (1978)
ANALYSIS OF A FUZZY LOGIC CONTROLLER
李福安 (1992)
DECOMPOSITION OF STEINBERG GROUPSChinese Science Bulletin, 37
M. Ying (2003)
Reasoning about probabilistic sequential programs in a probabilistic logicActa Informatica, 39
H. Lee-Kwang, Yoon-Seon Song, Keon-Myung Lee (1994)
Similarity measure between fuzzy sets and between elementsFuzzy Sets and Systems, 62
(1992)
Fuzzy reasoning under approximate match
(1991)
Value approximation of fuzzy systems, 39
Y. Zheng, Changshui Zhang, Xing Yi (2004)
Mamdanian logic2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542), 2
S. Bai (1997)
Q-convergence of nets and weak separation axioms in fuzzy latticesFuzzy Sets Syst., 88
M. Ying (1994)
A logic for approximate reasoningJournal of Symbolic Logic, 59
Guojun Wang (1999)
On the Logic Foundation of Fuzzy ReasoningInf. Sci., 117
(2009)
171-174 Fuzzy Inf
(2003)
Computing with words and perceptions -a paradigm shift in computing and decision analysis
S. Bai (1997)
Q-convergence of ideals in fuzzy lattices and its applicationsFuzzy Sets Syst., 92
Guo-Jun Wang (1998)
Fuzzy continuous input-output controllers are universal approximatorsFuzzy Sets Syst., 97
C. Pappis (1991)
Value approximation of fuzzy systems variablesFuzzy Sets and Systems, 39
Fuzzy Inf. Eng. (2009) 1: 91-101 DOI 10.1007/s12543-009-0007-z ORIGINAL ARTICLE Ya-lin Zheng · Bing-yuan Cao · Guang Yang · Yong-cheng Bai Received: 18 May 2008/ Revised: 20 December 2008/ Accepted: 19 January 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract It can reﬂect the nature of approximate reasoning and meet more appli- cation expectations to design the approximate reasoning matching schemes and the corresponding algorithms with similarity relationQ instead of equivalence relationR. In this paper, based on similarity relation Q, we introduce type V matching scheme and corresponding approximate reasoning type V Q−algorithm with the given input A and knowledge A → B. Besides, we present completeness of type V and its perfection on knowledge base K inQ−logic C in this paper. Keywords Approximate reasoning · Formula mass · Knowledge mass·Q−logic · Q−algorithm ·Q−completeness 1. Introduction For some applications of approximate reasoning, it is too restrict to use the equiva- lence R in formula set F (S). If an approximate reasoning matching scheme is de- signed with similarity Q in F (S), it may be more reasonable to reﬂect the nature of approximate reasoning and meet the application expectations in many aspects. The diﬀerence is that the intersections of any components of family F (S)/Q = {A |A∈F (S)} Ya-lin Zheng () · Guang Yang Research Center of Intelligent Information Technology, Department of Computer Science and Technology, Faculty of Software, Dongguan University of Science and Technology, Dongguan 523808, P.R.China e-mail: bgzhengyl@dgut.edu.cn Bing-yuan Cao School of Mathematics and Information Sciences, Guangzhou University, Guangzhou Higher Education Mega Center, Guangzhou 510006, P.R.China e-mail: caobingy@163.com Yong-cheng Bai Institute of Information Science, Faculty of Mathematics and Information Science, Shanxi University of Science and Technology, Hanzhong 723001, P.R.China 92 Ya-lin Zheng · Bing-yuan Cao · Guang Yang · Yong-cheng Bai (2009) made of corresponding formula mass A = {B|B∈F (S), AQB} are unnecessarily to be empty in the situation of more general similarityQ. 2. The construction of Q−formula mass andQ−logic ˆ ˜ In classic propositional logic C = (C, C), for any similarity relation Q in formula set F (S), it satisﬁes A →B ⊆A → B Q Q Q for each implication A → B ∈ H , where ∗ ∗ ∗ ∗ A →B = {A → B |A ∈A , B ∈B }, Q Q Q Q (F (S),Q) is called Q−space; ˆ ˜ C = (C, C,Q) is calledQ−logic. In Q−logic C , for each formula A ∈F (S), A is called Q−formula mass with Q Q the kernel of A, or just Q−mass in short. Specially, for every theorem A → B ∈ Ø , A → B is called Q−knowledge mass with the kernel of knowledge A → B. 3. Type V simple approximate reasoning based onQ−logic C Consider the simple approximate reasoning model inQ−logic C A → B , (1) ∗ ∗ where A → B ∈ Ø is called knowledge; A ∈F (S) is called input; B ∈F (S) is called output or approximate reasoning conclusion. If input A is contained in Q−formula mass A with the kernel of A, there is the antecedent of knowledge A → B, that is A ∈A . It means that input A can activate the knowledge A → B. Regard B ∈B as the approximate reasoning conclusion, which is the type V Q−solution of the sim- ple approximate reasoning with regard to input A under knowledge A → B. If input A is not contained in Q−formula mass A with the kernel of A, there is the antecedent of knowledge A → B, that is A A . Q Fuzzy Inf. Eng. (2009) 1: 91-101 93 ∗ ∗ It means that input A can not activate knowledge A → B. And A is not considered in this situation, meaning there is no type VQ−solution with regard to input A under knowledge A → B. This algorithm is named type V Q− algorithm of simple approximate reasoning. Theorem 3.1 In Q−Logic C , considering the simple approximate reasoning with ∗ ∗ regard to input A under knowledge A → B, if B is one of the type V Q−solutions, we have ∗ ∗ A → B ∈A → B . ∗ ∗ It means that A → B is always contained inQ−knowledge massA → B with the kernel of A → B, that is to say, the approximate knowledge we obtained isQ−similar to the standard knowledge A → B. Proof Suppose that the simple approximate reasoning with regard to input A under knowledge A → B has a type V Q−solution B , then ∗ ∗ A ∈A , B ∈B , Q Q therefore ∗ ∗ A → B ∈A →B . Q Q However, according to the construction of Q−logic C , we know A →B ⊆A → B , Q Q Q therefore ∗ ∗ A → B ∈A → B . Obviously, the solution, response and examination domains of the simple approxi- mate reasoning type VQ−algorithm with regard to input A under knowledge A → B are (V ) Dom (A → B) =A , (V ) Codom (A → B)=B , (V ) Exa (A → B) =A → B . Theorem 3.2 In Q−logic C , type V Q−algorithm of simple approximate reason- ing(3.1) is an MP reappearance algorithm. Proof Because similarityQ is reﬂexive, meaning AQA, BQB, we have A∈A , B∈B . Q Q Theorem 3.3 InQ−logic C , simple approximate reasoning type VQ−algorithm is classic reasoning algorithm, if and only if similarity Q in C is the identity relation on formula set F (S). 94 Ya-lin Zheng · Bing-yuan Cao · Guang Yang · Yong-cheng Bai (2009) Proof In Q−logic C , simple approximate reasoning type V Q−algorithm is a clas- sic reasoning algorithm, if and only if “for each knowledge A → B, it can be activated ∗ ∗ by input A , if and only if A = A”, if and only if we have A = {A} for every formula A ∈F (S), and if and only if similarity relation Q of F (S)isan identity relation. 4. Type V multiple approximate reasoning based onQ−logic C Consider a multiple approximate reasoning model inQ−logic C A → B 1 1 A → B 2 2 ... ... ... , (2) A → B n n where K = {A → B|i = 1, 2,··· , n}⊆ Ø đis called knowledge base, A ∈F (S)is i i called an input and B ∈F (S) is called an output or approximate reasoning conclu- sion. If there exists an i∈{1, 2,··· , n} satisfying A ∈A , i Q we say input A can activate knowledge A → B in knowledge base K . i i If there exists an i∈{1, 2,··· , n} satisfying A A , i Q we say input A cannot activate knowledge A → B in knowledge base K . i i Suppose there exists a non-empty subset E of {1, 2,··· , n}, for each i ∈ E, in- put A can activate knowledge A → B in knowledge base K ; while for any j ∈ i i {1, 2,··· , n}− E, input A can not activate knowledge A → B in knowledge baseK . j j Then let B ∈B i Q and ∗ ∗ B = B , i∈E which satisfy B ∈ B , i Q i∈E Fuzzy Inf. Eng. (2009) 1: 91-101 95 ∗ ∗ ∗ where B is a certain aggregation of {B |i ∈ E}. B , the conclusion of approx- i i i∈E imate reasoning, is called multiple approximate reasoning type V Q−solution with regard to input A under knowledge baseK = {A → B|i = 1, 2,··· , n}. i i Suppose input A can not activate any knowledge A → B in knowledge base i i K = {A → B|i = 1, 2,··· , n}, which means for each i∈{1, 2,··· , n} we have i i A A . i Q Then, we say input A can not activate knowledge baseK = {A → B|i = 1, 2,··· , n}. i i In this case, we do not consider input A , that is, there dose not exist multiple approx- imate reasoning type V Q−solution with regard to input A under knowledge base K = {A → B|i = 1, 2,··· , n}. i i This algorithm is called type VQ−algorithmof multiple approximate reasoning. Theorem 4.1 InQ−logic C , there exists type VQ−solution to the multiple approx- imate reasoning with regard to input A under knowledge base K = {A → B|i = i i 1, 2,··· , n}, if and only if there exists at least an i∈{1, 2,··· , n} satisfying A ∈A , i Q that is, input A can activate at least one knowledge A → B in knowledge base K . i i Theorem 4.2 In Q−logic C , there exists type V Q−solution B of the multiple approximate reasoning with regard to input A under knowledge base K = {A → B|i = 1, 2,··· , n}, if and only if A ∈ A . i Q i=1 Theorem 4.3 InQ−logic C , if there exists the multiple approximate reasoning type ∗ ∗ V Q−solution B with regard to input A under knowledge base K = {A → B|i = i i 1, 2,··· , n}, there exists at least an i∈{1, 2,··· , n} satisfying B ∈B , i Q which means B isQ−similar to B . Theorem 4.4 InQ−logic C , if there exists the multiple approximate reasoning type ∗ ∗ V Q−solution B with regard to input A under knowledge base K = {A → B|i = i i 1, 2,··· , n}, we have B ∈ B . i Q i=1 Theorem 4.5 In Q−logic C , if there exists the multiple approximate reasoning V ∗ ∗ Q−solution B with regard to input A under knowledge base K = {A → B|i = i i 1, 2,··· , n}, there exists at least an i∈{1, 2,··· , n} satisfying ∗ ∗ A → B ∈A → B , i i Q 96 Ya-lin Zheng · Bing-yuan Cao · Guang Yang · Yong-cheng Bai (2009) ∗ ∗ that is, A → B isQ−similar to knowledge A → B . i i Proof Because there exists type VQ−solution B to multiple approximate reasoning with regard to input A under knowledge base K = {A → B|i = 1, 2,··· , n}, there is i i at least an i∈{1, 2,··· , n} satisfying ∗ ∗ A ∈A , B ∈B , i Q i Q we have ∗ ∗ A → B ∈A →B . i Q i Q However, because of the construction ofQ−logic C , A →B ⊆A → B , i Q i Q i i Q we have ∗ ∗ A → B ∈A → B . i i Q Theorem 4.6 In Q−logic C , if there exists multiple approximate reasoning type ∗ ∗ V Q−solution B with regard to input A under knowledge base K = {A → B|i = i i 1, 2,··· , n}, we have ∗ ∗ A → B ∈ A → B . i i Q i=1 Theorem 4.7 In Q−logic C , the solution, response and examination domains of the multiple approximate reasoning type V Q−algorithm, with regard to input A under knowledge baseK = {A → B|i = 1, 2,··· , n} are given by i i (V ) Dom (K ) = A , i Q i=1 (V ) Codom (K )= B , i Q i=1 (V ) Exa (K ) = A → B . i i Q i=1 5. Type V completeness and type V perfection of knowledge base K of Q−logic ˆ ˜ Endow the formula setF (S) in classic propositional logic C = (C, C) with a similarity relationQ, where for each implication A → B ∈ H we have A →B = A → B . Q Q Q ˆ ˜ Then, (F (S),Q) is called regular Q−space, and C = (C, C,Q) is called regular Q−logic. InQ−logic C , the type V approximate knowledge closure of knowledge base K = {A → B|i = 1, 2,··· , n} i i Fuzzy Inf. Eng. (2009) 1: 91-101 97 is K = A → B . A→B∈K For each knowledge A → B in knowledge base K , Q−knowledge mass A → i i i B with the kernel of A → B in Q−space (F (S),Q) is also called type V approxi- i Q i i mate knowledge mass with the kernel of A → B . i i Therefore, we say in Q−logic C , the type V approximate knowledge closure K of knowledge base K = {A → B|i = 1, 2,··· , n} i i is the union of a ﬁnite number of type V knowledge massA → B (i = 1, 2,··· , n), i i Q that is K = A → B . i i Q i=1 InQ−logic C , knowledge base K = {A → B|i = 1, 2,··· , n} i i is called type V complete, if type V approximate knowledge closure K of K covers the theorem set Ø , namely, Ø ⊆K . Knowledge base K is called type V perfect, if type V approximate knowledge closureK ofK is coincident with theorem set Ø , that is K = Ø . Theorem 5.1 InQ−logic C , knowledge base K = {A → B|i = 1, 2,··· , n} i i denotes type V complete, if and only if each theorem in Ø is contained in type V knowledge mass with the kernel of a knowledge inK . Proof K is type V complete, if and only if Ø ⊆K , if and only if Ø ⊆ A → B i i Q i=1 and if and only if for each theorem A → B ∈ Ø , there exists an i ∈{1, 2,··· , n} satisfying A → B∈A → B . i i Q Theorem 5.2 In regularQ−logic C , knowledge base K = {A → B|i = 1, 2,··· , n} i i 98 Ya-lin Zheng · Bing-yuan Cao · Guang Yang · Yong-cheng Bai (2009) is type V complete, if and only if the antecedent of each theorem A → Bin Ø can ac- tivate a certain knowledge in knowledge base K and the consequence is the multiple approximate reasoning type V Q−solution with regard to input A under knowledge baseK . Proof According to theorem 5.1, knowledge baseK is type V complete, if and only if each of the theorems A → B in Ø must be contained in type V knowledge mass with the kernel of a certain knowledge inK , that is to say, there is an i∈{1, 2,··· , n} satisfying A → B∈A → B . i i Q However, according to the construction of regularQ−logic C A → B = A →B . i i Q i Q i Q Therefore, knowledge base K is type V complete, if and only if for each theorem A → B in Ø , there is an i∈{1, 2,··· , n} satisfying A → B∈A →B , i Q i Q that is, A∈A , B∈B . i Q i Q It is equal to say A activates a certain knowledge A → B in knowledge base K i i according to type V Q−algorithm and B can be taken as the multiple approximate reasoning type VQ−solution with regard to A under knowledge base K . Theorem 5.3 In regularQ−logic C , knowledge base K = {A → B|i = 1, 2,··· , n} i i ∗ ∗ is type V perfect, if and only if K is type V complete, and A → B is always a theorem in logic C for each input A which can active the knowledge in knowledge baseK according to type V Q−algorithm, that is ∗ ∗ A → B ∈ Ø , where B is multiple approximate reasoning type V Q−solution with regard to input A under knowledge base K . Proof ⇒ Assume knowledge base K is type V perfect given by K = Ø . b ∗ Because of Ø ⊆K , K is type V complete. Suppose A is an input which can activate the knowledge in K according to type V Q−algorithm, there must exist an i∈{1, 2,··· , n} satisfying ∗ ∗ A ∈A , B ∈B , i Q i Q Fuzzy Inf. Eng. (2009) 1: 91-101 99 that is, ∗ ∗ A → B ∈A →B , i Q i Q where B is the multiple approximate reasoning type V Q−solution with regard to input A under knowledge base K . But because of the construction of Q−logic C , we know A →B = A → B . i Q i Q i i Q Therefore ∗ ∗ A → B ∈A → B , i i Q ∗ ∗ meaning A → B is contained in knowledge mass A → B with the kernel of i i Q the knowledge A → B in K . According to the construction of type V knowledge i i closure of K K = A → B , i i Q i=1 we know ∗ ∗ b A → B ∈ A → B = K . i i Q i=1 However, since that knowledge base K is type V perfect, we know K = Ø , therefore ∗ ∗ A → B ∈ Ø . ∗ ∗ Thus we have proved that A → B stands always for a theorem in logic C. ⇐ Assume that knowledge baseK is type V complete, and for any input A which ∗ ∗ can activate the knowledge in K according to type V Q−algorithm, A → B is ∗ ∗ ∗ always a theorem in logic C, that is A → B ∈ Ø , where B is multiple approximate reasoning type V Q−solution with regard to input A under knowledge base K . BecauseK is type V complete, we know Ø ⊆K . Now, we prove K ⊆ Ø . ∗ ∗ b In fact, for any A → B ∈K , according to the construction of type V approximate knowledge closure of K K = A → B , i i Q i=1 we know there exists an i∈{1, 2,··· , n} satisfying ∗ ∗ A → B ∈A → B , i i Q 100 Ya-lin Zheng · Bing-yuan Cao · Guang Yang · Yong-cheng Bai (2009) ∗ ∗ that is, A → B is always contained in type V knowledge mass A → B with the i i Q kernel of the knowledge A → B in K . But according to the construction of regular i i Q−logic C , we know A → B = A →B , i i Q i Q i Q therefore ∗ ∗ A → B ∈A →B , i Q i Q that is ∗ ∗ A ∈A , B ∈B . i Q i Q It means A is an input which can activate knowledge A → B in knowledge base i i K according to type V Q−algorithm. B can be taken as the multiple approximate reasoning type V Q−solution with regard to input A under knowledge K . From the ∗ ∗ assumption, we know that, A → B always denotes a theorem in logic C, that is, ∗ ∗ A → B ∈ Ø . ∗ ∗ b Because of the arbitrariness of A → B inK , we know K ⊆ Ø , thereby we have proved that knowledge baseK is type V perfect, that is K = Ø . 6. Conclusions We do research on the approximate reasoning in logic frame. Based on the similarity relationQ in formula setF (S), we illustrate the matching scheme and corresponding algorithms for approximate reasoning. We also discuss the completeness of type V and its perfection on knowledge base in Q−logic C . The construction of Q−logic C ensures that the approximate reasoning conclusion B is Q−similar to B and the ∗ ∗ obtained approximate knowledge A → B is Q−similar to A → B. It reﬂects the nature of approximate reasoning and satisfy more application expectations if we de- sign approximate reasoning matching scheme and corresponding algorithms by using similarity relation Q instead of equivalence relation R . 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Fuzzy Information and Engineering – Taylor & Francis
Published: Mar 1, 2009
Keywords: Approximate reasoning; Formula mass; Knowledge mass; Q -logic; Q -algorithm; Q -completeness
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