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Similarity Mass and Approximate Reasoning

Similarity Mass and Approximate Reasoning 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 . We hope that the ideas, ex- pressions and methods to this paper could become one of the logical standards of approximate reasoning. Acknowledgments Thanks to the support by National Natural Science Foundation of China (No. 70771030 and No. 70271047). References 1. Costas P. Pappis, Nikos I. Karacapilidis (1993) A comparative assessment of measures of similiarity of fuzzy values. Fuzzy Sets and Systems 56: 171-174 Fuzzy Inf. Eng. (2009) 1: 91-101 101 2. Costas P. Pappis (1991) Value approximation of fuzzy systems. Fuzzy Sets and Systems 39: 111-115 3. Guo-Jun Wang (1998) Fuzzy continuous input-output controllers are universal approximators. Fuzzy Sets and Systems 97: 95-99 4. Guo-Jun Wang (1999) On the logic foundation of fuzzy reasoning. Information Science 117: 47-88 5. Guo-Jun Wang, Hao Wang (2001) Non-fuzzy versions of fuzzy reasoning in classical logics. Infor- mation Sciences 138: 211-236 6. Guo-Jun Wang, Yee Leung (2003) Intergrated semantics and logic metric spaces. Fuzzy Sets and Systems 136: 71-91 7. Meng Guangwu (1993) Lowen’s compactness in L-fuzzy topological spaces. Fuzzy Sets and Systems 53: 329-333 8. Meng Guangwu (1995) On countably strong fuzzy compact sets in L-fuzzy topological spaces. Fuzzy Sets and Systems 72: 119-123 9. Shi-Zhong Bai (1997) Q-convergence of ideals in fuzzy lattices and its applications. Fuzzy Sets and Systems, 92: 357-363 10. Shi-Zhong Bai (1997) Q-convergence of nets and week separation axiom in fuzzy lattices. Fuzzy Sets and Systems 88: 379-386 11. Dug Hun Hong, Seok Yoon Hwang (1994) A note on the value similarity of fuzzy systems variables. Fuzzy Sets and Systems 66: 383-386 12. Hyung Lee-Kwang, Yoon-Seon Song, Keon-Myung Lee (1994) Similarity measure between fuzzy sets and between elements. Fuzzy Systems and Sets 62: 291-293 13. Shyi-Ming Chen, Ming-Shiow Yeh, Pei-Yung Hsiao (1995) A comparison of similarity measures of fuzzy values. Fuzzy Sets and Systems 72: 79-89 14. Shyi-Ming Chen (1995) Measures of smilarity between vague sets. Fuzzy Sets and Systems 74: 217-223 15. Thomas Sudkamp (1993) Similarity, interpolation, and fuzzy rule construction. Fuzzy Sets and Sys- tems 58: 73-86 16. W.J.M. Kickert, E.H.Mamdami (1978) Analysis of a fuzzy logic controller. Fuzzy Sets and Systems 1: 29-44 17. E. Czogala, J. Leski (2001) On equivalence of approximate reasoning results using different interpre- tations of fuzzy if-then rules. Fuzzy Sets and Systems 117: 279-296 18. L.A.Zadeh (2003) Computing with words and perceptions – a paradigm shift in computing and deci- sion analysis. Proceedings of International Conference on Fuzzy Information Processing, Tsinghua University Press, Springer Verlag 19. Mingsheng Ying (1994) A logic for approximate reasoning. The Journal of Symbolic Logic 59: 830-837 20. Mingsheng Ying (1992) Fuzzy reasoning under approximate match. Science Bulletin 37: 1244-1245 21. Mingsheng Ying (2003) Reasoning about probabilistic sequential programs in a probabilistic logic. Acta Informatica 39: 315-389 22. Yalin Zheng (2003) Stratified construction of fuzzy propositional logic. Proceedings of International Conference on Fuzzy Information Processing, Tsinghua University Press, Springer Verlag 1-2: 169- 23. Yalin Zheng, Changshui Zhang, Xing Yi (2004) Mamdaniean logic. Proceedings of IEEE Interna- tional Conference on Fuzzy Systems, Budapest, Hungary 1-3: 629-634 24. M.Last, Y.Klein, and A.Kandel (2001) Knowledge discovery in time series databases. IEEE transac- tions on systems, man, and cybernetics 31: 160-169 25. Y.Q.Zhang, M.D.Fraser, R.A.Gagliano, and A.Kandel (2000) Granular neural networks for numerical- linguistic data fusion and knowledge discovery. IEEE transactions on neural networks 11: 658-667 26. Ronald R. Yager (2004) On some new classes of implication operators and their role in approximate reasoning. Information Sciences 167: 193-216 27. Ronald R.Yager and V.Kreinovich (2003) Universal approximation theorem for uninorm-based fuzzy systems modeling. Fuzzy Sets and Systems 140: 331-339 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Similarity Mass and Approximate Reasoning

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Abstract

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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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Mar 1, 2009

Keywords: Approximate reasoning; Formula mass; Knowledge mass; Q -logic; Q -algorithm; Q -completeness

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