Download Full Issue in PDF - Academy Publisher

More documents

Recommendations

Info

1440 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 induction hypothesis, we get T Δφ →γ , From generalization rule we can get T ( ∀x)( Δφ →γ) , being Δ φ, γ are closed formula and from (U19), we can get T Δφ →( ∀x) γ , i.e. T Δφ →ψ . So the theorem holds. IV. CONCLUSION In this paper a predicate calculus formal deductive system ∀ h (0 1] based on the propositional system UL − ∈ , UL − h∈ (0, 1] for 1-level universal AND operator is built up. We prove the system ∀ UL − h∈ (0, 1] is sound. The deduction theorem is also given. Next we will discuss the completeness of system ∀ h (0 1] . UL − ∈ , ACKNOWLEDGMENT This work is partially supported Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 12JK0878) and Doctor Scientific Research Foundation Program of Xi'an Polytechnic University. REFERENCES [1] M. Gueffaz, S. Rampacek, C. Nicolle, “Temporal Logic To Query Semantic Graphs Using The Model Checking Method”, Journal of Software, vol. 7(7), pp. 1462-1472, 2012. [2] W. Jiang, “The Application of the Fuzzy Theory in the Design of Intelligent Building Control of Water Tank”, Journal of Software, vol 6(6), pp. 1082-1088, 2011. [3] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht/London, 2000. [4] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht/London, 1998. [5] R. Cignoli, F. Esteva, L. Godo, A. Torrens, “Basic fuzzy logic is the logic of continuous t-norms and their residua”, Soft computing, vol. 4, pp. 106-112, 2000. [6] F. Esteva, L.Godo, “Monoidal t-normbased logic: towards a logic for left-continous t- norms”, Fuzzy Sets and Systems, vol. 124, pp. 271–288, 2001. [7] U. Hohle, “Commutative, residuated l-monoids”, in: Non- Classical Logics and Their Applications to Fuzzy Subsets, U. Hohle, E. P. Klement, (eds.), Kluwer Academic Publishers, Dordrecht/London, pp. 53-106 , 1995. [8] G.J. Wang, Non-classical Mathematical Logic and Approximate Reasoning, Science Press (in Chinese), Beijing, 2000. [9] D.W. Pei, G. J. Wang, “The completeness and applications of the formal system L*”, Science in China (Series F) vol. 45, pp. 40–50, 2002. [10] D.W. Pei, “First-order Formal System K* and its Completeness”, Chinese Annals of Mathematics (Series A), vol. 23 (6), pp. 675–684, 2002. [11] H.C. He, et al, Universal Logic Principle, Science Press (in Chinese), Beijing, 2001. [12] H.C. He, et al, “Continuous-valued logic algebra-Studies on the basic of mathematical dialectical propositional logic”, IEEE International Conference on GrC, IEEE Press, pp: 194-199, 2010. [13] J.L. Chen, H.C. He, C.X. Liu, M.X. Luo. “Integrity studies on 0-level universal operation models of flexible logic”, Journal of Beijing University of Posts and Telecommunications, vol. 34 (4): 10-13, 2011. [14] Y.F. Fan, H.C. He, L.R. Ai, “N-norm on [0, ∞) and method for calculating generalized self-correlation coefficient k”, Journal of Northwestern Polytechnical University, vol. 28 (2): 270-275, 2010. [15] Y.C. Ma, H.C. He, “A Propositional Calculus Formal Deductive System ULh ∈ (0 , 1] of Universal Logic”, Proceedings of 2005 ICMLC, IEEE Press, pp: 2716-2721, 2005. [16] Y.C. Ma, H.C. He, “The Axiomatization for 0-level Universal Logic”, Lecture Notes in Artificial Intelligence, vol. 3930, pp. 367-37, 2006. [17] Y.C. Ma, H.C. He, “Axiomatization for 1-level Universal AND Operator”, The Journal of China Universities of Posts and Telecommunications, vol. 15 (2), pp. 125-129, 2008. [18] Y.C. Ma, Q.Y. Li, “A Propositional Deductive System of Universal Logic with Projection Operator”, Proceedings of 2006 ISDA, IEEE Press, pp: 993-998, 2006. [19] Q.Y. Li, Y.C. Ma, “The predicate system based on schweizer-sklar t-norm and its soundness”, Journal of Computational Information Systems, vol. 7 (15), p 5600- 5607, 2011. [20] Q.Y. Li, T, Cheng, The Predicate System Based on Schweizer-Sklar t-norm and Its Completeness. Lecture Notes in Electrical Engineering, vol. 107: 201-209, 2011, [21] Y.C. Ma, H.C. He, “Predicate Formal System ∀ULh ∈ [0.75 , 1] and and its completeness”, Computer Engineering and Applications, vol. 46 (34): 17-20, 2010. [22] Y.C. Ma, H.C. He, “Predicate Formal System ∀ULh ∈ [0.75 , 1] and its Soundness”, Computer Science, vol. 38 (5): 178-180, 2011. [23] Y.C. Ma, H.C. He, “Predicate formal system based on 0- level universal AND operator and its soundness”, Application Research of Computers, vol. 28 (1): 84-86, 2011. [24] Y.C. Ma, H.C. He, “Predicate Formal System Based on 0- level Universal and Operator and its Completeness”, Journal of Chinese Computer Systems, vol. 32 (10): 2105- 2108, 2011. Yingcang Ma, is a professor in school of science, Xi'an Polytechnic University. He received the PhD. degree from School of Computer Science, Northwestern Polytechnical University, in July 2006. His main researchinterests are in the areas of fuzzy set, rough set and non-classical mathematical logic. Hucan He, male, Professor and Ph.D. tutor from the Department of Computer Science and Engineering of Northwestern Polytechnical University, interested in the foundation and application of AI, universal logic and uncertainties reasoning. © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1441 Analysis of Boolean Networks using An Optimized Algorithm of Structure Matrix based on Semi-Tensor Product Jinyu Zhan University of Electronic Science and Technology of China, Chengdu, China Email: zhanjy@uestc.edu.cn Shan Lu and Guowu Yang University of Electronic Science and Technology of China, Chengdu, China Email: 617999242@qq.com, guowu@uestc.edu.cn Abstract—The structure matrix based on semi-tensor product can provide formulas for analyzing the characteristics of a Boolean network, such as the number of fixed points, the number of circles of different lengths, transient period for all points to enter the set of attractors and basin of each attractor. However, the conventional method of semi-tensor product gains the structure matrix through complex matrix operations with high computation complexity. This paper proposes an optimized algorithm which gains the structure matrix through the truth table reflecting the state transformation of Boolean networks. The effectiveness and feasibility of our optimized approach are demonstrated through the analysis of a practical Boolean network of the mammalian cell. Index Terms—semi-tensor product, Boolean network, structure matrix, truth table I. INTRODUCTION The Boolean network, introduced firstly by Kauffman [1], and then developed by [2][3][4][5][6][7][8] and many others, becomes a powerful tool in describing, analyzing, and simulating the cell network. It was shown that the Boolean network plays an important role in modeling cell regulation, because they can represent important features of living organisms [9][10]. It has received the most attention, not only from the biology community, but also physics, system science, etc. The structure of a Boolean network is described in terms of its cycles and the transient states that lead to them. Several useful Boolean networks have been analyzed and their circles have been revealed [11][12]. It was pointed in [13] that finding fixed points and circles of a Boolean network is an NP hard problem. Semi-tensor Manuscript received May 24, 2012; revised June 1, 2012; accepted July 1, 2012. This work was supported by the Fundamental Research Funds for the Central Universities of China under Grant No. ZYGX2009J062 and the National Natural Science Foundation of China under Grant No. 60973016. Corresponding author: Jinyu Zhan, email: zhanjy@uestc.edu.cn product of matrix (STP), presented by Cheng [14]. Using STP, a Boolean network equation can be expressed as a conventional discrete time linear system which contains complete information of the dynamics of a Boolean network. Analyzing the structure matrix of a Boolean network, precise formulas are obtained to determine the number of fixed points and numbers of all possible circles of different lengths. But the conventional method to calculate the structure matrix of a Boolean network, presented in [15][18][19][21], is very complex. In this paper, a optimized algorithm is proposed to calculate the structure matrix. Unlike existing methods, our approach gets the structure matrix of a Boolean network not through the complex matrix operations but through the truth table which reflects the state transformation of the Boolean network. Compared with the conventional method, our approach can greatly reduce the calculation complexity. The methods for analyzing the characteristics of a Boolean network are given. The analysis of a practical Boolean network of the mammalian cell shows that our approach is effective and efficient. The rest of the paper is organized as follows. Section II gives a brief introduction to semi-tensor product of matrices, matrix expression of logic and dynamics of Boolean network. The conventional method to calculate the structure Matrix of a Boolean Network is given in Section III and our approach is proposed in Section IV. Section V gives the methods to analyze the characteristics of Boolean networks through the structure matrix. Section VI gives a practical Boolean network of the mammalian cell to show the effectiveness and feasibility of our approach. Finally, some conclusions are drawn in Section VII. II. EXPRESSION OF BOOLEAN NETWORKS IN SEMI-TENSOR PRODUCT A. Semi-tensor Product This section is a brief introduction to semi-tensor product (STP) of matrices. STP of matrices is a © 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.6.1441-1448
Page 1 and 2:
Journal of Computers ISSN 1796-203X
Page 3:
Corn Moisture Measurement using a C
Page 6 and 7:
1378 JOURNAL OF COMPUTERS, VOL. 8,
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
Page 14 and 15:
Page 16 and 17:
Page 18 and 19: 1390 JOURNAL OF COMPUTERS, VOL. 8,
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 261:
Call for Papers and Special Issues
Page 264:
(Contents Continued from Back Cover
show all

Download Full Issue in PDF - Academy Publisher

Create successful ePaper yourself

Delete template?

Save as template?