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Singular Boolean networks: Semi-tensor product approach

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Abstract

Singular Boolean networks are introduced in this paper. Via semi-tensor product of matrices and the matrix expression of logical functions, two kinds of the condensed algebraic expressions of singular Boolean networks are obtained. The normalization problem of singular Boolean networks is addressed; that is, under what condition singular Boolean networks can be converted into normal Boolean networks with algebraic restrictions. Then one sufficient condition and one necessary and sufficient condition are derived for the normalization problem. Furthermore, the solvability of singular Boolean networks is discussed and the concept of admissible initial values of singular Boolean networks is presented. Finally, fixed points and cycles of singular Boolean networks are also investigated.

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References

  1. Kauffman S. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol, 1969, 22: 437–467

    Article  MathSciNet  Google Scholar 

  2. Davison E H, Rast J P, Oliveri P, et al. A genomic regulatory network for development. Science, 2002, 295: 1669–1678

    Article  Google Scholar 

  3. Harris S E, Sawhill B K, Wuenche A, et al. A model of transcriptional regulatory networks based on biases in the observed regulations rules. Complexity, 2002, 7: 23–40

    Article  Google Scholar 

  4. Bansal M, Belcastro V, Ambesi-Impiobato A, et al. How to infer gene networks from expression profiles. Mol Syst Biol, 2007, 3: 1–10

    Google Scholar 

  5. Drossel B, Mihaljev T, Greil F. Number and length of attractors in a critical kauffman model with connectivity one. Phys Rev Lett, 2005, 94: 088701

    Article  Google Scholar 

  6. Pal R, Datta A, Dougherty E R. Optimal infinite-horizon control for probabilistic Boolean networks. IEEE Trans Signal Process, 2006, 54: 2375–2387

    Article  Google Scholar 

  7. Cheng D, Qi H. A linear representation dynamics of Boolean networks. IEEE Trans Automat Control, 2010, 55: 2251–2258

    Article  MathSciNet  Google Scholar 

  8. Cheng D, Qi H, Li Z. Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach. London: Springer-Verlag, 2011

    Book  Google Scholar 

  9. Cheng D, Qi H. Controllability and observability of Boolean control networks. Automatica, 2009, 45: 1659–1667

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng D, Li Z, Qi H. Realization of Boolean control networks. Automatica, 2010, 46: 62–69

    Article  MathSciNet  MATH  Google Scholar 

  11. Cheng D, Qi H, Li Z, et al. Stability and stabilization of Boolean networks. Int J Robust Nonlinear Control, 2011, 21: 134–156

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng D. Disturbance decoupling of Boolean control networks. IEEE Trans Automat Control, 2011, 56: 2–10

    Article  MathSciNet  Google Scholar 

  13. Cheng D, Zhao Y. Identification of Boolean control networks. Automatica, 2011, 47: 702–710

    Article  MathSciNet  MATH  Google Scholar 

  14. Cheng D, Qi H, Li Z. Model construction of Boolean networks via observed data. IEEE Trans Neural Netw, 2011, 22: 525–536

    Article  Google Scholar 

  15. Li F, Sun J. Controllability of Boolean control networks with time delays in states. Automatica, 2011, 47: 603–607

    Article  MATH  Google Scholar 

  16. Li F, Sun J. Observability of Boolean control networks with state time delays. IEEE Trans Automat Control, 2011, 22: 948–954

    Google Scholar 

  17. Liu Z, Wang Y, Li H. Disturbance decoupling of multi-valued logical networks. In: Proc. of 30th Chinese Control Conference, Yantai, 2011. 93–96

    Google Scholar 

  18. Li H, Wang Y, Liu Z. Simultaneous stabilization of Boolean control networks via semi-tensor product method. In: Proc. of 30th Chinese Control Conference, Yantai, 2011. 6386–6390

    Google Scholar 

  19. Li F, Sun J. Stability and stabilization issue of probabilistic Boolean network. In: Proc. of 30th Chinese Control Conference, Yantai, 2011. 6380–6385

    Google Scholar 

  20. Dai L. Singular Control Systems. Berlin: Springer-Verlag, 1989

    Book  MATH  Google Scholar 

  21. Cheng D, Zhao Y, Xu X. Mix-valued logic and its applications. J Shandong Univ Nat Sci, 2011, 46: 32–44

    MathSciNet  MATH  Google Scholar 

  22. Ljung L, Söderström T. Theory and Practice of Recursive Identification. Cambridge: MIT Press, 1983

    MATH  Google Scholar 

  23. Cheng D. Semi-tensor product of matrices and its applications-a survey. In: The International Congress of Chinese Mathematicians (ICCM), Hangzhou, 2007. 641–668

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    JunE Feng & Juan Yao

  2. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Peng Cui

Authors
  1. JunE Feng
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  2. Juan Yao
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  3. Peng Cui
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Correspondence to JunE Feng.

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Feng, J., Yao, J. & Cui, P. Singular Boolean networks: Semi-tensor product approach. Sci. China Inf. Sci. 56, 1–14 (2013). https://doi.org/10.1007/s11432-012-4666-8

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  • DOI: https://doi.org/10.1007/s11432-012-4666-8

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