Investigation of the global dynamics of cellular automata using Boolean derivatives

  • Authors:

  • Pabitra Pal Choudhury;Sudhakar Sahoo;Mithun Chakraborty;Subir Kumar Bhandari;Amita Pal

  • Affiliations:

  • Applied Statistics Unit, Indian Statistical Institute, Kolkata, 700108, India;Department of CSEA, Silicon Inst. of Tech., Silicon Hills, Patia, Bhubaneswar-751024, India;Department of Electronics and Telecommunication Engg., Jadavpur University, Kolkata-700032, India;Bayesian and Interdisciplinary Research Unit, Indian Statistical Institute, Kolkata, 700108, India;Bayesian and Interdisciplinary Research Unit, Indian Statistical Institute, Kolkata, 700108, India

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2009

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Abstract

Global dynamics of a non-linear Cellular Automaton (CA), is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In this paper, efforts have been made to systematize non-linear CA evolutions in the light of Boolean derivatives and Jacobian matrices. A few new theorems on Hamming Distance between Boolean functions as well as on Jacobian matrices of cellular automata are proposed and proved. Moreover, a classification of Boolean functions based on the nature of deviation from linearity has been suggested with a view to grouping them together to classes/subclasses such that the members of a class/subclass satisfy certain similar properties. Next, an error vector, which cannot be captured by the Jacobian matrix, is identified and systematically classified. This leads us to the concept of modified Jacobian matrix whereby a quasi-affine representation of a non-linear cellular automaton is introduced.