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Sensitivity of Elementary Cellular Automata to Their Inputs

The sensitivity of an elementary cellular automaton (CA) to its inputs is defined as the space-averaged proportion of the cells in the neighborhoods of the CA's cells , that is, ), that affect the state of during the subsequent time step. This proportion can be expressed as
,
where denotes the number of cells upon which the CA is built and represents the , entry in an Jacobian matrix that can be one if a modification of the state of at the time step implies a perturbation of 's state during the subsequent time step and is zero otherwise. Since for an elementary CA, , constitutes a tridiagonal matrix and if and only if for every neighbor of , and this holds for every of the CA. It has been shown that can be employed to find an upper bound on the maximum Lyapunov exponent of elementary CA.

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More information on the sensitivity of CA to their inputs can be found in [1], [2], and [3].
References
[1] J. M. Baetens and B. De Baets, "Phenomenological Study of Irregular Cellular Automata Based on Lyapunov Exponents and Jacobians," Chaos, 20, 033112, 2010. doi:10.1063/1.3460362.
[2] J. M. Baetens and B. De Baets, "On the Topological Sensitivity of Sellular Automata," Chaos, in press, 2011.
[3] F. Bagnoli, R. Rechtman, and S. Ruffo, "Damage Spreading and Lyapunov Exponents in Cellular Automata," Physics Letters A, 172, pp. 34–38. doi:10.1016/0375-9601(92)90185-O.
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