Description of Reversibility of 9-Cyclic 1D Finite Linear Cellular Automata with Periodic Boundary Conditions
Hasan Akin
We consider a family of one-dimensional (1D) finite linear cellular automata (FLCA) with radius 4 on the periodic boundary conditions (PBC, shortly) over a finite field 𝔽𝒑 where 𝒑 ≥ 2 is prime number. We show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata. We solve the reversibility problem for 9-Cyclic 1D FLCA with radius 4 and periodic boundary conditions. Computing the determinant of relevant rule matrices for some finite number of cells and any given values of the coefficients of the local rules, we test the reversibility of the cellular automaton.
Keywords: Cellular automata, reversibility, determinant, Toeplitz matrix
MSC2020: 68Q80, 37B15, 15B05, 11C20