Studies on integrable cellular automata

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I investigated (1) discrete integrable equations over finite fields and (2) a classification of an elementary cellular automaton.(1) (i) Replacing a parameter with a value in finite fields, one can uniquely determine the values of the dependent variables in finite fields and infinity. Thanks to this property, I obtained the time evolution and N soliton solutions of the (generalized) discrete KdV equations over finite fields. (ii) I defined the notion of almost good reduction (AGR), which is an arithmetic analogue of passing the singularity confinement test, and proved that the discrete Painleve II equation (dPII) has AGR. By AGR, I obtained the time evolution and a solution of the dPII over finite fields. I had the similar results for qPII, qPIII, qPIV and dKdV.(2) I suggested a classification of an elementary cellular automaton by singular value decomposition. Though class 4 resembles class 3 of Wolfram's classification, one can distinguish them by using the above method.