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A one-dimensional two-state number-conserving cellular automaton (NCCA) is acellular automaton whose states are 0 or 1 and where cells take states 0 and 1and updated their states by the rule which keeps overall sum of statesconstant. It can be regarded as a kind of particle based modeling of physicalsystems and has another intuitive representation, motion representation, basedon the movement of each particle. We introduced a kind of hierarchicalinterpretation of motion representations to understand the necessary patternsize to each motion. We show any NCCA of its neighborhood size n can behierarchically represented by NCCAs of their neighborhood size from n-1 to 1.

A hierarchical structure in the motion representation of 2-state number-conserving cellular automata