On radius 1 nontrivial reversible and number-conserving cellular automata
Résumé
Reversibility and number-conservation are widely studied physics-like constraints for cellular automata (CA). Although both seem to be 'natural' constraints for a CA, it was conjectured that one-dimensional reversible and number-conserving CA (RNCCA) only has a limited computing ability. Particularly in the case of radius 1/2 (2-neighbor), it was shown that the class of RNCCA is equal to a trivial class of CA, so called shift-identity product cellular automata (SIPCA). But recently it was also shown that a RNCCA of neighborhood size four is computation-universal. In this paper, we list radius 1 (3-neighbor) RNCCAs up to 4-state by exhaustive search. In contrast to the radius 1/2 case, there are three new types of nontrivial RNCCA rules in the case of 4-state. We also show that it is possible to compose new nontrivial RNCCAs by modifying a SIPCA even when the state number is larger than four.