Towards the Definition of Conservation Degree for One-Dimensional Cellular Automata Rules

Decidability of the number-conservation property of one-dimensional cellular automata rules can be established by necessary and suficient conditions given by Boccara and Fuk. Nevertheless, a related open question would ask for a definition of the intermediate conservation degree of a rule. Based upon those conditions, a theoretical measure is formulated for the intermediate conservation degree for one-dimensional cellular automata rules, based upon Boccara-Fuks conditions. However, its appropriateness for the target quantity is not verified, according to a corresponding empirical measure that is defined. Two additional empirical measures are then defined, representing alternative interpretations to the quantity. Results of computational experiments are discussed involve all approaches, and possible relations between the theoretical and experimental measures are investigated. The paper advances the conceptual structure of the theme, by suggesting that the theoretical approach is not adequate for the intended objective, at the same time that the latter two empirical alternatives suggest what is really at stake, thus pointing at what their theoretical counterparts should account for.