Correlations and Random Information in Cellular Automata

Infin ite one-dimens ional cellu lar automata are st udied using information theory. The average informat ion per cell is divided into cont ribut ions from different correlation lengths and random vari ations (measure entropy). It is shown that the measure entropy is no n-increasing in time for deterministic rules, and constant for ru les which are one-to-one mappings of their first or last argument (almost reversible rules). For probabilistic rules, there is no such general law I but for almost reversible rules where the states are randomly shifted, it is proven that the system evolves towards the maximally disordered state, independent of initial conditions. It is discussed how some of the information-theoretical concepts are rela t ed to analogous conce pts in algorithmic informati on theory, and an equality between algorithmic informat ion an d measure entropy is proved . Numerical an d analy t ical examples are given for specific rul es .