Complexity of Forecasting in a Class of Simple Models

A bstract . We deal in this paper wit h the di fficul ty of per for ming opt imal or nearl y optimal forecasts of discrete symbol sequences generated by very simp le models . T hese are spatial sequences gene rated by elementary one-dimensional cellular automata. after one time step , wit h com pletely ran dom input st rings . T hey have p ositi ve entropy and t hus cannot be ent irely predict ed . Mak ing forecas ts which are optimal wit hin th is Li mita t ion is proven to be sur prisingly difficult. Scaling laws with new ano malous exponent s are found bot h for opt imal forecast s and for forecasts which are nearly opti mal . The same remark s hold not only for forecastin g but also for data. compression.