The maximum order complexity of sequence ensembles

Abstract

In this paper we extend the theory of maximum order complexity from a single sequence to an ensemble of sequences. In particular, the maximum order complexity of an ensemble of sequences is defined and its properties discussed. Also, an algorithm is given to determine the maximum order complexity of an ensemble of sequences linear in time and memory. It is also shown how to determine the maximum order feedback shift register equivalent of a given ensmble of sequences, i.e. including a feedback function. Hence, the problem of finding the absolutely shortest (possibly nonlinear) feedback shift register, that can generate two or more given sequences with characters from some arbitrary fmite alphabet, is solved. Finally, the consequences for sequence prediction based on the minimum number of observations are discussed.