DelftIn this paper the problem of finding the absolutely shortest (possibly nonlinear) feedback shift register, which can generate a given sequence with characters from some arbitrary finite alphabet, is considered. To this end, a new complexity measure is defined, called the maximum order complexity. A new theory of the nonlinear feedback shift register is developed, concerning elementary complexity properties of transposed and reciprocal sequences, and feedback functions of the maximum order feedback shift register equivalent. Moreover, Blumer's algorithm is identified as a powerful tool for determining the maximum order complexity profile of sequences, as well as their period, in linear time and memory. The typical behaviour of the maximum order complexity profile is shown and the consequences for the analysis of given sequences and the synthesis of feedback shift registers are discussed.

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.

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