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.

On the maximum order complexity of subsequences of the Thue–Morse and Rudin–Shapiro sequence along squares

“…k > 2 and k even, 7 · 2 ℓ+3 , k = 3, ((2 k−1 − 1)2 ℓ+2 , k > 3 and k odd, we get a contradiction to (7), (8), (9) or (10) This problem may be out of reach and we state some weaker problems. It is known that the subword complexity is maximal if d = 2 [12].…”

Sequences With High Nonlinear Complexity

“…368-369]). For k ≥ q − 1 all nonlinear complexities N (k) (s) of a fixed s are the same and equal to the maximum-order complexity M(s) = N (q−1) (s) introduced by Jansen [4], [5]. Connections between the Lempel-Ziv complexity and the maximum-order complexity were studied in [4], [9], [10].…”

On Nonlinear Complexity and Shannon’s Entropy of Finite Length Random Sequences

“…Among all these complexity measures, the following four measures may be the most important ones. They are linear complexity, Lempel-Ziv complexity, eigenvalues and nonlinear complexity, respectively [16][17][18][19][20][21][22][23][24][25][26]. Among these four measures, linear complexity and Lempel-Ziv complexity have achieved widely studied.…”