# Spectral Studies of Automata

• J. H. Loxton
Chapter
Part of the Algorithms and Combinatorics 8 book series (AC, volume 8)

## Abstract

It is reasonable to argue that the simplest sequences are the periodic sequences. There are at least two ways to make this truism meaningful. The n-th term of a periodic sequence can be determined once we know the period and the integer n and this amounts to approximately log2 n bits of information for large n. Also, the number of different words of length n in a periodic sequence is bounded. Periodic sequences minimise both of these characteristics. At the other end of the scale are the random sequences. We can take a random sequence to be one for which the most efficient way to find the n-th term is to make a list of the first n terms, requiring at least n bits of information. This implies another essential aspect of randomness that, for every n, the words of length n in the sequence are equidistributed.

## Keywords

Functional Equation Spectral Study Arithmetic Progression Finite Automaton Periodic Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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