Spectral Studies of Automata
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.
KeywordsFunctional Equation Spectral Study Arithmetic Progression Finite Automaton Periodic Sequence
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