Surveying some notions of complexity for finite and infinite sequences (Functions in Number Theory and Their Probabilistic Aspects)
Year: 2012
Type: article
Abstract: We survey some ways of approaching the complexity of finite or infinite sequences.We emphasize in particular the notion of inconstancy of a sequence. §1. IntroductionGiven a finite or infinite sequence, how to recognize that it is ( ( \ m a t h r m { c } \ m a t h r m { o } \ m a t h r m { m } \ m a t h r m { p } \ m a t h r m { l } \ m a t h r m { i } \ m a t h r m { c } \ m a t h r m { a } \ m a t h r m { t } \ m a t h r m { e } \ m a t h r m { d } (or complex), ((random, chaotic?How to compare the ((complexity, the random- ness, the chaos of two sequences?Is it reasonable to consider that the sequence 0000. . . is ((simpler than the sequence 010101. . ., and that this last sequence is sim- pler than a binary eventually periodic sequence like 000001001001. . .?Also should a random sequence be complicated?Among the many studies devoted to these complicated questions, we will survey quickly some notions of complexity for sequences taking only finitely many values; then we will end with a notion related to an old the- orem of Cauchy and Crofton and recently adressed in a paper of L. Maillard-Teyssier and the author: the inconstancy of a sequence. §2. Algorithmic complexityOne way of measuring the complexity of a sequence is to evaluate how difficult it is to generate that sequence.The difficulty is measured, e.g., by the quantity of instruc- tions needed to generate the sequence (typically the length of a computer program), or
Author: Jean‐Paul Allouche
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1560798486