Locally catenative sequence explained

In mathematics, a locally catenative sequence is a sequence of words in which each word can be constructed as the concatenation of previous words in the sequence.[1]

Formally, an infinite sequence of words w(n) is locally catenative if, for some positive integers k and i1,...ik:

w(n)=w(n-i1)w(n-i2)\ldotsw(n-ik)forn\gemax\{i1,\ldots,ik\}.

Some authors use a slightly different definition in which encodings of previous words are allowed in the concatenation.[2]

Examples

The sequence of Fibonacci words S(n) is locally catenative because

S(n)=S(n-1)S(n-2)forn\ge2.

The sequence of Thue–Morse words T(n) is not locally catenative by the first definition. However, it is locally catenative by the second definition because

T(n)=T(n-1)\mu(T(n-1))forn\ge1,

where the encoding μ replaces 0 with 1 and 1 with 0.

References

  1. Book: Rozenberg , Grzegorz . Salomaa, Arto . Handbook of Formal Languages . Springer . 1997 . 262 . 3-540-60420-0.
  2. Book: Allouche , Jean-Paul . Shallit, Jeffrey . Automatic Sequences . Cambridge . 2003 . 237 . 0-521-82332-3.