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| | In [[mathematics]], a '''locally catenative sequence''' is a sequence of [[word (mathematics)|words]] in which each word can be constructed as the concatenation of previous words in the sequence.<ref>{{cite book |
| | | last = Rozenberg |
| | | first = Grzegorz |
| | | coauthors = Salomaa, Arto |
| | | title = Handbook of Formal Languages |
| | | publisher = Springer |
| | | date = 1997 |
| | | pages = 262 |
| | | isbn = 3-540-60420-0}}</ref> |
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| | Formally, an infinite sequence of words ''w''(''n'') is locally catenative if, for some positive integers ''k'' and ''i''<sub>1</sub>,...''i''<sub>''k''</sub>: |
| | :<math>w(n)=w(n-i_1)w(n-i_2)...w(n-i_k) \text{ for } n \ge \max(i_1, ... i_k) \, .</math> |
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| | Some authors use a slightly different definition in which encodings of previous words are allowed in the concatenation.<ref>{{cite book |
| | | last = Allouche |
| | | first = Jean-Paul |
| | | coauthors = Shallit, Jeffrey |
| | | title = Automatic Sequences |
| | | publisher = Cambridge |
| | | date = 2003 |
| | | pages = 237 |
| | | isbn = 0-521-82332-3}}</ref> |
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| | ==Examples== |
| | The sequence of [[Fibonacci word]]s ''S''(''n'') is locally catenative because |
| | :<math>S(n)=S(n-1)S(n-2) \text{ for } n \ge 2 \, .</math> |
| | |
| | The sequence of [[Thue-Morse sequence|Thue-Morse words]] ''T''(''n'') is not locally catenative by the first definition. However, it is locally catenative by the second definition because |
| | :<math>T(n)=T(n-1)\mu(T(n-1)) \text{ for } n \ge 1 \, ,</math> |
| | where the encoding μ replaces 0 with 1 and 1 with 0. |
| | |
| | ==References== |
| | <references/> |
| | |
| | [[Category:Formal languages]] |
| | [[Category:Combinatorics on words]] |
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:
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
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
where the encoding μ replaces 0 with 1 and 1 with 0.
References
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- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534