Repetition-Free Words
Abstract Diamond Implementation Diamond Paper
Blanchet-Sadri Abraham Elara

First applet: Given a full word w, a positive integer p, and a non-negative integer d, this implementation outputs a d-valid, p-periodic partial word contained in w if any such word exists. A d-valid partial word is one that has no two holes within distance d.

Second applet: Given a morphism and a seed string that determine a fixed point, together with a non-negative integer n, this implementation outputs the list of all length n factors of this fixed point.

Usage: Morphisms are inputted by specifying the alphabet (as a list with no delimiter) in the "alphabet" field and the images of these letters (as a space-delimited list in the corresponding order) in the "images" field. The morphism defined by a→bc, b→bd, c→ca, d→cb is provided as an example.

There are several restrictions on the morphism and seed: The morphism must be such that the image of each letter has length at least 2. Furthermore, the seed must be a prefix of its image. (This ensures that with repeated application of the morphism to the seed, the word grows exponentially and converges to a fixed point.)

Tip: If your morphism maps some letter to a word of length less than 2, try raising the morphism to a higher power, since this new morphism will generate the same fixed point. For example, consider the morphism defined by a→abc, b→ac, c→b. The square of this morphism is defined by a→abcacb, b→abcb, c→ac, which sends every letter to a word of length at least 2.



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Acknowledgement: This material is based upon work supported by the National Science Foundation under Grant No. DMS-0754154. The Department of Defense is also gratefully acknowledged.
Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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