Abstract
We consider Billiard Words on alphabets with k = 3 letters: such words are associated to some 3-dimensional positive vector \({\overrightarrow{\alpha}=({\alpha_1},{\alpha_2},{\alpha_3})}\). The language of these words is already known in the usual case, i.e., when the α j are linearly independent over \({\rm \rule{.33em}{0ex}\rule{.08em}{1.52ex}\kern -0.33em Q}\), and so for the \({\alpha}_{j^{-1}}\)’s. Here we study the language of these words when there exist some linear relations. We give the complexity of Billiard Words in any case, which has asymptotically a ”polynomial-like” form, with degree less or equal to 2. These results are obtained by geometrical methods.
AMS Classification. 68R15.
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Borel, J.P. (2006). Complexity of Degenerated Three Dimensional Billiard Words. In: Ibarra, O.H., Dang, Z. (eds) Developments in Language Theory. DLT 2006. Lecture Notes in Computer Science, vol 4036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779148_35
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DOI: https://doi.org/10.1007/11779148_35
Publisher Name: Springer, Berlin, Heidelberg
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