TY - GEN
T1 - The Degrees of Squares is an Atom
AU - Endrullis, Jörg
AU - Grabmayer, Clemens
AU - Hendriks, Dimitri
AU - Zantema, Hans
PY - 2015
Y1 - 2015
N2 - We answer an open question in the theory of degrees of infinite sequences with respect to transducibilityby finite-state transducers. An initial study of this partial order of degrees was carried out in [1], but many basic questions remain unanswered. One of the central questions concerns the existence of atom degrees, other than the degree of the ‘identity sequence’ 100101102103 · · ·. A degree is called an ‘atom’ if below it there is only the bottom degree 0, which consists of the ultimately periodic sequences. We show that also the degree of the ‘squares sequence’ 1001011041091016 · · · is an atom. As the main tool for this result we characterise the transducts of ‘spiralling’ sequences and their degrees. We use this to show that every transduct of a ‘polynomial sequence’ either is in 0 or can be transduced back to a polynomial sequence for a polynomial of the same order.
AB - We answer an open question in the theory of degrees of infinite sequences with respect to transducibilityby finite-state transducers. An initial study of this partial order of degrees was carried out in [1], but many basic questions remain unanswered. One of the central questions concerns the existence of atom degrees, other than the degree of the ‘identity sequence’ 100101102103 · · ·. A degree is called an ‘atom’ if below it there is only the bottom degree 0, which consists of the ultimately periodic sequences. We show that also the degree of the ‘squares sequence’ 1001011041091016 · · · is an atom. As the main tool for this result we characterise the transducts of ‘spiralling’ sequences and their degrees. We use this to show that every transduct of a ‘polynomial sequence’ either is in 0 or can be transduced back to a polynomial sequence for a polynomial of the same order.
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U2 - 10.1007/978-3-319-23660-5_10
DO - 10.1007/978-3-319-23660-5_10
M3 - Conference contribution
SN - 9783319236599
VL - 9304
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 109
EP - 121
BT - Combinatorics on Words - 10th International Conference, WORDS 2015, Proceedings
PB - Springer/Verlag
T2 - 10th International Conference on Words, WORDS 2015
Y2 - 14 September 2015 through 17 September 2015
ER -