k-synchronized sequence

In mathematics and theoretical computer science, a k-synchronized sequence is an infinite sequence of terms s(n) characterized by a finite automaton taking as input two strings m and n, each expressed in some fixed base k, and accepting if m = s(n). The class of k-synchronized sequences lies between the classes of k-automatic sequences and k-regular sequences.

Definitions

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As relations

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Let Σ be an alphabet of k symbols where k ≥ 2, and let [n]k denote the base-k representation of some number n. Given r ≥ 2, a subset R of   is k-synchronized if the relation {([n1]k, ..., [nr]k)} is a right-synchronized[1] rational relation over Σ × ... × Σ, where (n1, ..., nr)   R.[2]

Language-theoretic

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Let n ≥ 0 be a natural number and let f:   be a map, where both n and f(n) are expressed in base k. The sequence f(n) is k-synchronized if the language of pairs   is regular.

History

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The class of k-synchronized sequences was introduced by Carpi and Maggi.[2]

Example

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Subword complexity

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Given a k-automatic sequence s(n) and an infinite string S = s(1)s(2)..., let ρS(n) denote the subword complexity of S; that is, the number of distinct subwords of length n in S. Goč, Schaeffer, and Shallit[3] demonstrated that there exists a finite automaton accepting the language

 

This automaton guesses the endpoints of every contiguous block of symbols in S and verifies that each subword of length n starting within a given block is novel while all other subwords are not. It then verifies that m is the sum of the sizes of the blocks. Since the pair (nm)k is accepted by this automaton, the subword complexity function of the k-automatic sequence s(n) is k-synchronized.

Properties

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k-synchronized sequences exhibit a number of interesting properties. A non-exhaustive list of these properties is presented below.

  • Every k-synchronized sequence is k-regular.[4]
  • Every k-automatic sequence is k-synchronized. To be precise, a sequence s(n) is k-automatic if and only if s(n) is k-synchronized and s(n) takes on finitely many terms.[5] This is an immediate consequence of both the above property and the fact that every k-regular sequence taking on finitely many terms is k-automatic.
  • The class of k-synchronized sequences is closed under termwise sum and termwise composition.[6][7]
  • The terms of any k-synchronized sequence have a linear growth rate.[8]
  • If s(n) is a k-synchronized sequence, then both the subword complexity of s(n) and the palindromic complexity of s(n) (similar to subword complexity, but for distinct palindromes) are k-regular sequences.[9]

Notes

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  1. ^ Frougny, C.; Sakarovitch, J. (1993), "Synchronized rational relations of finite and infinite words", Theoret. Comput. Sci., 108: 45–82, doi:10.1016/0304-3975(93)90230-Q
  2. ^ a b Carpi & Maggi (2010)
  3. ^ Goč, D.; Schaeffer, L.; Shallit, J. (2013). Subword complexity and k-synchronization. Lecture Notes in Computer Science. Vol. 7907. Editors Béal MP., Carton O. Berlin: Springer. ISBN 978-3-642-38770-8.
  4. ^ Carpi & Maggi (2010), Proposition 2.6
  5. ^ Carpi & Maggi (2010), Proposition 2.8
  6. ^ Carpi & Maggi (2010), Proposition 2.1
  7. ^ Carpi & Maggi (2010), Proposition 2.2
  8. ^ Carpi & Maggi (2010), Proposition 2.5
  9. ^ Carpi, A.; D'Alonzo, V. (2010), "On factors of synchronized sequences", Theoret. Comput. Sci., 411 (44–46): 3932–3937, doi:10.1016/j.tcs.2010.08.005

References

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