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
As relations
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 [math]\displaystyle{ \mathbb{N}^{r} }[/math] is k-synchronized if the relation {([n1]k, ..., [nr]k)} is a right-synchronized[1] rational relation over Σ∗ × ... × Σ∗, where (n1, ..., nr) [math]\displaystyle{ \in }[/math] R.[2]
Language-theoretic
Let n ≥ 0 be a natural number and let f: [math]\displaystyle{ \mathbb{N} \rightarrow \mathbb{N} }[/math] 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 [math]\displaystyle{ \{(n, f(n))\} }[/math] is regular.
History
The class of k-synchronized sequences was introduced by Carpi and Maggi.[2]
Example
Subword complexity
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
- [math]\displaystyle{ \{(n, m)_k \mid n \geq 0 \text{ and } m = \rho_S(n)\}. }[/math]
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 (n, m)k is accepted by this automaton, the subword complexity function of the k-automatic sequence s(n) is k-synchronized.
Properties
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
- ↑ 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
- ↑ 2.0 2.1 Carpi & Maggi (2010)
- ↑ Goč, D.; Schaeffer, L.; Shallit, J. (2013). Subword complexity and k-synchronization. Lecture Notes in Computer Science. 7907. Editors Béal MP., Carton O.. Berlin: Springer. ISBN 978-3-642-38770-8.
- ↑ Carpi & Maggi (2010), Proposition 2.6
- ↑ Carpi & Maggi (2010), Proposition 2.8
- ↑ Carpi & Maggi (2010), Proposition 2.1
- ↑ Carpi & Maggi (2010), Proposition 2.2
- ↑ Carpi & Maggi (2010), Proposition 2.5
- ↑ 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
- Carpi, A.; Maggi, C. (2010), "On synchronized sequences and their separators", Theoret. Informatics Appl. 35 (6): 513–524, doi:10.1051/ita:2001129, http://www.numdam.org/item/ITA_2001__35_6_513_0/.
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