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.

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 N r {\displaystyle \mathbb {N} ^{r}} is k-synchronized if the relation {([n₁]_k, ..., [n_r]_k)} is a right-synchronized^[1] rational relation over Σ^∗ × ... × Σ^∗, where (n₁, ..., n_r) ∈ {\displaystyle \in } R.^[2]

Let n ≥ 0 be a natural number and let f: N → N {\displaystyle \mathbb {N} \rightarrow \mathbb {N} } 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 { ( n , f ( n ) ) } {\displaystyle \{(n,f(n))\}} is regular.

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

Given a k-automatic sequence s(n) and an infinite string S = s(1)s(2)..., let ρ_S(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

{ ( n , m ) k ∣ n ≥ 0  and  m = ρ S ( n ) } . {\displaystyle \{(n,m)_{k}\mid n\geq 0{\text{ and }}m=\rho _{S}(n)\}.}

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.

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]

• Carpi, A.; Maggi, C. (2010), "On synchronized sequences and their separators", Theoret. Informatics Appl., 35 (6): 513–524, doi:10.1051/ita:2001129.