Strongly k-recursive sequences

TL;DR

The paper investigates the landscape of base-$k$ sequence classes by introducing strongly $k$-recursive sequences and establishing their relation to $k$-automatic and $k$-regular sequences. It proves that every $k$-automatic sequence is strongly $k$-recursive but the converse fails, and shows that strongly $k$-recursive is a proper subclass of $k$-regular. A key part is the explicit family $g_{k,\ell}(n)$, which is $k$-regular and has varied synchronization properties (synchronized iff $k=\ell$), yet is not strongly $k$-recursive or $k$-recursive when $k\neq \ell$. The work uses Cobham-type arguments and automated verification with the Walnut prover to illustrate identities and to contrast behavior of these classes.

Abstract

Drawing inspiration from a recent paper of Heuberger, Krenn, and Lipnik, we define the class of strongly k-recursive sequences. We show that every k-automatic sequence is strongly $k$-recursive, therefore k-recursive, and discuss that the converse is not true. We also show that the class of strongly k-recursive sequences is a proper subclass of the class of k-regular sequences, and we present some explicit examples. We then extend the proof techniques to answer the same question for the class of k-recursive sequences.

Figures (2)

• Figure 1: $g_{2,2}(n)$ is $2$-synchronized.
• Figure 2: $g_{k,k}(n)$ is $k$-synchronized, $k > 2$.

Theorems & Definitions (17)

• Example 1
• Theorem 2
• Theorem 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6
• proof