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On the pseudorandomness of Parry--Bertrand automatic sequences

Pierre Popoli, Manon Stipulanti

TL;DR

This work investigates the pseudorandomness of a broad family of morphic sequences built from Parry–Bertrand numeration systems, focusing on even-order correlation measures. By blending automata theory with abstract numeration systems, it extends previous results to the $U_\beta$-automatic setting and shows that binary $U_\beta$-automatic sequences have large even-order correlations: $C_{2k}(\mathbf{s},N) \gg N$ for all $k\ge 1$, implying these sequences are not pseudorandom. The authors also demonstrate a genuine difference between even- and odd-order correlations in certain morphic sequences and provide an outline of the proof using a product automaton and carefully constructed morphisms. The findings illuminate the limitations of pseudorandomness in morphic/automatic frameworks and suggest directions for extending the approach to broader numeration systems and sequence classes.

Abstract

The correlation measure is a testimony of the pseudorandomness of a sequence $\infw{s}$ and provides information about the independence of some parts of $\infw{s}$ and their shifts. Combined with the well-distribution measure, a sequence possesses good pseudorandomness properties if both measures are relatively small. In combinatorics on words, the famous $b$-automatic sequences are quite far from being pseudorandom, as they have small factor complexity on the one hand and large well-distribution and correlation measures on the other. This paper investigates the pseudorandomness of a specific family of morphic sequences, including classical $b$-automatic sequences. In particular, we show that such sequences have large even-order correlation measures; hence, they are not pseudorandom. We also show that even- and odd-order correlation measures behave differently when considering some simple morphic sequences.

On the pseudorandomness of Parry--Bertrand automatic sequences

TL;DR

This work investigates the pseudorandomness of a broad family of morphic sequences built from Parry–Bertrand numeration systems, focusing on even-order correlation measures. By blending automata theory with abstract numeration systems, it extends previous results to the -automatic setting and shows that binary -automatic sequences have large even-order correlations: for all , implying these sequences are not pseudorandom. The authors also demonstrate a genuine difference between even- and odd-order correlations in certain morphic sequences and provide an outline of the proof using a product automaton and carefully constructed morphisms. The findings illuminate the limitations of pseudorandomness in morphic/automatic frameworks and suggest directions for extending the approach to broader numeration systems and sequence classes.

Abstract

The correlation measure is a testimony of the pseudorandomness of a sequence and provides information about the independence of some parts of and their shifts. Combined with the well-distribution measure, a sequence possesses good pseudorandomness properties if both measures are relatively small. In combinatorics on words, the famous -automatic sequences are quite far from being pseudorandom, as they have small factor complexity on the one hand and large well-distribution and correlation measures on the other. This paper investigates the pseudorandomness of a specific family of morphic sequences, including classical -automatic sequences. In particular, we show that such sequences have large even-order correlation measures; hence, they are not pseudorandom. We also show that even- and odd-order correlation measures behave differently when considering some simple morphic sequences.
Paper Structure (9 sections, 7 theorems, 14 equations, 3 figures)

This paper contains 9 sections, 7 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. On combinatorics on words
  4. On generalized automatic sequences
  5. Representation of real numbers
  6. Main results
  7. Linearly recurrent sequences
  8. Proof of \ref{['thm:main']}
  9. Discussion and future work

Key Result

theorem 1

Let $b\ge 2$ be an integer and let $\mathbf{s}$ be a $b$-automatic binary sequence generated by a DFAO with $m$ states. Then $C_2(\mathbf{s},N) \ge \frac{N}{b(m+1)}$ for all $N\ge b(m+1)$.

Figures (3)

  • Figure 1: A DFAO producing the sum-of-digit function $\mathbf{s}_F$ modulo $2$ in the Zeckendorf numeration system $F$.
  • Figure 2: A representation of the situation depicted in the proof of \ref{['thm:main']}.
  • Figure 3: The product $\mathcal{P}$ between a DFA for $L_{U}$ (top) and a DFAO for the morphic sequence $\tau(\mathbf{x})$ (right) where $U$, $\mathbf{x}$, and $\tau$ are all defined in \ref{['ex:example non Parry but Bertrand']}.

Theorems & Definitions (12)

  • theorem 1: MW18
  • theorem 2
  • proposition 1: Rigo14-2
  • theorem 3
  • proof
  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • ...and 2 more