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Chung-Graham and Zeckendorf representations

Rob Burns

TL;DR

The paper studies how Zeckendorf and Chung-Graham Fibonacci-based representations relate, using Walnut automata to formalize conversions and operations. It builds automata for Chung-Graham representations (cgval), CG addition (cgadd), and the split/convert pipeline to Zeckendorf forms (cgsplit, fibrepmsd, fibreplsd, cgrep), and verifies their interaction with the Zeckendorf system. It then shows that Fibonacci-synchronised functions, such as $f(n)=\\lfloor \\phi n\\rfloor$, can be transferred to Chung-Graham form via the cg bridge (fibcg), yielding a CG-synchronising automaton (cgphin). These results provide a rigorous, automated framework for comparing and composing multiple Fibonacci-based numeration systems and their synchronisation properties.

Abstract

We examine the relationship between the Chung-Graham and Zeckendorf representations of an integer using the software package {\tt Walnut}.

Chung-Graham and Zeckendorf representations

TL;DR

The paper studies how Zeckendorf and Chung-Graham Fibonacci-based representations relate, using Walnut automata to formalize conversions and operations. It builds automata for Chung-Graham representations (cgval), CG addition (cgadd), and the split/convert pipeline to Zeckendorf forms (cgsplit, fibrepmsd, fibreplsd, cgrep), and verifies their interaction with the Zeckendorf system. It then shows that Fibonacci-synchronised functions, such as , can be transferred to Chung-Graham form via the cg bridge (fibcg), yielding a CG-synchronising automaton (cgphin). These results provide a rigorous, automated framework for comparing and composing multiple Fibonacci-based numeration systems and their synchronisation properties.

Abstract

We examine the relationship between the Chung-Graham and Zeckendorf representations of an integer using the software package {\tt Walnut}.

Paper Structure

This paper contains 4 sections, 2 theorems, 6 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Chung-Graham representation in Walnut.
  3. Converting strings into Chung-Graham form
  4. Synchronisation

Key Result

Theorem 1

Any positive integer $n$ can be expressed uniquely as a sum of Fibonacci numbers with $a_i \in \{0,1\}$ and $a_i a_{i+1} = 0$ for all $i$.

Figures (6)

  • Figure 1: Automaton which accepts valid Chung-Graham representations.
  • Figure 2: Automaton which converts between the Zeckendorf and Chung-Graham representations.
  • Figure 3: Automaton which converts an msd binary string into the Zeckendorf representation.
  • Figure 4: Automaton which converts an lsd binary string into the Zeckendorf representation.
  • Figure 5: Synchronised automaton for the function $\lfloor \phi n \rfloor$ in Zeckendorf form.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: Zeckendorf's theorem
  • Theorem 2: Chung and Graham