A Fibonacci analogue of the two's complement numeration system

TL;DR

A Fibonacci-equivalent of the two's complement notation is introduced and it is shown that addition in this numeration system can be performed by a deterministic finite-state transducer.

Abstract

Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of the two's complement notation and we show that addition in this numeration system can be performed by a deterministic finite-state transducer. The result is based on the Berstel adder, which performs addition of the usual Fibonacci representations of nonnegative integers and for which we provide a new constructive proof. Moreover, we characterize the Fibonacci-equivalent of the two's complement notation as an increasing bijection between $\mathbb{Z}$ and a particular language.

Figures (3)

• Figure 1: The Berstel adder $\mathcal{B}$ is a 10-state Mealy machine with 30 transitions illustrated as solid edges with initial state $\mathtt{0}\mathtt{0}\mathtt{0}.0$. The Mealy machine $\mathcal{T}$ is obtained by adding a new state start that replaces $\mathtt{0}\mathtt{0}\mathtt{0}.0$ as initial state and adding three additional transitions shown with dashed edges.
• Figure 2: The language $\mathop{\mathrm{rep}}\nolimits_{\mathcal{F} c}(\mathbb{Z})$ can be represented as a tree of integers where each word $\mathop{\mathrm{rep}}\nolimits_{\mathcal{F} c}(n)$ labels the path from the node start to the node $n\in\mathbb{Z}$. The labeling of the nodes by integers illustrates a breadth-first traversal of the upper and lower half of the tree.
• Figure 3: The edges represent the directed labeled graph $G$, which can be folded (after merging equivalent states) into the Mealy machine $\mathcal{Z}$ equivalent to the Berstel adder $\mathcal{B}$. A vertex reached from a path $u$ has an ellipse shape if and only if $u$ is minimal with respect to the radix order within the equivalence class $[u]_\equiv$ and has a rectangle shape if $u\equiv v$ for some word $v<_{\mathop{\mathrm{\rm rad}}\nolimits} u$.

Theorems & Definitions (54)

• Theorem A
• Lemma 2.1
• proof
• Definition 2.2: Neutral prefix
• Proposition 2.3
• Definition 2.4: Fibonacci complement numeration system
• Definition 2.5: Pad function
• Definition 2.6: Numeration system $\mathcal{F} c$ for $\mathbb{Z}^2$
• Definition 2.7: Sum of two words
• Lemma 2.8