A Fibonacci analogue of the two's complement numeration system

TL;DR

This paper introduces a Fibonacci-based analogue of the two's complement system, enabling addition via a finite-state transducer and establishing a bijection between integers and a specific language.

Contribution

It presents a novel Fibonacci-equivalent of two's complement notation and demonstrates how addition can be performed with a finite-state transducer, extending classical binary methods.

Findings

Addition in the Fibonacci system can be performed by a finite-state transducer.

The paper provides a new constructive proof for the Berstel adder.

It characterizes the Fibonacci two's complement as a bijection between integers and a language.

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.