Issue |
RAIRO-Theor. Inf. Appl.
Volume 57, 2023
|
|
---|---|---|
Article Number | 12 | |
Number of page(s) | 23 | |
DOI | https://doi.org/10.1051/ita/2023007 | |
Published online | 18 December 2023 |
A Fibonacci analogue of the two’s complement numeration system
1
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, 33400 Talence, France
2
FNSPE, CTU in Prague, Trojanova 13, 120 00 Praha, Czech Republic
* Corresponding author: sebastien.labbe@u-bordeaux.fr
Received:
10
January
2023
Accepted:
3
August
2023
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 ℤ and a particular language.
Mathematics Subject Classification: 68Q45 / 11A63 / 11B39
Key words: Two’s complement / numeration system / transducer / Fibonacci
© The authors. Published by EDP Sciences, 2023
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