A Fibonacci analogue of the two's complement numeration system
arXiv:2205.02574 · doi:10.1051/ita/2023007
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 and a particular language.
arXiv commentv3: 21 pages, 3 figures, 3 tables. v4: 24 pages, added a new section characterizing the Fibonacci's complement numeration system as an increasing bijection. v5: changes after review