Daniel Etiemble
2019 · arXiv
A demonstration that e=2.718 rounded to 3 is the best radix for computation is disproved. The MOSFET-like CNTFET technology is used to compare inverters, Nand, adders, multipliers, D Flip-Flops and SRAM cells. The transistor count ratio between ternary and binary circuits is generally greater than the log(3)/log(2) information ratio. The only exceptions concern a circuit approach that combines two circuit drawbacks (an additional power supply and a circuit conflict between transistors) and only when it implements circuits based on the ternary inverter. For arithmetic circuits such as adders and multipliers, the ternary circuits are always outperformed by the binary ones using the same technology.
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The concept of radix economy assumes that hardware complexity for each logic element is proportional to the number of logic levels. In practice this isn't true, and base 2 is best. >Ternary circuits: why R=3 is not the Optimal Radix for Computation https://arxiv.org/abs/1908.06841 Previous HN discussion: https://news.ycombinator.com/item?id=38979356
It's not so clear ternary would actually be an improvement: at least in terms of how we can implement circuits today, ternary's radix efficiency is offset by the inefficiency of implementing ternary logic. It might work if you have a new way of constructing logic circuits where this efficiency difference doesn't appear. (see https://arxiv.org/abs/1908.06841)
> Balanced Ternary can be mathematically elegant way to represent numbers When discussing ternary computation, I always see people talking about representation and storage. In these aspects, ternary can indeed be more elegant and efficient than binary. I never see any discussion about the complexity of the physical circuits, though. From the little I have seen (see [1]) it seems that using ternary circuits for computation will never be a good idea. Even though you'd need fewer wires to carry information, that advantage is more than cancelled by the increased complexity of the computation circuits themselves (those papers compare specifically binary and ternary adders and multipliers). [1]: https://arxiv.org/pdf/1908.06841.pdf https://arxiv.org/pdf/1908.07299.pdf
Also, this efficiency is generally offset by the increased inefficiency of the implementations of the operations[1]. For communication any number of symbols may be optimal depending on the nature of the channel, but for computation it definitely looks like binary is better, at least until we move to a fundamentally different way of constructing logic. [1] https://arxiv.org/abs/1908.06841
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