Balanced ternary, otherwise known as "maths for moties"? I'm afraid I
lack time currently[0] to decently investigate this system; for now,
just a few scattered impressions:
On the one hand, binary divisions can certainly be accused of
over-simplification. In the special cases where it's possible to make
a true binary decision, one has "not not a = a". But for most things
in life, things have an inside, and outside, and a bunch of stuff
that's right on the border between the two[1], and for these more
common ternary cases the strongest logic one can settle for is just
"not not not a = not a"[2].
On the other hand, the use of binary in computers isn't due to
historical accident alone. At the software level, one tries to
minimize branching[3] -- and a three way decision takes twice as many
decisions as a two way[4]. "Fine", one might say, "it's always much
easier to make highly parallel decisions in hardware than software".
But, here, too, it has a price: area is the figure of merit for chips,
and a generic binary operation on two trits would take 9/4's the area
of the equivalent op on bits. So, at least for computers, it's not
enough for a given number of digits to be shorter, but they'd have to
be shorter in proportion to the increase in resources required[5]. Are
balanced ternary numbers, to a given precision, under half the length
of the equivalent binary?
On the gripping hand, maybe there is some computational embodiment that
more naturally lends itself to 3x3 decision tables? Is it possible to
compute with triplet spin states? (consider this under the caveat that
it's all very well for Feynmann to wave his hands around and say "all
we need is a Hamiltonian that looks kind of like this...", but the rest
of us would be better off finding at least one such H before counting
too much on their existence)
-Dave
[0] still owing silk an explanation of what we are up to and why I have
taken a sabbatical from retirement (work is the curse of the thinking
classes...)
[1] consider Buridan's ass (asinus sapiens?) -- the medieval version of
Brouwer's fixed points. I've also run across an interesting slogan for
judges (originally due to the Louisiana courts, but since confirmed in
other jurisdictions), concerning the existence of legal fixed points.
Be just. If you can't be just, be arbitrary.
[2] applications of this logic to the ability to (not so) lossily
"round-trip" data between different formats, or to the implementation
of Thompson's compiler trojan, is left as an exercise for the reader.
[3] especially with modern architectures. Speculative execution is all
well and fine, but in the presence of branches, it's grabbing an
exponential by the wrong end.
[4] one way -- no decisions -- is even better, but not usually
possible. Part of "vectorizing" code involves recognizing when there
are enough patterns underlying a given possibility tree that it is, in
reality, only making one-way decisions, and hence really a constant
twig in disguise.
[5] Turing noted with this issue in specifying his eponymous tape -- he
pointed out that angels (who enjoy, it is assumed, working out
algorithms when they're not occupied in dancing on pinheads...) could
get by with a finite tape of one square with an infinitely complex
symbol, but that he, a mortal, preferred to deal with the case of an
infinite tape of squares with a finite number of different symbols.
