Re: [math-fun] Could change of base (binary —> ternary) speed up computation?
Mike Speciner wrote: ----- What does c have to do with it? ----- Nothing, as far as I can tell. But a logarithm must have a base. ----- And why do you think a trit would take the same amount of space as a bit? ----- I was just counting storage *locations* for bits vs. trits. —Dan ----- On 06-Jan-19 16:59, Dan Asimov wrote:
Since useful quanputers might be a long ways off, maybe some other stunt could speed up current computation as we know it.
If trits could be stored, retrieved, and copied as easily as bits, there would be less *space* needed by a factor of of log_c(2)/log_c(3) = .6309+, where c = zeta(3).
Perhaps someone knowledgeable can estimate how much trits could actually speed up computation. Surely this has been much studied by now.
But the practical question is, Is there a practical way to build trit-based processors? Would it be easier if they could be as large as a room, or larger?
Mike Speciner wrote: ----- What does c have to do with it? -----
Nothing, as far as I can tell. But a logarithm must have a base.
It defaults to base e unless otherwise specified. Indeed, how do you define log_c(x) if not by log(x) / log(c)...?
But the practical question is, Is there a practical way to build trit-based processors? Would it be easier if they could be as large as a room, or larger?
As far as I know, *balanced* ternary can be implemented more efficiently than binary. Moscow State University had a machine called 'Setun' that ran on balanced ternary and three-valued logic, and when it was eventually replaced with an equally powerful binary machine, the latter was more expensive: https://en.wikipedia.org/wiki/Setun Whether or not this is still true with modern semiconductors is another question entirely, and one best answered by an expert in semiconductors. I suspect that even if balanced ternary turns out to still be better, no-one will build a balanced ternary computer (other than, say, a specialised coprocessor for deep learning) because so much software implicitly assumes little-endian binary computers with 8-bit bytes. Best wishes, Adam P. Goucher
On Sun, Jan 6, 2019 at 5:58 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Nothing, as far as I can tell. But a logarithm must have a base.
It defaults to base e unless otherwise specified.
Indeed, how do you define log_c(x) if not by log(x) / log(c)...?
I'd define it as the unique continuous function such that f(xy) = f(x) + f(y) and f(c) = 1. You can prove existence as the appropriate multiple of Int_0^x (1/x) dx. e doesn't have to show up unless you want to figure out what the constant you need to multiply by is. As an aside, how do you deal with sentences like the immediately proceeding one, which start with a mathematical expression, which means they can't start with a capital letter, which looks funny to my eye (even if typeset, so that the e is in math-italics). I usually rephrase; in this case, I might have said "The constant e" or "Euler's e" instead of just "e" to start the sentence. Is there some "style guide for mathematicians", and what does it say about starting a sentence with a formula? Andy
there’s the book “mathematical writing” by knuth, roberts, and larrabee, and if memory serves, i think it does say that starting a sentence with a mathematical symbol is to be avoided On Mon, Jan 7, 2019 at 6:42 AM Andy Latto <andy.latto@pobox.com> wrote:
On Sun, Jan 6, 2019 at 5:58 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Nothing, as far as I can tell. But a logarithm must have a base.
It defaults to base e unless otherwise specified.
Indeed, how do you define log_c(x) if not by log(x) / log(c)...?
I'd define it as the unique continuous function such that f(xy) = f(x) + f(y) and f(c) = 1. You can prove existence as the appropriate multiple of Int_0^x (1/x) dx. e doesn't have to show up unless you want to figure out what the constant you need to multiply by is.
As an aside, how do you deal with sentences like the immediately proceeding one, which start with a mathematical expression, which means they can't start with a capital letter, which looks funny to my eye (even if typeset, so that the e is in math-italics). I usually rephrase; in this case, I might have said "The constant e" or "Euler's e" instead of just "e" to start the sentence. Is there some "style guide for mathematicians", and what does it say about starting a sentence with a formula?
Andy
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participants (4)
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Adam P. Goucher -
Andy Latto -
Dan Asimov -
Thane Plambeck