On 14 Feb 2014, at 21:12, meekerdb wrote:
On 2/14/2014 8:14 AM, Bruno Marchal wrote:
With some definition of the abacus, it is Turing universal. With
others it is not.
The slide rules is not Turing universal. You can add and multiply
approximation of natural numbers only, or, if you want, you can
analogically add and multiply the real numbers, and that is not
Turing universal. (That is not entirely obvious).
That's an interesting point to me (I own a collection of circular
slide rules). Of course you can add and subtract on a slide rule as
well as multiply, divide, exponentiate, and compute the value of
other functions encoded on the rule (sin, tan), but the rule doesn't
do it by itself; you provide the sequence of operations consisting
of reading a cursor and moving the rule. So why would that not be
Turing universal?
Because you work in first-order analysis, where you can add and
multiply real numbers, but still cannot differentiate
1,
0000000000000000000000000000000000000000000000000000001001011110111...
and 1.
Another way to see that slides rules are not universal is in trying to
define a compiler FORTRAN--> slides rules. In that case, ... well I
don't know, if you have the sin, you might be able to find the natural
numbers, by solving sin(2pi*x) = 0, but you will get analogical
natural numbers, and not clear digital 0, 1, 2, ...
May be slide rule + your moves + infinitely good eye sight might be
Turing universal.
May be slides rules can be Turing Universal "in a ring", in the sense
of Blum, Shub and Smale. I am not sure.
Bruno
Brent
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