| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 739.1 | | VINO::JMUNZER | | Fri Jul 24 1987 16:19 | 11 |
| It seems to me that computers only get to represent a finite number
of values, and that you could invent conventions for those values
to be considered integer, or integer_times_two_to_the_minus_something
(normal floating point), or real, or anything else.
For instance, we could represent rationals by storing numerators and
denominators. Or complex numbers by storing polynomials to represent
their roots.
John
|
| 739.2 | Representation is easy. Using them is harder. | SQM::HALLYB | Like a breath of fresh water... | Fri Jul 24 1987 18:09 | 11 |
| > Or complex numbers by storing polynomials to represent their roots.
Well, algebraic complex numbers anyway.
Of course you could also use pairs of rationals to represent intervals
of the reals as several systems have done. Or triplets (x,y,r) to
represent neighborhoods of radius r about the complex number x+yi.
For that matter, you could define internal symbols for pretty much
anything you see printed in a math textbook. _How_ you manipulate
them is another matter.
|
| 739.3 | The troouble is that computers don't do math | AKQJ10::YARBROUGH | Why is computing so labor intensive? | Fri Jul 24 1987 18:42 | 18 |
| This note should better be titled 'computer arithmetic philosophy'.
Note for starters that computers *do not perform arithmetic* over any infinite
domain, including the Integers, but only over finite rings of a bounded size;
and even that may not be entirely correct around the edges. Therefore
computers cannot perform arithmetic over the Reals.
On any computer, one can choose to represent a finite number of Numbers (in the
math sense); standard computer arithmetic circuits limit these to Rationals,
since every n-ary fraction of finite length is rational. With the standard
interpretation, is there any way of writing a correct procedure of the form
Boolean function is_real (x : Numeric)
IF <x represents a rational Number>
THEN return false
ELSE return true;
that returns 'true'?
|
| 739.4 | Incomputable madness | ANGORA::JANZEN | Tom LMO2/O23 2965421 | Fri Jul 24 1987 18:45 | 9 |
| Hey, wait a minute. Wasn't this the whole idea of a Turing machine and
the study of computability? The theoretical purpose of the Turing machine
wasn't to manipulate binary, like in all the magazine examples, but
mathematical symbols of all types, including integral signs, wasn't it?
So that it could be shown that there were incomputable problems, and
some assumed premises (Goedel?), right?
huh?
speak to me!
Tom
|
| 739.5 | All I have is this itty bitty 32 meg... | AKQJ10::YARBROUGH | Why is computing so labor intensive? | Fri Jul 24 1987 19:10 | 4 |
| >Hey, wait a minute. Wasn't this the whole idea of a Turing machine and
>the study of computability?
As I recall the definition, Turing machines operate on infinite 'tapes'...
|
| 739.6 | | CLT::GILBERT | Builder | Fri Jul 24 1987 19:54 | 5 |
| Within the constraints of memory limitations, computers can perform
arithmetic over the reals. If we allow the results of a computation
to be 'error -- exceeds implementation limits', and that result is
given infrequently for practical problems, then yes, we can say that
the computer performs arithmetic over the reals.
|
| 739.7 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Fri Jul 24 1987 23:49 | 6 |
| Computers are Turing machines. A few lines of code and a message to
the operator of the form "Mount next/previous tape." is all you need to
get past finite limitations. Supplying the media is your problem.
-- edp
|
| 739.8 | REad the book first, then talk | ANGORA::JANZEN | Tom LMO2/O23 2965421 | Mon Jul 27 1987 12:40 | 4 |
| "Computability and Unsolveability" published by Dover, makes clears the
true application of the Turing machine model was to derive theorems
on mathematical symbols, and the limits of provability.
Tom
|