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in reply to Re: about Coolness, Impatience and Complexity
in thread about Coolness, Impatience and Complexity
Good point. Turing machines can do any polynomial-time computation in polynomial time, but they might take a higher order polynomial length of time than some other type of machine. I was trying to compose a snappy reply, and got sloppy.
Still, when we talk about "polynomial time", what we really mean is "polynomial in the length of the problem using a reasonable encoding scheme." Not "polynomial in the number of entities in the input set," though we sometimes ignore the distinction. If the point and the line in your geometrical computation are specified with 512 bit numbers, that's a larger problem than if they were only 16 bits. You can't reduce that - it's information theory.
Any physical machine you build is going to have a finite number of bits of processing power available to it. We can't make computers out of infinitesimally thin Euclidian line segments. A quantum computer might be able to consider a virtually unlimited number of possibilities at the same time, but it has a limited number of qubits and they have to collapse down into one answer before it can be printed out on the screen.
If we start talking about machines that are impossible to build, are we still doing computer science? I don't say this to be facetious, but just to ask what computer science really is. |
Re: about Coolness, Impatience and Complexity
by Abigail (Deacon) on Jul 13, 2001 at 13:59 UTC
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Still, when we talk about "polynomial time", what we really mean is
"polynomial in the length of the problem using a reasonable encoding
scheme." Not "polynomial in the number of entities in the input set,"
though we sometimes ignore the distinction. If the point and the line in
your geometrical computation are specified with 512 bit numbers, that's
a larger problem than if they were only 16 bits. You can't reduce that -
it's information theory.
No, what we mean is "polynomial in the size of the input". That could be
bits if your computational model asks for that. If your computational
model is geometry, then your input might consist of points and lines.
Not bits. Bits don't play part in geometry. They might play a part in an
implementation of the model, but not in the computational model itself.
Yes, this requires a level of abstraction, but that's why we call it
"science". ;-)
Any physical machine you build is going to have a finite number of bits
of processing power available to it. We can't make computers out of
infinitesimally thin Euclidian line segments. A quantum computer might
be able to consider a virtually unlimited number of possibilities at
the same time, but it has a limited number of qubits and they have to
collapse down into one answer before it can be printed out on the screen.
All true, but not really relevant. If we have an O (f (N))
geometric algorithm (I graduated in the closely related fields of
datastructures and computational geometry, hence bringing up geometry
over and over again) and we want to implement it on a real machine,
we might have to multiply or divide by a factor of log N (or
something else) somewhere to get a running time expressed in the
number of bits of the input. [1] But we'll have the same
conversion for another algorithm. Hence, the relation of running time
(and space) complexities in one computational model stay the same in
the other, assuming you use the same implementation. Of course, in a
typical implementation numbers are limited to 32, 64,
128 or some other number of bits, which is a constant disappearing in
the big-Oh anyway.
If we start talking about machines that are impossible to build, are we
still doing computer science? I don't say this to be facetious, but just
to ask what computer science really is.
Well, it was you who brought up Turing Machine. ;-) Now, have
you ever seen a Turing Machine? Not only are they impossible to build,
if they would exist they are quite impractical to do real computation
on. But much of the theory of Computer Science is founded on Turing
Machines. Take away Turing Machines, and Computer Science no longer is
the Computer Science we all know.
Do not forget the words of one of the most famous Computer Scientists
of all times, Edsgar Dijkstra: computer science is as much about
computers as astronomy is about telescopes. Computer Science is
a big misnomer; the French word is much better: Informatique
(I hope I spelled that right), which is described by the French
academy of Science as the science of processing, possibly by using
automata, of information, where information is the is knowledge and
data. I'm sure it sounds a lot better in French. ;-) There are a few
European departments that use the term "Computing Science" when
translating their department to English. I prefer Computing Science
over Computer Science as well.
[1]I'm using a lot of handwaving here. An algorithm
on N numbers each consisting of M bits (and hence having size
N*M) doesn't need to have the same running time when it's run
on an input of M numbers each consisting of N bits (which also
has size N*M).
-- Abigail
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Now, have you ever seen a Turing Machine? Not only are they
impossible to build, if they would exist they are quite
impractical to do real computation on.
Bosh. The only reason we can't make a perfect Turing machine
is that we can't actually give it an infinite length of
tape. We have a well developed theory that tells us what
computations can and can't be done by a Turing machine with
a finite length of tape. In many cases, it's easier to talk
about one with an infinite length, so we do that instead,
but at least we know what we're getting ourselves into.
Sorry, but I don't agree with Dijkstra. It's true that
astronomy is not about telescopes. Astronomy is about stars
and nebulae and planets and things, and telescopes are just
the tools that let us study them. But computer science is
about computing, and things that compute are computers, by
definition. So computer science is all about computers -
not about PCs and Macs and Sparc workstations in particular,
but about computers in general.
And what is computing? Pushing information around. The
flow of information is governed by information theory, which
tells us that information is very nearly a physical thing.
We usually deal with information in the form of bits, but
it doesn't have to be stored that way. A given quantity of
information is going to take up a finite minimum amount of
time and space, no matter what you do with it. Getting as
close as possible to the minimum is an engineering problem
that we'll be workng on forever. But it's important to
keep in mind that we're dealing with stuff that's not
infinitely compressible. Information is solid.
Of course, in a typical implementation numbers are limited
to 32, 64, 128 or some other number of bits, which is a
constant disappearing in the big-Oh anyway.
In many cases, you can make the number of bits disappear
into the big O, because the numbers don't need to be all
that large. However, the reason this conversation started
is that I was talking about implementing arbitrary
precision arithmetic, which is not one of those cases.
If I'm planning on using a raid array to hold the bits of a
number, I'd better not count on the number of bits
disappearing into a constant factor. Maybe I want to
multiply two 10-terabyte numbers together for some reason.
For you to claim that you have a geometrical computing
model that ignores the limits of information theory and can
do it with the same amount of effort as 2*2 is, quite
frankly, silly.
I don't think there's as much distinction as you think
between "computer science" and "computing science". Before,
I said, "why talk about computers that can't possibly be
built?" Now, I say, "why talk about computations that can't
possibly be performed?"
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We have a well developed theory that tells us what computations can and
can't be done by a Turing machine with a finite length of tape.
Please stick to the subject. The subject isn't computability, a domain in
which Turing Machines frequent, but run time complexity of algorithms,
of which you claimed where measured how fast they can be done on Turing
Machines - but then you later claimed that any device that couldn't be
build was irrelevant. But yeah, we could build a machine with a finite
tape. Now, how many of them do you have laying around?
But computer science is about computing, and things that compute are
computers, by definition.
Not everything that is about computing needs to be done by a "thing"!
Are you dismissing the majority of physics as well because the models they
use don't exist in the real world? It's Computer SCIENCE
not electronics.
And what is computing? Pushing information around. The flow of information
is governed by information theory, which tells us that information
is very nearly a physical thing. We usually deal with information in
the form of bits, but it doesn't have to be stored that way. A given
quantity of information is going to take up a finite minimum amount
of time and space, no matter what you do with it. Getting as close as
possible to the minimum is an engineering problem that we'll be workng
on forever. But it's important to keep in mind that we're dealing with
stuff that's not infinitely compressible. Information is solid.
So what? Pushing bits around is indeed an engineering problem. Just
like getting measurements of stars, nebulae and planets is an
engineering problem, and has nothing to do with astronomy. Computer
Science isn't about how to move bits around - Computer Science is
about where to move "the bits" to. And note that I'm
using quotes here, because Computer Science doesn't necessary deal with
physical bits. It all depends on your computational model what your
basic units are.
I don't think there's as much distinction as you think between "computer
science" and "computing science".
I'm sorry, but did I say there was any difference? I don't think so.
All I said was that some people think it's a better description of
the same thing.
Before, I said, "why talk about computers that can't possibly be
built?" Now, I say, "why talk about computations that can't possibly
be performed?
Sir, you have never been a scientist, you are not a scientist, and
you never will be a scientist. You just don't get it. If people always would
have had that attitude, we would still live in the stone age.
-- Abigail
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What is CS about?
I agree with Abigail, CS is not about computers. CS is
about how we work together to organize ideas and concepts
to be able to specify the operation of something incredibly
complex. It is not about the incidental details of what
is actually going to happen. It is about the principles
and details that humans need to understand to get it to
happen.
And so from the point of view of the programmer it is
fundamentally irrelevant whether the underlying
computation is being carried out by a Turing machine, a
RISC processor, an x86 chip, or a Lisp machine. For a human
to get things to happen it is essential that the human stop
thinking at a low level and start creating useful
abstractions. And understanding and manipulating those
abstractions means forgetting about the operations that
are being performed and thinking about the process of
humans trying to comprehend and direct the process.
You don't believe me? Well what is Dijkstra best known
for? Go To Considered
Harmful. Read it. It is a famous paper. When the ACM
decided to put some famous papers that had appeared in
their magazines online, it was the second one they chose.
It started a famous debate, and even people who have not
read and understood the paper know that goto is
something you are supposed to avoid.
You haven't read it yet? Please do. Or else what I am
about to say will make no sense.
OK, you just read one of the most influential papers ever
written in CS. And not just influential in the abstract.
It completely revolutionized the practice of programming.
If there is an essence to the study of CS, that paper of
all papers should reflect it.
What did it have to do with computers?
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