

Major unsolved problems in theoretical computer science? - riffraff
http://cstheory.stackexchange.com/questions/174/major-unsolved-problems-in-theoretical-computer-science

======
Permit
Reading over these reminded me how poor I am at discrete mathematics. I've
taken one class so far, in which I did fairly poorly and I was wondering if
anyone had suggestions on where one could turn for practice or learning
material for this sort of math/CS theory?

~~~
liuhenry
My recommendation is "Concrete Mathematics", by Graham, Knuth, and Patashnik.
([http://www.amazon.com/Concrete-Mathematics-Foundation-
Comput...](http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-
Science/dp/0201558025))

It is excellently written, but by no means a light read. If you put in the
time and effort, you will come out on the other side with a great foundation,
and deep understanding and mastery of the topics.

However, it's definitely not an introductory book, and some background in math
will be very helpful.

I'd also check out these threads:

Ask HN: Best Mathematics book for complete noobie?
<http://news.ycombinator.com/item?id=755043>

Ask HN: good math books <http://news.ycombinator.com/item?id=665029>

Ask HN: Learning advanced math <http://news.ycombinator.com/item?id=1753029>

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dkarl
One answer links to a list of open questions in the lambda calculus. All the
others are defined in terms of computational complexity. As a math major,
computational complexity was the first topic in computer science that caught
my interest (as opposed to programming, which I'd enjoyed for years) but I
didn't know it was virtually the whole show! Are there open problems beyond
complexity theory and the lambda calculus?

~~~
riffraff
honestly, I have no clue what the boundaries are between disciplines and what
lies under the umbrella of "computer science", but I recall many open problems
related to distributed systems.

Although now that I think of it, those may also have been (re)phrased in terms
of computational complexity.

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jamesrcole
I see 'what is computation?' and 'what is information?' are not listed there.
I'm curious what HN thinks of these questions. Do you think they should be on
there or not, and why?

I think they should.

What makes something computation? What is it about a physical process that
means it is performing computation? We do not have good answers to this
question. (Turing Machines are a model of a computational device, not an
account of what computation is).

We often characterise computation as information processing, but precisely
what does this mean? What is information? There is such a lack of clarity
surrounding this notion which seems so fundamental to CS. People often think
that Information Theory provides an answer, but it doesn't really enlighten us
about what information is and what its nature is.

I think part of the thing here is that theoretical computer science gets
equated with maths, whereas these are not mathematical problems (or at least I
do not think they are).

~~~
bermanoid
_People often think that Information Theory provides an answer, but it doesn't
really enlighten us about what information is and what its nature is._

Information theory provides a definition. That definition turns out to be an
extremely useful abstraction in a lot of mathematical theorems, and to some
extent matches the intuitions we have about information.

If there's a problem with the definition, and it doesn't match up with your
idea of what information _really_ is, then it's easy to come up with new
definitions; if they're useful, they might even take off.

But it's very hard for a field to answer questions like "what is X, and what
is its nature?", because it's not clear what kind of answer would suffice, for
_any_ value of X.

~~~
jamesrcole
It provides a measure of quantity, not a definition of what it is.

I don't think this is a matter of definition.

Here's an analogy. Back before we had an understanding of matter in terms of
atomic details (or anything deeper than that), people could have disputed
about what is a liquid and what is a solid and what is a gas. Different people
might have given different definitions of these in terms of qualities that
differentiated them from each other ("solids maintain their shape", "a gas
does not weigh much" etc).

But we now know that this question was not a matter of definition but a matter
of understanding what was actually going on underneath the phenomenal
appearances.

Understanding what a liquid is not a matter of definition, it's a matter of
understanding how the phenomena actually works. And in this case we have even
been able to show the categories we thought existed weren't even literally
different categories.

Returning to the question of what information is, the idea that it's just
something to be defined in one of a number of ways is an assumption.

~~~
jaekwon
I could ask you what matter is, what an atom is, or what space / time are, and
I wouldn't be satisfied with your definition of matter -- no matter what. I
just assume that our universe is running in a Turing machine, and that all
laws are derived mathematically. I bet we could find simple emergent
computational phenomena that model the laws of relativity. It's simpler than
trying to work backwards like you, and it lets me sleep at night.

It's programs and programmers all the way down. Or up, if you're Kardashevly
inclined. The neat thing about information is that it can probably pass
through boundaries of cosmological scale. The neat thing about intelligence is
that it emerges, and that it _wants_ to pass through boundaries of
cosmological scale. I feel that general / special relativity has something to
say about this -- that these laws permit some sort of communication across
scales, and that the universe is but a bunch of astronomers and molecular
physicists nerding it out. The photons that are helping you read this text is
actually composed of subatomic Voyager Golden Phonograph Records with Carl
Sagan's signature on each one. Billions and billions of them, into a sea of
intergalactic neurons causing a glorious dawn of awakening in your head.

\- copyright 2012 Jae Kwon

~~~
jamesrcole
_I could ask you what matter is, what an atom is, or what space / time are,
and I wouldn't be satisfied with your definition of matter._

I've said it plenty of times already in this thread, but I do not think
questions of "what is X" are answered by giving a definition.

The are _empirical_ questions, and your job is to try and figure out as best
you can how the details work. Reality chooses what the answers are, it's not a
matter of you choosing the definition you like.

~~~
jaekwon
I agree with you, but I'm suggesting that on the other end of the axis, away
from empirical observation, is our ability to use information to affect
destiny and cosmology in a grand (or not so grand) scale. I doubt that reality
exists independent of intelligence, which makes reality a difficult beast to
understand, like a naturally intractable problem. And if you insist on
observing empirically, you're just collapsing cosmological scale invariant
alternatives, but you can never figure it out completely. In fact it probably
just keeps getting more complex as you keep trying, even considering the
occasional unifying theories.

Anyways, I agree with everything that you've said, I just don't believe we can
fully know how the phenomena actually works -- there are infinitely many ways
in which it could work, all the time. To observe phenomena empirically in
order to figure out how things _further_ work, is to further isolate your
existence into a remote section of the possible universes (and _sometimes_
open your known universe up to new opportunities). I prefer to spend my time
making reality rather than being chosen by it, which feels more like an active
enterprise. In the end it doesn't matter. Have faith in science, or your
ability to shape your world, or whatever that engages your computer noggin. :)

\-- copyright 2012 Jae Kwon

