Why Philosophers Should Care About Computational Complexity (2011) [pdf]

http://www.scottaaronson.com/papers/philos.pdf

by eru • 10 years ago

177 54 10 years ago

scottaaronson.com

alberto_ol• 10 years ago

already subitted 8 times. last time https://news.ycombinator.com/item?id=9061744

eru6• 10 years ago

Thanks! I was actually surprised this got accepted as a new submission.

klez• 10 years ago

That was more than a year ago. Is there a (hard) rule regarding reposts?

From the FAQ

If a story has had significant attention in the last year or so, we kill reposts as duplicates.

So having more than a year passed I guess this is ok?

Tomte• 10 years ago

Yes, it is. And it wasn't meant as criticism, I think.

The pointers to past discussions are helpful if there was a real discussion then.

I strongly dislike all those "but look, it has been submitted before (but nobody ever commented)", because it adds absolutely nothing, but in this case it's different.

sp332• 10 years ago

You can click the "past" button under the link to see all the times it has been submitted.

eru6• 10 years ago

Also compare the The Ghost in the Quantum Turing machine (http://www.scottaaronson.com/papers/giqtm3.pdf) by the same author, and his marvellous blog at http://www.scottaaronson.com/blog/ .

There's also a slatestarcodex post about Scott Aaronson: http://slatestarcodex.com/2015/01/01/untitled/

ExpiredLink• 10 years ago

Or is it rather that mathematicians should care about philosophy?

conceit• 10 years ago

Many do. Foundations of mathematics is a specialized but deeply philosophical topic. Many would be interested orthogonal to maths, too.

robinhoodexe• 10 years ago

Not related to the content of the PDF, but I seriously dislike the width of those margins. While LaTeX look marvelous as always, there's just too much text per page.

eru6• 10 years ago

PDFs are a pretty stupid format for anything but printing on paper. (And even there, postscript is better.)

Alas, they are sort-of a standard.

tome• 10 years ago

I highly recommend okular. It trims whitespace around the margins.

stared9• 10 years ago

As an ex-physicist (quantum physics, now - data science), I am still puzzled why computer scientist consider computational complexity something metaphysical. Or even worse - a rule which universe should care about; look up considerations of non-local boxes and computational complexity (a generalization of quantum correlations; see Popescu-Rohrlich box).

First, even the CS's nightmare (where P=NP) may mean little practical difference (e.g. the simplest 'NP' takes O(n^1000)).

Second, for many problems even O(n^2) is too much (think of big data problems). (Also, given that computation requires physical resources, you cannot scale it arbitrarily.) So even in P the actual exponent matters a lot.

One interesting, physical case is the spontaneous emission of a photon from an excited atom. It looks like an exponential decay, but the theory shows that the decay have to be polynomial (I don't remember it's power, if more like 1/t^3 or 1/t^6, but not too high). Up to my knowledge, we haven't detected the polynomial tail (because when it is, the probabilities are so low).

So, is there a physical effect that cares about computational complexity? Or a phenomenon in which it matters that we can simulate it in "only" O(n^100), not O(2^n)?

amelius• 10 years ago

As an ex-physicist (quantum physics, now - data science), I am still puzzled why computer scientist consider computational complexity something metaphysical.

Please don't generalize one or a few computer scientists to the whole group.

By the way, a similar thing can be said for the halting problem. We may not be able to prove mechanically whether any program halts, but if we can do it for any program appearing in practice, that may be good enough. Still the proof is of immense theoretical value.

stared9• 10 years ago

Please don't generalize one or a few computer scientists to the whole group.

s/computer scientist/some theoretical computer scientists

(You are right that I missed a quantifier; this regex would also fix a typo.)

Still the proof is of immense theoretical value.

I enjoy mathematics, also in its pure form. Sure, there are many things nice, valuable and beautiful. Some are of practical use.

Yet, misusing it is a problem. Some otherwise great minds (like Roger Penrose) make some questionable claims about consciousness based on unjustified implications for the real world.

eru6• 10 years ago

We may not be able to prove mechanically whether any program halts, but if we can do it for any program appearing in practice, that may be good enough.

Alas, we can't even do that. Eg I have a simple program that systematically looks for a counterexample to Fermat's Last Theorem, and halts if it finds one.

We know now that this simple and comparatively natural program will never halt. But the proof is crazy.

johncolanduoni2• 10 years ago

First, even the CS's nightmare (where P=NP) may mean little practical difference (e.g. the simplest 'NP' takes O(n^1000)).

Sure, it may. But there are precious few examples in the wild where we're dealing with crazy powers like that. I think the bigger (but still quite small) failure in P's ability to measure "ease" of computation is that it only considers the worst case. A good example is graph isomorphism. There was recently a development of a quasipolynomial algorithm for graph isomorphism, which had the best bound we've found for the problem yet. However, with our existing algorithms it was extremely difficult to find examples where they didn't run in polynomial time, even when actively looking for them. But again, this is the exception, not the rule.

Second, for many problems even O(n^2) is too much (think of big data problems). (Also, given that computation requires physical resources, you cannot scale it arbitrarily.) So even in P the actual exponent matters a lot.

Very true, but TCS spends a lot of time developing new algorithms to shrink exponents as well. A lot of the methods and skills needed for making better algorithms to get problems into P also apply to moving them around within P. And many of the same tactics for trying to figure out exactly how P and NP fit together apply to trying to figure out the same for P and BQP.

Since you have a background in quantum physics and (I assume) care about that field, are you aware that quite a bit of the TCS community works on quantum problems? There have been plenty of results that stand to be very physically relevant. Some good examples are the recent (quite rapid) progress in analyzing symmetric cryptography schemes' vulnerability to attacks by a quantum computers[1]. Interestingly, many are starting from the same jumping off that asymmetric attacks did; namely Simon's algorithm (an important stepping stone towards Shor's algorithm).

[1] https://arxiv.org/abs/1602.05973

stared9• 10 years ago

Don't take me wrong - TCS is an important discipline, with both interesting practical and theoretical results.

The only thing I am criticizing here is the _metaphysical_ implications of computational complexity classes (and also, jargon which is misleading for not TCS people - e.g. O(n^100) s not "efficient" or "fast"). Furthermore, as you pointed, some problems have practical solutions which usually work efficiently, even if not always or give approximate, but still - useful, solution.

There have been plenty of results that stand to be very physically relevant

What I meant was not "can be applied to physical systems" (I agree!), but "is a foundation for physical laws" (I am highly skeptical and I am yet to see an example).

(Though, to be honest, I was more in quantum info that quantum computing; and right now I am more into... quantum JavaScript: http://quantumgame.io/)

gone352• 10 years ago

So, is there a physical effect that cares about computational complexity? Or a phenomenon in which it matters that we can simulate it in "only" O(n^100), not O(2^n)?

Firewalls[1,2], for starters.

[1] http://arxiv.org/abs/1301.4504v4

[2] http://arxiv.org/abs/1402.5674

stared9• 10 years ago

Thanks! Any summary? (I admit, GRT is not something I am deep into.)

todd8• 10 years ago

Interestingly, Knuth believes P=NP. See his interview, in particular his answer to question 17:

http://www.informit.com/articles/article.aspx?p=2213858&WT.m...

im3w1l• 10 years ago

"The universe believes in encryption." - Assange

Most cryptography rests on open problems so I don't think the question is settled yet. In either case, the answer is of immense importance.

chriswarbo• 10 years ago

I am still puzzled why computer scientist consider computational complexity something metaphysical.

In what sense do you mean "metaphysical"?

Personally, I would say computational complexity is both "physical", in the sense that it gives fundamental limits to what can/cannot be done in the physical world using some amount of resources (say, anything obeying Blum's axioms). I would say it's also "metaphysical", in the sense that those limits are derived purely mathematically, and are "imposed on" the physical world; rather than, for example, by observing the physical world and imposing consistency with those observations on some mathematical model. I think of these computational "laws of nature" in a similar way to the "law of nature" that e^(i*pi) + 1 = 0.

In other words, we can imagine (AKA develop a consistent mathematical model of) a world operating by Newtonian physics; we can do the same for a world operating by special or general relativity; and for the standard model; we can do it for worlds with N dimensions; with Euclidean, hyperbolic or spherical geometry; and so on. All of these "make sense", in a self-consistent way, and it's up to physicists to figure out which of those models best describes our physical world.

On the other hand, there don't seem to be self-consistent models of universal systems which can solve their own halting problem, or models which don't posess the hierarchical complexity classes we've found (although, of course, the machine models may differ, we may introduce hierarchies of halting oracles, etc.). In that sense, these seem to be "laws of nature" which are intrinsic to our models, by virtue of their mathematical definition, rather than being imposed from the outside in order to agree with observations.

ProfChronos• 10 years ago

The section about closed timelike curves is, I think, a good example of why computational complexity has metaphysical consequences but struggles to bridge the two - see what Aaronson writes about the "grandfather paradox".

nabla9• 10 years ago

So, is there a physical effect that cares about computational complexity?

The No-SuperSearch Postulate (there is no physical means to solve NP-complete problems in polynomial time) is not purely mathematical conjecture. It's also a claim about the laws of physics. The postulate includes P != NP as a special case, but it is stronger.

If you assume that the postulate is true, you can use it to explain things like why quantum mechanics is linear, why adiabatic systems have small spectral gaps, why timeline curves don't exist, black hole information paradox, impossibility of relativity computers, why QM-amplitudes are not directly observable. The question of how much physics you could derive from No-super-search postulate is interesting.

stared9• 10 years ago

And it is assumption I find... strange. If it really works, it would be interesting.

QM is linear for many other reasons (the simplest related to not being able communicate faster than light); smallness of spectral gaps is AFAIK a conjuncture.

And in general, playing with causality will bring many other problems.

pdonis3• 10 years ago

the theory shows that the decay have to be polynomial

I thought this was the other way around: the theory predicts exact exponential decay, but experiments aren't entirely clear that that is always the case. Depending on which textbook or paper you read, some say the deviations from exact exponential decay are just experimental error, while others say they are evidence of additional effects that the exponential decay law does not account for.

stared9• 10 years ago

It's clearly not that way you say.

From quantum field theory, you get polynomial decay and it's quite fundamental (related to causality, holomorphic functions (Kramers–Kronig relations)) and if it were not working, a lot of physics would need rewriting.

EDIT:

I remember it from a lecture, many years ago. Some googling brings me to this thread with links to papers:

http://physics.stackexchange.com/questions/7008/long-time-de...

stared9• 10 years ago

Wow, I got >=3 downvotes. Is the knowledge about non-exponential spontaneous emission so obvious?

eru6• 10 years ago

Alas, please disregard early downvotes.

jgeada• 10 years ago

Well, there is the Landauer's principle (which is related to Maxwell's daemon): there is a physical lower limit to the cost of computations. https://en.wikipedia.org/wiki/Landauer%27s_principle

So yes, there is a physical cost to non-reversible computation.

kanzure• 10 years ago

see also:

"Ultimate physical limits to computation" http://arxiv.org/pdf/quant-ph/9908043v3.pdf

"Limits on fundamental limits to computation" http://arxiv.org/pdf/1408.3821v2

"The physics of information processing superobjects: daily life among the Jupiter brains" http://diyhpl.us/~bryan/papers2/The%20physics%20of%20informa...

mmalone• 10 years ago

Personally, I consider it metaphysical because it says something about provability (since, deep down, proofs and programs are the same thing). Truth beyond proof (or at least beyond what's feasible to prove) is bizarre and mysterious.

Metaphysics (and physics) is concerned with uncovering the fundamental truths of the universe. Computational complexity indicates this may be unachievable.

Natanael_L• 10 years ago

Looks like this is extremely relevant - the Halting problem and Turing completeness applied to quantum physics:

http://phys.org/news/2015-12-quantum-physics-problem-unsolva...

http://www.nature.com/nature/journal/v528/n7581/full/nature1...

janan11• 10 years ago

Very interesting. I'll be honest, I expected this to be vague and self-important but it was very concrete and made a lot of discussed-to-death topics interesting again. He sort of lost me about halfway through the paper but I suppose that's because he has a PhD and I don't.

GlobalTraveler2• 10 years ago

Is there anyone on hacker news that could explain me secion 6 in this article? Overall the article is very interesting but I am not sure if the waterfall argument is valid.

The waterfall conjecture was stated as : 'Given any chess-playing algorithm A that accesses a "waterfall oracle" W, there is an equally good--chess-playing algorithm A', with similar time and space requirements that does not access W.'

From this we can assume that the article is a metaphor for us as external observer given semantics for computation. Thus by providing the reduction we showed that the computation in and of itself does not have semantics. However, why would we assume there to exists A'? Even if there is A', can't we reduce A -> A', and if we can reduces W to A, then by proxy W -> A'?

Could somebody help me?

solusipse• 10 years ago

Well, author tries to prove it's not. And, btw, oracle is something more than just an external observer, it can be used, for example, to solve the halting problem in classes for which it would be impossible with nondeterministic machines.

GlobalTraveler2• 10 years ago

In my hastly typing i forgot to add 'in' to valid.

Could you elaborate on the oracle approach? Because I interpreted it as being the external observer in this case. From computational complexity I know that an oracle solves NP problems in polytime by a nondetermnistic turing machine, but I am not sure how this oracle stuff loops back to the philosphical argument.

The proof provided by Aaronson is really opague as to what and how it proves that the argument is invalid.

eru6• 10 years ago

The main problem with that section is that the author tries to engage with a particular objection (the waterfall argument) that is itself a bit obscure.

bitL• 10 years ago

I think philosophers should focus on novel contemporarily-unfalsifiable ideas, like are we living in a simulation?; if you are a person outside time with instant access to all moments in the history (past-present-future), how much power do you have?; how would you create time (continuous step? adaptive resolution depending on what beings inside time decide to observe?) etc. Some of these ideas later might become falsifiable and move science forward, even if they sound super sci-fi or even insane right now. Or would be a nice background for a movie/game with some non-trivial thoughts already applied.

unabst2• 10 years ago

Unfalsifiability was a philosophical breakthrough first, and it is now the de facto test to see if an idea has any real world consequence.

To say philosophers should focus on unfalsifiable ideas is to say philosophers should be antipragmatic and inconsequential. One could easily argue that many are, but to encourage it? That's not necessary.

We need to focus on everything, and think about everything. We shouldn't ask "our philosophers" to "think" about things. We should all learn how to think and become better philosophers for ourselves, and find a way to openly contribute the progress of our thoughts.

gumby• 10 years ago

To say philosophers should focus on unfalsifiable ideas is to say philosophers should be antipragmatic and inconsequential.

Philosophy is the origin of all the sciences. Starting with Mathematics, as various topics became tractable they became self-standing disciplines, leaving the non-tractable ones behind. Thus inherently philosophers are the ones who work on the really hard problems.

(True, epistemology remains in philosophy but I would argue that the tractable parts moved into mathematics).

AI suffers from the same problem -- once something becomes well accepted people say, "well THAT isn't AI; AI is <some other intractable domain>".

LionessLover• 10 years ago

like are we living in a simulation?

What does that even mean? The entire question only makes sense from a human point of view.

Why Philosophers Should Care About Computational Complexity (2011) [pdf]

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