Church-Turing Thesis Cannot Possibly Be True [video]

https://www.microsoft.com/en-us/research/video/church-turing-thesis-cannot-possibly-be-true/

by espeed • 8 years ago

174 147 8 years ago

olliej10• 8 years ago

Are there Paper/slides anywhere?

I shall repeat my complaint about AV only material on HN: not everyone (wants to/can/has time to) watch and listen to 10 minutes of reading spread out to 50 minutes.

It continues to frustrate me.

gabrielhn• 8 years ago

Listen at 2x speed and skip ahead?

syrrim• 8 years ago

listen at 2x speed

I'm trying 1.5x, but then when he is speaking it is too fast

skip ahead

I skipped the introduction, but the standstill periods are too short, and you have no way of knowing when something interesting will happen, so skipping ahead entails missing things. This is where a copy of the slides is particularly useful: you can identify those slides you don't comprehend, or of particular interest, and sit through the speaking just for these portions.

olliej10• 8 years ago

Doesn’t work for people who can’t listen.

Doesn’t work for people who can’t see.

Doesn’t work for people in environments where the above two are essentially true.

25 minutes is still longer than 10 minutes (transcript, slides would be even shorter although less informative), and I was being conservative about reading speed.

drb91• 8 years ago

How do you scan forward for the interesting stuff? How do you search for a quote a friend sent you? What if you’re deaf? Or blind and can’t read the slides?

Just transcribe it and include the slides! If you can afford the speaker you can afford to make it accessible.

smhost• 8 years ago

here are the captions printed out: https://txt.fyi/+/933087e8/

nabla9• 8 years ago

I think these are relevant:

http://web.eecs.umich.edu/~gurevich/Opera/209a.pdf

http://web.eecs.umich.edu/~gurevich/Opera/210.pdf

olliej10• 8 years ago

Can we replace the article link with these? :)

azhenley• 8 years ago

Would be cool if a web app existed that could take a video and produce not only a transcript but also a series of intelligently selected images from the video, basically producing a full document that stands alone.

olliej10• 8 years ago

I feel like transcribe + pull out slides would be a generally useful thing for conferences. Even a specialized piece of software that has the original slides and just records when slide changes happen and inserts the appropriate slide into the transcription.

I’m convinced this would help conference videos that frequently do just have points where just the slide is shown (WWDC videos have this a lot - where frequently the speaker isn’t being shown). There should clearly be a standardized video codec for exactly this ;)

Rainymood• 8 years ago

Pay me and I'll do it for you. If there is enough demand I'll look for ways to automate it.

marssaxman• 8 years ago

Seriously! I don't understand why people waste so much effort presenting information in such a high-overhead, low-bandwidth format. Wouldn't it be easier to just post the notes and skip all the AV folderol?

titzer• 8 years ago

An echo of the Youtube effect; it's less work for the producer. Just turn on the cameras and start babbling, no editing, annotating, summarizing, or other further work required.

(tbc not faulting the speaker here).

dragonwriter• 8 years ago

Yeah, speed aside, audio-video is much better for conveying emotion than information.

hinkley• 8 years ago

My coworkers are currently on this same bent and it drives me mad.

They’d rather make a video than write an architecture document. And what happens when someone has a question or complaint about the middle of the video? How do you come back in a week to review the second to last part?

Learn to take screenshots and do some light typesetting. Spell it out for people (in my case, they are literally getting paid to do the work).

TimTheTinker2• 8 years ago

Overview pasted here:

————

The thesis asserts this: If an algorithm A computes a partial function f from natural numbers to natural numbers then f is partially recursive, i.e., the graph of f is recursively enumerable.

The thesis has been formulated in 1930s. The only algorithms at the time were sequential algorithms. Sequential algorithms were axiomatized in 2000. This axiomatization was used in 2008 to prove the thesis for sequential algorithms, i.e., for the case where A ranges over sequential algorithms.

These days, in addition to sequential algorithms, there are parallel algorithms, distributed algorithms, probabilistic algorithms, quantum algorithms, learning algorithms, etc.

The question whether the thesis is true in full generality is actively discussed from 1960s. We argue that, in full generality, the thesis cannot possibly be true.

berti• 8 years ago

We argue that, in full generality, the thesis cannot possibly be true.

If you could post a text summary of this part, we'd be very grateful.

TimTheTinker2• 8 years ago

I don't have the time to put that together, but here's a somewhat cleaned up (i.e. times and some newlines removed) paste of the closed captions. This may be helpful if you're good at scanning a lot of text.

https://gist.github.com/RoyTinker/1a7e27cb4dbd468b45586a380f...

olliej10• 8 years ago

I’m still unclear on whether probabilistic algorithms are “algorithms” in the strict sense - in that definition an algorithm should have deterministic behavior, which a probabilistic method does not have. But let’s say your “random” is a deterministic prng- does that then make the PA a deterministic algorithm?

Basically the distinction can be best illustrated with bozo/bogo sort. The definition many people use is:

1. Randomly shuffle the input

2. See if it is sorted, if not go to 1.

The problem, as explained to me by an algorithms lecturer a long time ago, is that the algorithm is not guaranteed to finish, and more over the same input may or may not result in it finishing on different runs.

Per that lecturer a “correct” bozo sort is

1. Generate a list of every permutation of the input

2. Iterate the list until you find a sorted entry and return that.

It has computable best case, worst case, and average case (all terrible). It is guaranteed that the same input will always produce the same output (terminate vs not terminate).

Obviously you could optimize this if you had an algorithm that could shuffle an input such that you were guaranteed that if you shuffled it’s own output you were guaranteed to produce every ordering of the original input. Then you would get closer to: shuffle; check; repeat. But still note that there is no random involved :)

hyperion20104• 8 years ago

I think that the intuition here has to do with the dimensionality of the problem, in the sense that we have proved that in 3 dimensions a random walk will never return to it's starting point, bit in two dimensions it will. Mapping all probabilistic algorithms into demensional random walk space is probably extremely difficult though.

rcfox2• 8 years ago

Do you have a better source than "an anonymous lecturer once said..."? I've never heard such a strict definition of algorithm before.

Randomized algorithms are often used to avoid falling into worst-case scenarios and improve run-time, like with quicksort, or to avoid settling in local minima like with simulated annealing.

rdlecler1• 8 years ago

Technically speaking, even physical computers have error rates and are, therefore probabilistic. With large distributions, your mean approaches your expected value -- the same kind of analytical result you get from boolean logic. Similarly, biological systems run on probabilistic algorithms and the fact that two twins starting from the same DNA can be so remarkably similar after millions of steps it a testament to the power of probabilistic algorithms.

rocqua3• 8 years ago

See [1], which is about the 'complexity class' Bounded-error Probabilistic Polynomial time. Similar to the P=NP conjecture, there is a BPP=P conjecture. Also consider [2], which concerns variations of the Church-Turing thesis. Notably, it mentions probabilistic computation.

To answer your original question, indeed probabilistic algorithms are an extension of normal algorithms. One way to make the extension (and retain the property that algorithms give the same output on the same input) is to add a 'random oracle'.

[1] https://en.wikipedia.org/wiki/BPP_(complexity) [2] https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis#V...

hyperion20104• 8 years ago

Without having looked at the details, why do I feel like all of the other types of algorithms should be able to be shown to be equivalent to the sequential case under the assumption of local realism? That is to say, if you don't consider that an algorithm must be implemented in this universe then I suspect that many things are possible. Am I missing something here?

sometime• 8 years ago

Conversely, those algorithms which cannot be converted to sequential possibly only run on exotic Turing Machines like ones with infinite bands or infinite parallelism and those likely don't have a physical equivalent because it looks like our universe only supports finite computation. In other words, the algorithms which violate Church-Turing might not be interesting at all, or only for analysis in an idealized setting.

daveFNbuck12• 8 years ago

These days, in addition to sequential algorithms, there are parallel algorithms, distributed algorithms, probabilistic algorithms, quantum algorithms, learning algorithms, etc.

These can all be converted into sequential algorithms with some overhead. Is there an actual proof in the video or just an argument that something that can't be made sequential must exist?

macleginn• 8 years ago

As was proved recently, there are quantum algorithms that can not: https://www.quantamagazine.org/finally-a-problem-that-only-q...

jonathanstrange5• 8 years ago

A parallel OR will terminate even if one of the subclauses does not terminate, whereas sequential OR might not terminate in this case. So if you define an algorithm as a computation that terminates, parallel OR cannot be fully emulated by sequential OR.

The problem is that, as far as I know, sequential algorithms form a uniform class, because e.g. equivalence between different sequential models like Turing machines and the untyped lambda calculus can be proved. AFAIK, this is not possible for parallel models of computation, there are different models of them with different expressivity. (I'm not an expert on that, so take this with a grain of salt and please correct me if I'm wrong!)

mwilcox• 8 years ago

What about the Church–Turing–Deutsch principle? https://en.wikipedia.org/wiki/Church%E2%80%93Turing%E2%80%93...

tlb• 8 years ago

That one's pretty aspirational. At present, we have no idea how to build a supercomputer that simulates exactly even a cubic nanometer-femtosecond of space. It's tempting to believe that we might someday figure out that physics is digital, but there is no workable theory for how.

kowdermeister2• 8 years ago

Not sure what your definition of simulation, but there is already one done on a supercomputer:

https://www.youtube.com/watch?v=J3xLuZNKhlY

scythe• 8 years ago

The talk begins:

ANNOUNCER: So it's my distinct pleasure, and privilege, to host Yuri for a talk on a topic I have discussed with him since I was even a student: models of computation. And through the years, we have interacted in various ways, on both fundamentals of models of computation and then abstract state machine applications, and at this point, it's also probably the last talk that Yuri will give at this institution, as he is moving to Michigan next week. But I have no doubts that there will be many opportunities to interact with Yuri in the future. I hear that there's a birthday party coming up in a couple of years, and it was always a pleasure to participate in these events. So, without more adieu, please go ahead.

YURI: Thank you very much. It's a pleasure for return for a day.

[First slide appears]

My scholarly friend Nachum Dershowitz and I published a long paper
"A natural axiomatization of computability and proof of Church's thesis", Bulletin of Symbolic Logic, 2008
where we proved Church's thesis for classical algorithms.
Here classical is not the counterpart to quantum as is common in quantum literature. By classical algorithms we mean traditional algorithms from antiquity to Turing's time, also known as sequential algorithms.
The claim that the Church-Turing thesis cannot be true in general is implicit in:
"What is an Algorithm?" SOFSEM 2012, Springer LNCS 7147 (2012)
The main goal of this talk is to clarify things and argue for the claim.

YURI: In 2008 Nachum Dershowitz, my very learned scholarly friend and myself, we wrote this paper, we proved Church's thesis, and Turing's thesis, but on very technical-- they are slight differences, it's a technicality -- for classical algorithms. Now what do we mean by classical algorithms? Now quantum people hijack the word "classical" -- it became to mean non-quantum, but our meaning was classical in the sense: algorithms from time immemorial, certainly, until 1950s, 1960s. So, this is the algorithm that Church and Turing had in mind when they put forward their theses. So, another name for these classical algorithms was traditional algorithms, or sequential. I will call them mostly sequential, because the first sort of non-classical, non-traditional algorithms were parallel, and so there was this -- there was traditional or non-sequential. You can generalize, and generalize, then it's not completely obvious why it should be even true.

YURI: I had this argument implicitly published years ago, but as you see:

[Second slide appears]

There is a a problem with long scholarly papers: it takes time to read them, and there is a tendency to skim.
Peter Shor: The Dershowitz-Gurevich paper says nothing about probabilistic or quantum computation. It does write down a set of axioms about computation, and prove the Church-Turing thesis assuming those axioms. However, we're left with justifying the axioms. Neither probabilistic nor quantum computation is covered by these axioms (they admit this for probabilistic computation, and do not mention quantum computation at all), so it's quite clear to me these axioms are false in the real world, even though the Church-Turing thesis is probably true.
Nov, 22, 2010. https://cstheory.stackexchange.com/questions/what-would-it-m...
The great hero of quantum computing has not read the paper.

===========================================================

I hope this will help you decide whether or not to make time to watch the video, although I agree it would be helpful to have a complete transcription available.

joe_the_user5• 8 years ago

This seems like really exotic stuff.

It's one of those talks that seems to talk about things I know but then immediately jumps to theories and definitions I have never heard of. "Sequential algorithms were axiomatized in 2000" - This "axiomization" was apparently performed by the author of the talk, my Googling says. All links about things talked in this talk just seem to lead to other texts and such by this author. Might not be a good sign but maybe it's just a very hard to grasp realm.

Anyway, broadly, the Church-Turing thesis is not a theorem and cannot be proven true or false (so said my professors and Wikidepia). Effectively, it is a definition of computation that has so far been accepted because nothing fundamentally different and more powerful than a Turing machine has been exhibited (there is a theorem that Turing machines, partial recursive functions and the lambda calculus are equivalent - and then there's the jump to "these are really general and I think that about covers it" but still, not a theorem, not provable).

My hunch is the author is mixing in "higher-order" logic "version of Church-Turing" into their argument. Which is basically "cheating", switching to an effectively different question without making things clear. However, I simply don't know. This is basically opaque to my mere MA-level maths.

joe_the_user5• 8 years ago

OK, yeah, reading more[1] so I think the author is indeed considering things like computation on real numbers where, for example, the value of square root of two can only be approximated by a Turing machine but where ruler-and-compass construction hypothetically can construct this exactly.

So if you toss out the idea of computation involving digital computers, you can exhibit something not (exactly) Turing computable. If you have a chaotically behaving dynamic system and call it a computer, then you have something where computation becomes something in the eye of the beholder.

I'm not sure what questions this answers but there you are.

[1] https://www.researchgate.net/publication/221512843_What_Is_a...

perl4ever• 8 years ago

"the value of square root of two can only be approximated by a Turing machine but where ruler-and-compass construction hypothetically can construct this exactly."

Ruler and compass are made of atoms though. Or, if not atoms, planck units or something.

daveFNbuck12• 8 years ago

A Turing machine can compute the square root of two exactly. It can't represent it exactly as a fixed point number, but that's not the only way to compute something. There's a pretty standard definition of computable numbers [1] and it includes the square root of two.

[1] https://en.wikipedia.org/wiki/Computable_number

fnrslvr• 8 years ago

The burden is on the specifier of the compass and straight-edge algorithm to explain a reasonable mechanism for carrying out the steps exactly, to infinite precision. Without this, the procedure can only be expected to give a reliable approximation of sqrt(2) to some finite precision, which a Turing machine can match. A similar observation applies to chaotic systems.

In general, a chaotic system can only be said to "solve" a problem if it can be configured to reliably arrive at (an encoding of) the solution for a given instance, despite any chaotic behaviour exhibited by the system. It wouldn't be reasonable to, for example, claim that a Turing machine cannot solve the problem of predicting the behaviour of a chaotic system with some particular initial conditions, if you couldn't reasonably conifgure the chaotic system itself to do that either.

remcob• 8 years ago

In some sense the ruler-and-compass constructions are even more computable than the natural numbers: Any (first-order) statement in Euclidean geometry is either true or false and there is an algorithm (on a Turing machine) that can compute it. Natural numbers don't have this property due to Gödel incompleteness.

In practice, anything you do with a ruler and compass has an equivalent representation in polynomials that is exactly computable.

Al-Khwarizmi• 8 years ago

Not an expert on the Church-Turing thesis, but I suppose to find a counterexample, one has to be rigorous in defining the problem. And defining the problem includes defining the output.

If the problem is "compute and output the decimal representation of the square root of two", then neither the human nor the Turing machine can do it in finite time, because that decimal representation is infinite. In fact, as far as I know this falls out of the scope of the Church-Turing thesis because most definitions of algorithms include that they must run in a finite number of steps, so there is no algorithm to do that.

If the problem is "compute and output a symbolic representation of the square root of two" (in some symbol system where it is finite), then the human and the machine can both do it.

If the problem is "compute and output a segment whose length is sqrt(2) times the length of this segment"... I don't think that's what computation is about. You might as well claim the Church-Turing thesis is false because a Turing machine cannot wave its hand.

jonathanstrange5• 8 years ago

Ah, okay, so it boils down to the question whether real numbers are physically realizable.

There is a debate about that in the philosophy of physics and mathematics. AFAIK, according to the current state of physics real numbers cannot be realized within a finite amount of space, at least Gisin [1] recently argued that way and I've seen other arguments for this view before [2,3,4,5:167-170]. I also remember some argument based on thermodynamics but cannot find the reference.

[1] https://arxiv.org/pdf/1803.06824.pdf [2] https://arxiv.org/abs/math/0411418 [3] https://news.ycombinator.com/item?id=14080024 [4] https://ptolemy.berkeley.edu/~eal/pnerd/blog/are-real-number... [5] https://www.sciencedirect.com/science/article/pii/S030439750...

tathougies2• 8 years ago

Square root of two represented exactly in Haskell (turing equivalent):

newtype Real = Real { isGreaterThan :: Rational -> Bool }

sqrt2 :: Real sqrt2 = Real (\r -> r * r < 2)

Doing arithmetic on these numbers is a more complicated matter. But this representation is exact.

rocqua3• 8 years ago

Effectively, it is a definition of computation that has so far been accepted because nothing fundamentally different and more powerful than a Turing machine has been exhibited.

Note that there is a commonly accepted extension that is more powerful than a Turing machine: Probabilistic Turing machines. I'm guessing the proof of equivalence generalizes to partial recursive functions and the lambda calculus if you add a 'random oracle'.

An interesting counter-argument is that real random oracles don't exist, and that in reality, we can simulate probabilistic computation using Pseudo Random Number Generators. So:

* True probabilistic computation isn't possible * And anyway, deterministic computation gets close enough to probabilistic computation through PRNGs

Alternatively, one might argue one could build a 'real' random oracle by reading from a 'true' random source. E.g. Cosmic microwave background, atmospheric noise, quantum fluctuations or a video of lava lamps. There is a difference between a random source and a random oracle, but with sufficient storage, that can be patched.

yters• 8 years ago

Any PTM can be simulated by a DTM.

OscarCunningham3• 8 years ago

More powerful in what sense? The computable functions are the same for either type of machine.

tom_mellior3• 8 years ago

the Church-Turing thesis is not a theorem and cannot be proven true or false

This is not quite correct: It cannot be proven true. But it could be proven false with a counterexample. The speaker would simply have to point to a concrete algorithm that can be performed by a person with pencil and paper but not by a Turing machine (or vice versa).

I haven't had the time to watch the video, but from the comments here it appears that the speaker did not provide such an example.

lmm• 8 years ago

The speaker would simply have to point to a concrete algorithm that can be performed by a person with pencil and paper but not by a Turing machine (or vice versa).

Humans do all sorts of things that Turing machines don't (e.g. a Turing machine can't hold a pencil), so the question there is what constitutes an "algorithm" which comes right back to the definition of computation. Even an example that was obviously "computation" in the intuitive sense wouldn't make the Church-Turing definition "false", it would just mean that it defines something different from what we thought it defines.

dooglius3• 8 years ago

If theoretical physics can find a complete model of all particles and spacetime, and we can show that a Turing Machine can emulate it, then that would suffice.

otabdeveloper1• 8 years ago

The speaker would simply have to point to a concrete algorithm that can be performed by a person with pencil and paper but not by a Turing machine (or vice versa).

No, this is a circular definition. We define an "algorithm" as something that can be performed by a Turing machine. So the Church-Turing thesis is a trivial tautology. (Though of course a useless one.)

andrewla2• 8 years ago

He does in fact provide at least one counterexample (around 18:55 in the video).

One is the real-number GCD algorithm; basically the same as Euclid's standard GCD algorithm, but with real numbers. He seems to be arguing that since Turing machines cannot represent real numbers, they cannot compute this algorithm. On the other hand, it can be easily computed with pen and paper, and he gives an example of a ruler-and-compass algorithm for evaluating it.

To me this seems a little wishy-washy; a Turing machine can approximate the problem to an arbitrary degree of precisions, just as a pen-and-paper approach is only capable to executing to a finite degree of approximation. Still, an "idealized" ruler and compass does provide a framework for describing an algorithm which is not directly Turing computable.

joe_the_user5• 8 years ago

This is not quite correct: No, my original statement is absolutely correct.

Church-Turing is an approach, a labeling, a thesis, not a theorem. Not being a theorem, it is not subject to proof.

One could exhibit a plausible argument that Church-Turing is wrong but that argument would not be a proof. It would not involve a sequence of mathematical argumentation that is standard for those artifacts which we call proofs. Just no.

Pointing to a concrete algorithm would not be a "proof". Maybe it would be very persuasive and the world would be persuaded to abandon Church-Turing. Still it would not be a proof.

Edit: I will admit my claim here involves a "mathematicians' thesis" that proofs are mathematical objects rather than just extra-strong arguments. So be it. I believe the whole reason Church-Turing is labeled a "thesis" rather than a theorem was because it's framers were using this distinction. And the distinction appears in the foundation of mathematics. Philosophers still argue for and against the continuum hypothesis even it's well not to be provable or disprovable - the argument is whether assuming it's true or falsehood produces good or bad mathematics. etc.

austincheney2• 8 years ago

I am so not a math person. From a math perspective perhaps this is important. From a computer perspective it completely misses the point.

This work from Alonzo Church gave us https://en.m.wikipedia.org/wiki/Lambda_calculus which in turn provides recursive function abstraction and thus lexical scope. While this model of computation is older than OOP it currently seems to be having a modern renaissance in its influence of language design.

austincheney2• 8 years ago

This many downvotes without any explanation is a true Reddit moment.

espeed• 8 years ago

NB: Better to refrain from saying stuff like "it completely misses the point" unless you are certain you understand the point beyond the level presented.

starpilot• 8 years ago

I understood all of it.

abdulhaq• 8 years ago

A Turing machine can compute root 2 but it takes an infinite amount of time to do so. A ruler and compass can do it near instantaneously, but it takes forever to measure it.

comboy• 8 years ago

It also takes an infinitely precise ruler and other tech which at some level of precision is not in any way more trivial than an infinite amount of time.

You can compute as many digits as you want, but when trying to measure it, at some point Heisenberg (or Planck to be more precise) will stop you.

rain1• 8 years ago

This is pretty misleading and click-baity! The formulation they use is quite esoteric and not what people usually think of the term to mean. The Church-Turing Thesis really just states that turing completeness is the strongest type of computational class we can expect to build (finite approximations of) in our universe. And this does seem to be true. I hope the "exciting" title of the talk they chose doesn't cause a lot of people to misunderstand this.

Verdex_3• 8 years ago

There is a series of talks by Yuri Gurevich on Channel 9 where he talks about algorithms. While I haven't watched this particular video (yet), I did watch the other series a few times (3 videos IIRC). Anyway, in those videos Yuri seems to basically be saying that he doesn't think all algorithms from now to eternity will always be representable with turing machines. Which seems a bit more reasonable than the title here (Church-Turing cannot be true). It's a different way of saying when an old scientist says something is possible he's probably right and if he says something is impossible then he's probably wrong (only inverted ... ie saying church turing will never be irrelevant is hedging your bets against human progress).

All that being said, I don't buy what Yuri is selling. Even quantum computing can be simulated with a classical computer (with exponential slowdown), so I'm not seeing what's going to turn out to be more fundamental at least in a theoretical sense. Practically people might not care (yes technically you can simulate that graphics card in lambda calculus, but practically you would never do that and the guys building these things probably don't care about the church turing thesis), but the effective computability definition feels pretty effective. Either way, I'm not going to loose too much sleep over it. At the end of the day I'm more interested in looking for ways to be more effective and I'm not really interested in whether or not what I'm doing fits into the current definition of effective computability.

avodonosov3• 8 years ago

FWIW, Yuri Gurevich is famous for the Abstract State Machines formalism.

avodonosov3• 8 years ago

This talk is Yuri Gurevich's answer to Peter Shor's (famous for Shor's algorithm) comment [1] regarding a previous Gurevich work. In that previous work Dershovich and Gurevich prove the Church-Turing thesis formally, using an axiomatisation of what "algorithm" is, which captures the meaning assumed in Church and Turing times.

But Shor complains their axiomatisation does not include probabilistic algoriths, quantum algorithms.

In this talk Gurevich argues that in full generality of the term "algorithm" the thesis can not be true.

1 - https://cstheory.stackexchange.com/questions/88/what-would-i...

stevebmark• 8 years ago

Why would you share this? It's incomprehensible.

Church-Turing Thesis Cannot Possibly Be True [video]

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