Hypercomputation: Computing more than the Turing machine (2002)
arxiv.org
arxiv.org
> Under the banner of "hypercomputation" various claims are being made for the feasibility of modes of computation that go beyond what is permitted by Turing computability. In this article it will be show n that such claims fly in the face of the inability of all currently accepted physical theories to deal with infinite precision real numbers. When the claims are viewed critically, it is seen that they amount to little more than the obvious comment that if non-computable inputs are permitted, then non-computable outputs are attainable.
The original idea in Turing's thesis was that the Turning Machine was something that could actually be implemented, and indeed you can find many examples of people's side projects where such machines can be made and can actually compute things.
On the other hand, hypercomputation has no physical basis. So what's the point of using it to model anything real? It's next to useless.
[1]: http://www1.maths.leeds.ac.uk/~pmt6sbc/docs/davis.myth.pdf
The author repeatedly makes the point that we don't know, for sure, that this is the case. And that "we haven't built it yet", sans additional explanation, is pretty shoddy evidence for the non-existence of something.
And that, if such a thing did exist, its existence may well be "very fundamental to the structure of the universe and should not be ignored".
I agree with both you and the author.
I don't think we'll ever build device capable of general-purpose "harnessable" hypercomputation. And I don't think any of the ideas described in this paper are in any meaningful sense physically realizable.
But I also allow the possibility that I am wrong, especially on the first count. And if I am wrong, even just on the first count, then some of the theory surveyed in this paper might turn out to be useful.
If you can't measure anything with arbitrary accuracy then that seems more than enough reason to assume you can't go beyond finite (digital) accuracy.
For example, classical field equations apparently can behave in non-computable ways, so long as their initial conditions are sufficiently wierd. I'm not sure if QM as we undestand it can handle such fields: but then "as we understand it" is a bit too limiting.
So why haven't we observed these wierd non-computable fields? My guess is that it's because they don't exist. But it might be that we just haven't recognized them, perhaps some of those puzzling things in physics -- like QM perhaps itself is caused by naughty non-computable stuff going on where we haven't yet recognized it.
Is it sort of like the n-body problem of classic chaotic behavior?
Regarding the first question, I am only repeating pop-science here.
I've read (In something by Roger Penrose, presumably The Emperors New Mind, but maybe in The Road to Reality) that the Maxwell Equations are uncomputable only if the initial conditions are functions that don't have Fourier transforms.
So really it is the field dynamics that are non-computable. I might have been talking nonsense when I the fields themselves were non-computable. But then again, the kinds of functions that don't have a Fourier transforms are also very rough, jaggedy things that probably can't be approximated by any algorithm.
And speaking of rough jaggedy functions, that's what you get if you evolve a chatoic system to infinity. So maybe there is some connection after all, but it is well beyond my understanding.
So that kind of computability might be the mathematical algebraic computability (as in does this monstrosity have a closed form?), which can always be approximated (of course with fast accumulating errors).
The Bekenstein bound (https://en.wikipedia.org/wiki/Bekenstein_bound) and the Holographic Principle (albeit the latter not being a concrete part of physics yet) provide strong reasons as to why analog computing is not a viable path to the realization of "hypercomputation". Besides, a few loopholes nonwithstanding (https://pdfs.semanticscholar.org/a4b2/4409635c442b45f6fa4271...), the physical existence of legitimate hypercomputation stands in almost direct opposition to the Church-Turing thesis.
[1] https://plato.stanford.edu/entries/computation-physicalsyste...
It's not claimed sans explanation. Hypercomputation would break a lot of physics. Others have mentioned the Bekenstein Bound, but just consider just basic concepts like signal to noise ratios. You'd need infinitely precise detectors for hypercomputation, detectors capable of distinguishing literally any signal, not matter how small, among any amount of noise, no matter how overwhelming.
These properties simply stretch credulity, to say the least.
(I can understand why we might not be able to build a Turing machine because we don't have infinite tape, etc, but if a program never halts but merely needs finite space, how do you prove that, in general?)
I would think that deciding if a loop halts is equivalent to the halting problem (just wrap your program in a loop). Merely detecting loops seems insufficient.
Of course all this takes at least exponential time and so is outside our normal computability assumptions.
We know that a program with n bits of memory has at most 2^n states, so we can maintain an n bit counter that is incremented once on each program iteration. If this counter reaches (2^n)-1, then the program does not terminate. (Depending on how you define things, there is an off-by-one error here; just run the program a few times before you start counting).
That's why it's impossible to solve.
If it were finite (bounded), then you could do an enumeration of the possible programs of that bounded size, and you could make a program that simply looks up that list. (Sure, maybe the making of the list would be a bit problematic, because it'd take ... multiple eons, lesser gods would die, rise and die again during that time, but it's still finite.)
Noting this a few more states than a handful are possible. For example my laptop with bitpacking could track the visited or not visited flag for 64000000000 states without even using disk. Tracking a 33 bit FSM.
I agree though that linear bounded Turing machines having a solution for the halting problem is not actually that useful for real computers considering by the time a few registers had been iterated over the sun would have exploded.
To be clear, these things have formal definitions, and this statement is not correct.
1. A linear bounded automaton[0] is a Turing machine that can only overwrite symbols presented on its input tape as input. However, the definition of the automaton is still required to be finite, but it is required to operate on unboundedly large input tapes.
2. A finite state machine[1] is a model of computation where the sequence of input symbols are observed once and the machine is at all times in one of a finite number of states. It is equivalent to a TM that can only move right (and consequently cannot read anything it writes to the tape).
They are different, formally[2]. There are languages that a LBA can accept that a FSM cannot accept (famously, the strings of balanced parentheses cannot be recognized by a FSM).
[0]: https://en.wikipedia.org/wiki/Linear_bounded_automaton
[1]: https://en.wikipedia.org/wiki/Finite-state_machine
[2]: https://en.wikipedia.org/wiki/Pumping_lemma_for_regular_lang...
Mathematically, this sort of simplification is used a lot: when our model requires some sort of bound, we can work under the assumption of it being 'sufficiently large' that we can ignore edge-cases. Another example is the set of "real" numbers, which we can represent as decimals and assume a sufficiently large number of decimal places to avoid rounding. In fact, similar to the "tape factories" of a TM, we can think of each real number as having a "decimal-place factory" which produces new digits more quickly than we can read them (for example, during Cantor diagonalisation).
The infinities in hypercomputation don't seem to be providing such a simplification. Their 'sufficiently large' assumption seems to be the number of steps which can be executed in a unit of time, which avoids the edge-case of non-halting programs. I'm not sure that's a useful simplification.
Plenty of mathematics does not "model anything real", but you may still want to understand it. The paper starts with an analogy to non-Euclidean geometry, which once "had no physical basis", until we learned more about the universe and found that it did.
Mathematics isn't a pitch deck.
If anything is a problem it is the attractive name, that pushes non-technical journalists into paying attention to it.
Anyway, the bottom line was that it did an unbounded amount of work in a bounded amount of time.
I stopped paying attention to Dewdney after that.
EDIT: Whoops, that's pretty much the same thing as Zeno's paradox.
Isn't conventional wire delay RC quadratic?
The implementation of either is left as an exercise for the reader.
[1] https://en.wikipedia.org/wiki/Halting_problem
[2] https://www.amazon.com/Introduction-Theory-Computation-Micha...
But never say never. "Reality/logic/math" can be a lot stranger than you think.
There used to be a time when we thought that infinity ( aka countable infinity ) was the largest number. Turns how there are bigger infinities.
And of course quantum mechanics. The duality of light ( both a wave and a particle ).
I understand it's a Black Swan kind of problem, but I don't think we live in that reality. Please show me a contradiction. The ramifications would be stupendous, to say the least.
There used to be a time when we thought that infinity ( aka countable infinity ) was the largest number. Turns how there are bigger infinities.
Yes, and we know how to build Turing machines that use countable infinity. Please show me a machine that actually uses bigger infinities. Please hypercompute the digits of a real number that cannot be computed by a Turing machine.
And of course quantum mechanics. The duality of light ( both a wave and a particle ).
Universal quantum computers are faster than classical computers, but they are no more powerful (they are not known to be able to hypercompute).
Computer science is a fairly new field. It's within the realm of possibility that reality can be turned upside too. I'm not said it is or will be, but it can. Okay?
https://cs.stackexchange.com/a/4871
It reminds me of what Whitehead and Russell tried to achieve with Principia Mathematica, to try to eradicate paradoxes altogether using hierarchies of sets, until Gödel came along.
I use "solves" somewhat advisedly - we're postulating something that may very well not exist; it's just that it's not obviously inconsistent for it to exist in the way that it would be if it had an oracle for its own class of machine.
In this kind of context, you should provide a link. This kind of thing can spill over into "You're wrong because you haven't done the work to see that I'm right", and minimizing the work you're pushing off shows good faith as well as increasing clarity that we're talking about the same thing.
I think it's more likely we're talking about different things than either of us failing to understand the relevant proofs. I'll take a look, though, when you've provided the link (and ideally some explanation of what you see as important).
You can't use it to build a self-contradicting Turing Machine, because a hypercomputer cannot be emulated by a Turing Machine.
There is no "the" halting problem. Each class of machines has its own halting problem. The halting problem for turing machines is very hard. The halting problem for non-cyclic finite state automatons is very easy. A hypercomputer that can solve all halting problems is self-contradictory. A hypercomputer that can solve turing-or-weaker halting problems is not self-contradictory.
So yes, physical possibility is definitely a big factor here, even if it's not always stated explicitly. If it were impossible to build practical random-access memory, then that wouldn't be the standard model used for computational complexity.
Consider the big question of whether P=NP. The question itself is unrelated to physics, since P is defined as polynomial-time algorithms on deterministic Turing machines, and NP as polynomial-time on nondeterministic Turing machines. But the question gets a lot of attention in part because it's considered to be impossible to actually build a nondeterministic Turing machine. If that were untrue, then we could solve problems in NP in polynomial time even if P≠NP.
In theory, you could build something like a nondeterministic Turing machine (without infinite memory, but close enough, much like we consider real-world computers to be close enough to normal Turing machines) if you could, say, send information back in time, or if you could split the universe into arbitrary many parallel universes, then destroy all but one. Obviously, nobody has the slightest idea of how to do this, but at least there are some vague, crazy possibilities.
Hypercomputation is even worse. To accomplish it in anything like physics as we know it would require infinite time, space, and energy. In theory you can get infinite time by putting your computer into just the right orbit around a rotating black hole. Infinite space and energy are somewhat harder to come by. (You know you're in trouble when you need to do something harder than putting something into a precise orbit around a rotating black hole.)
There's always the possibility of new physics. Nothing says it's impossible for there to exist some fundamental particle which acts like a Turing oracle when you poke it just right. But the chances seem pretty poor.
In short, yes, barring any massive physics breakthrough, treat hypercomputation as a theoretical math construct, nothing more.
Please don't try it at home.
"The paradigm hypermachine is the O-machine....."
They claim to be able to solve NP-complete problems with polynomial resources using non-Turing architectures. I'd normally call them crackpots but they have serious pedrigree and have been published in Science and Nature. I'm still very skeptical, but don't have the chops to evaluate their caims.
tl;dr It's crap
> all proposals along these lines simply “smuggle the exponentiality” somewhere that isn’t being explicitly considered, exactly like all proposals for perpetual-motion machines smuggle the entropy increase somewhere that isn’t being explicitly considered. The problem isn’t a practical one; it’s one of principle.
"Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines
"We introduce a class of digital machines, we name Digital Memcomputing Machines, (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space, and energy)."
http://aip.scitation.org/doi/abs/10.1063/1.4975761?journalCo...
The actual paper is behind a paywall. I am not Scott Aaronson. I probably could not reasonably evaluate their claims even if I could read them. But Aaronson has somewhere posted a list of the weird things that would happen if you could solve an NP-complete problem in polynomial time, and that list provides a good reason to believe you can't.
A part of another abstract of one of their other papers (http://ieeexplore.ieee.org/document/7924376/?reload=true) is,
"Our method can be extended efficiently to any level of precision, since we prove that producing n-bit precision in the output requires extending the circuit by at most n bits. This type of numerical inversion can be implemented by DMM units in hardware; it is scalable, and thus of great benefit to any real-time computing application."
Which makes me suspect there are hard limits on the size of the problems their "hardware" can handle.
But it's fun to chase the graph thingys around on their home page.
Not sure why you'd sell that technology, instead of using it to effectively and invisibly conquer the world.
One of the most stressful parts of grad school in a new field (to me) was pouring over dozens of papers that I did not understand. I wish I had known then that you dont HAVE to understand every part of everything you read, especially when new to an academic field with few or no textbooks. Just be sure to watch how certain you are in your conclusions until you're comfortable with the material.
There's a famous problem in CS called the halting problem. The halting problem asks for a program that tells you whether an arbitrary program (turning machine) ever halts (a.k.a. stops executing). A function that could compute a solution to the halting problem is called a halting function.
Turning machines can't compute a solution to the halting problem.
Therefore, if you add an oracle for the halting problem to an otherwise normal turing machine, then the resulting model of computation is stronger than turing machines alone. And oracle is basically a magical box that answers your questions in O(1) time, nevermind how.
Merely assuming an oracle for the halting function is only one of many ways to build something that does things turning machines can't do.
For example, allowing infinities in various places where the classical theory of computation assumes finite sets also sometimes increases the computational power of machines. E.g., we normally assume that a turing machine's tape is uninitialized or initialize only a finite subset of the tape. But by initializing a turing machine with a carefully selected infinite tape, you can compute the halting function. Some other "now make the finite thing infinite and then code up the halting function" constructions can be found in this paper.
If you fix one of these various constructions, you can start to do all the classical CS theory stuff in the context of a slightly different machine. It's interesting to see results transfer, which break down, and what surprising new/nice results show up here but not in classical cs theory.
It's hard to say how useful any of this is because all of these constructions basically assume we already have at hand something that we have no idea how to construct in reality. I.e., the whole paper begins with "assume false" as far as us pragmatic engineering folk are concerned. But that's perhaps basically the same thing Euclidean geometers said during the emergence of non-Euclidean geometry, so perhaps this work will end up being as revolutionary as turing machines themselves and we just can't see exactly how yet.