Calculating A Theoretical Boundary To Computation 583
TMB writes "Lawrence Krauss and Glenn Starkman, astrophysicists at Case Western Reserve University (and in LK's case, author of a number of books including Physics of Star Trek), just submitted this nice little paper to Phys. Rev. Letters. It claims that in an accelerating universe, the existence of a future event horizon puts a fundamental physical limit on the total amount of calculation that can be done, even in an infinite time. This limit is much smaller than the traditional Hawking-Beckenstein entropy. Among other things, this implies that and Moore's Law must have a finite lifetime, here calculated to be 600 years, and that consciousness must be finite."
Roger Penrose (Score:3, Informative)
Re:Roger Penrose - linky link? (Score:2)
Re:Roger Penrose - linky link? (Score:5, Insightful)
Re:Idiot (Score:4, Insightful)
Re:Roger Penrose - linky link? (Score:5, Informative)
Ummm.... no. Godel proved that the axiomatic system of Russel's PM allows the construction statements which can neccessarily neither be proven true nor proven false. There are other axiomatic systems that can be complete and consistent; IIRC it was in fact Godel who proved that the first-order propositional calculus is complete and fully consistent. Godel's fork only attaches to systems that allow the construction of statements about statements; many propositional systems (like the first-order propositional calculus) do not.
Oy.... where to start? Kant's Critique of Pure Reason is nothing but 600 pages describing how people distinguish intuition from fact (though admittedly Kant was using "intuition" in a sense that we don't normally use it today). Descartes wrote his Meditations as an attempt to remove "intuition" (again, closer to Kant's sense of the word than ours, but still) from philosophy. Plato, of course, says nothing about the subject directly but narrates several dialectical processes about the subject.
Re:Shadows of the Mind (Score:5, Insightful)
I haven't read SotM, so I can't comment on that. My biggest problem with Penrose is that I personally am convinced that strong AI can be achieved, while Penrose believes that it is impossible. That does not mean that the book can't be good, but Penrose doesn't do a good job matching his arguments with those of his opponents.
He has a tendency to repeat arguments that strong AI supporters give, but in a way that they can be misunderstood, and then he misunderstands them and basically says they are stupid. Already in chapter one of ENM he starts ridiculing strong-AI supporters, without giving arguments. Later those arguments follow, but they are seriously flawed.
For example, while discussing Searle's Chinese Room experiment, he suggests that strong-AI supporters believe that "understanding Chinese" is in the book, and argues that it is stupid that a book can "understand" anything. But Turing's view (which is repeated by many others) is that the understanding is not in the book, but in the book + the human that reads the book. There is an emergent understanding of Chinese that comes from a book that describes how Chinese can be perfectly translated, and a human who strictly follows the rules that are written down in the book. This argument is mentioned by Penrose somewhere, but he just puts it aside as a very weak argument, that does not need refuting.
Writing like this annoys me immensly, and I suspect most my colleague AI researchers.
Incidentally, I think Penrose' work in mathematics is absolutely brilliant.
Re:Shadows of the Mind (Score:4, Insightful)
Penrose, however, simply denies that consciousness can be electronically replicated even in theory. When asked why, he waves his hands and shouts "Quantum physics!" Now while it's certainly possible that quantum phenomenon play some role in consciousness that can't be replicated by a computer, there's no real evidence for the position and Penrose is just arguing it a priori. It's sort of the metaphysical equivalent of Creationism.
Re:Roger Penrose (Score:5, Informative)
So, no, it isn't really like Penrose's work.
Re:Roger Penrose (Score:5, Interesting)
If a machine (human in this case) can simulate a single Turing machine, and a Turing machine can simulate it, then it is exactly as capable (though perhaps not as quick) as any other Turing machine.
The first part is easy to prove: Any student who has learned Automata Theory should be able to simulate a Turing machine in their head, though it will be VERY slow and tedious.
The second is harder, but there is no reason to think that a simulation of every particle that makes up a human, plus a small environment (air, ground, food, water) around her/him will successfully simulate consciousness. The fact that today's computers are not strong enough doesn't invalidate humans being bound to a Turing machine's capabilities.
Any Turing machine is computational, therefore if the applications of Turing's thesis to humans holds, humans, and every part of them, including consciousness, are computational.
As far as Heisenburg's uncertainty theorem and quantum mechanics goes, it can be inserted into the simulator using rand().
Godel's Incompleteness Theorem doesn't apply to Turing's Theorem. Godel is talking about that there exists inconsistencies in any sufficiently complex langage (ie., the statement "this statement is a lie."). It doesn't contradict Turing's Theorem, since to disprove Turing's Theorem, we'd need to find a Turing machine that is incapable of simulating another Turing machine. All Godel says is that there will be non-sensical or impossible states in any Turing machine, but the machine can still work. (the proof that they exist is that English syntax can be programmed into any Turing machine, and the "this statement is a lie." statement inputted into the machine).
And as far a philosophy goes, so what if I'm limited to 2^2^40 states. I'll never get anywhere near experiencing all of them in the life of the universe, assuming I live that long. And in the same way that computers can execute computer games with fantasy themes, a computation human has nothing interfering with dreaming, pretending, or religion (though it might point out the silliness of latter).
Re:Roger Penrose (Score:3, Informative)
Basically it is believed that it is very unlikely that you can do quantum simulation on a classical computer.
Re:Consciousness and Turing Machine (Score:3, Interesting)
All you have said is that each half of the brain is capable of consiousness independently. When both halves of the brain are present in a skull, then they act together via interconnections to produce a single consiousness.
You could say that complete brains consist of the
Re:Consciousness and Turing Machine (Score:3, Funny)
Re:Roger Penrose (Score:5, Informative)
So for all you
Re:Roger Penrose (Score:2)
a lot more science than you will ever do. It's got
to hurt to be called a nutbar by someone named
"TwistedGreen".
Re:Roger Penrose's argument is sound (Score:5, Funny)
Penrose is just a carbon chauvinist with a chip on his shoulder. I've never seen him once offer actual proof of any such conjecture, only carbonist assertions that he can magically understand something that silicon-based life cannot: our future silicon overlords have a special place in virtual Hell reserved for his uploaded consciousness.
Re:Roger Penrose's argument is sound (Score:4, Interesting)
Skolem's paradox basically boils down to the fact that within axiomatic set theory you can prove the existence of uncountable sets, but if a model exists for the theory then it is countable. Hence according to the axiomatic system, there exist uncountably many sets, but the system can really only manifest countably many sets. This is similar to how Godel showed that formal arithmetic cannot manifest proofs for every proposition the formal axiomatic system claims exist.
While Skolem's paradox is held amongst by many mathematicians as the first incompleteness proof, in my opinion, this proves something much more significant... it proves that axiomatic set theory is semantically inconsistent.
Set theory has been problematic from the start, and even after axiomatizing it in order to avoid simple inconsistency... set theory is still plagued with inconsistency of a different form: the formalisms don't hold any meaning - they are semantically inconsistent. What the formalisms say contradicts what the formalisms mean.
The problem is that lots of people don't understand math, and even many of those that do understand it, love it so much that they are unwilling to give up the flawed parts.
For me, mathemathics is constructive recursive mathematics
Exception (Score:5, Insightful)
Except, of course, for those using certain popular mind-expanding substances
Seriously though - it seems we are finding a new limit every day. Wasn't it last week that they theorized limitations on data storage, as well as data transmission speed?
Re:Exception (Score:4, Funny)
Dude...
I know, I totally stayed at a holiday inn express last night.
Re:Exception (Score:3, Interesting)
In fact there are limitations to everything. Even to our ability to determinie limitations.
Re:Exception (Score:2, Insightful)
Re:Exception (Score:2)
I feel the same way, however, in contrast, perhaps we need to know the limitations so instead of learning how to break them, we can find ways around them. If we rested on limitations set already, we'd be having great fun working inside the 640K box.
Re:Finite Consciousness doesn't follow (Score:4, Insightful)
Re:Finite Consciousness doesn't follow (Score:3, Insightful)
However, since the assumption that the mind is physical has not led to contradictions, and since humans understand physics better than theology (or whatever), scientists tend to stick with the atoms-are-all-you-get approach.
Nobody's arguing that materia
enough! (Score:4, Insightful)
Re:enough! (Score:3, Informative)
Law [m-w.com] 6 a : a statement of an order or relation of phenomena that so far as is known is invariable under the given conditions b : a general relation proved or assumed to hold between mathematical or logical expressions.
Re:enough! (Score:3, Interesting)
Exactly. Moore's law only works because it gave Intel (and these days, AMD, too) a goal for predictable release cycles. It has absolutely nothing to do with physics.
The idea of "consciousness" really doesn't either. No credible physicist would get involved in this k
Re:enough! (Score:2)
As such, it has no weight.
Unless you can weigh electricity.
Unfortunatly, I dont remember where I read this.
Re:enough! (Score:5, Informative)
I'm glad we know this now (Score:2)
Re:enough! (Score:5, Informative)
Re:enough! (Score:3, Insightful)
No it isn't. It gives an excellent measure of scale! The statement that Moore's law is limited by the universe to 600 years duration gives an idea of how unimaginably FREAKING FAST our technology is expanding at the moment. 600 years in the scale of all time is a really short time.
Also, it sorta shows how far we are from the limit in terms of what we are capable of at the moment.
J
Re:enough! (Score:5, Insightful)
I wish this discussion was in the scope of the computational power of my consciousness, though. I cannot begin to fathom the implications of matter being CREATED at the centre of the universe. I don't think I fully grasp my tax return filings (I'm filing in Poland this year), much less this discussion.
This was the old "Steady State" theory (Score:3, Informative)
The Slashdot effect horizon? (Score:4, Funny)
Sweet (Score:5, Interesting)
Re:Sweet (Score:5, Insightful)
If there's a limit to consiousness on the high end of an expanding universe then we should also be able to make educated guesses at the low end and then put a front time on the resulting "wave" of intelligence.
If we have a beggining of the "wave" then we should be able to make better educated guesses about the distribution of intelligence in the universe and possible level of advancement of any intelligent life we might find. We might discover, for example, that we're reletively advanced (came early in the wave) and that we're less likely to find more advanced life. On the other hand, we may find that we're late in the wave and thus likely surrounded by life much more advanced than us.
This could be a much better way of looking at extraterrestrial life than just guessing based on the number of stars.
TW
Distribution of Intelligent Life (Score:3, Interesting)
Physics of star trek (Score:3, Informative)
It's not a referer link, don't worry...
Re:Physics of star trek (Score:2)
Re:Physics of star trek (Score:3, Funny)
Infinite Wisdom? (Score:5, Funny)
So they are saying that, using fundamental physics and mathematics, they have proof that if somebody has infinite wisdon, the universe can not be expanding?
Re:Infinite Wisdom? (Score:5, Insightful)
Re:Infinite Wisdom? (Score:3, Informative)
Re:Infinite Wisdom? (Score:4, Interesting)
Paul
Re:Infinite Wisdom? (Score:3, Insightful)
All that aside, should God be the only infinite being, it rather bodes ill for the old 'Heaven' concept.
Re:Infinite Wisdom? (Score:3, Informative)
Re:Infinite Wisdom? (Score:3, Interesting)
How can eternal life be consistent with the second law of thermodynamics? All things tend toward disorder -- to create order requires energy, and an increase in entropy which outmatches the decrease in some localized area. Hence God must have an infinite supply of energy in order to give you eternal life. Where does this energy come from?
Re:That's what I call spooky action at a distance. (Score:3, Funny)
Re:Infinite Wisdom? (Score:3, Insightful)
The original calculation was made using a particular geometry of space-time. Assuming a different geometry, such as a connection to such an extension, would result in a different calculation.
Furthermore, its not clear your definition of "wisdom" is congruent to the definition of "consciousness" which is postulated in t
Re:Infinite Wisdom? (Score:3, Informative)
C: "I beat you senseless!"
A->C is not the same as ~C->~A.
A->C would be "If you look at me funny, I will beat you senseless!" taken as literal truth.
~C->~A would be "I will not beat you senseless if you do not look at me funny", and is not at all the same.
No, ~C->~A is "If I am not beating you senseless, you didn't look at me funny." Your statement above is ~A->~C, which, as you noted, is not logically equivalent to A->C.
Daniel
And in other news... (Score:5, Interesting)
This article contains a very large number of assumptions, which may well prove not to be the case (constant cosmological constant, no FTL communication/travel, no access to other universes etc. etc.). Still, an interesting intellectual exercise I suppose... ;-)
Re:And in other news... (Score:3, Insightful)
That's an interesting perspective. I haven't RTFA, but aren't those assumptions fairly reasonable? Considering that we have Einstein with a proof that faster than light is impossible, it doesn't seem unreasonable to assume it.
Maybe the articl
Re:And in other news... (Score:3, Interesting)
So yes, the assumptions (at least most of 'em) are pretty decent ones. As I said to a friend of mine once who was bitching about his CD player requiring gravity to operate, "yeah, and
Re:And in other news... (Score:4, Funny)
Re:And in other news... (Score:3, Informative)
What we have is a theory (GR) that says that conventional acceleration of a massive object to lightspeed requires infinite energy.
GR doesn't state that FTL communication via quantum entanglement (for instance) is impossible (though that effect is also yet to be demonstrated as an FTL effect). This effect was used to explain the "ansible" used in Orson Scott Card's books. FTL communication would completely invalidate the
Re:And in other news... (Score:3, Informative)
Re:And in other news... (Score:2)
I also don't believe it holds up in the face of quantum computing - one of the assumptions that is made is a maximum amount of information that can be moved across a channel in the presence of noise, from which they directly derive their limit. I'
arXiv reaches it's computational limit! (Score:5, Informative)
"Consciousness is finite?" (Score:3, Interesting)
Are we talking about the physical computational capacity of a headful of neurons, which is finite by definition unless you believe that the brain can somehow reach into unknown dimensions somewhat like early CPUs used bank shifting to increase their RAM range?
Or are we talking about the sensation we have of being alive, a sensation that is arguably simply generated by our brains as a mechanism to ensure our survival. Yes, the vaunted consciousness that reacts a full 1/4 second after the fact when we do most common actions such as crossing the road, kicking a ball, picking up a cup, or typing comments to Slashdot?
The definition of "consciousness" is seriously under debate and it's meaningless to discuss whether it's finite or infinite.
Most likely, consciousness is a sense, like sight or sound. Would you frame the discussion of your sense of smell in terms of computational power? No, me neither.
Mu.
Re:"Consciousness is finite?" (Score:2)
Well, I 'read' the article (skimmed over it, more like), and it has precisely this to say about finite conciousness:
In this case, if one treats consciousness, conservatively, as merely a form of computation, then one can derive a finite total lifetime for any civilization in an accelerating universe.
This conclusion results from the fact that in such a universe one ultimately has access to only a finite volume, even after an infinite time. In the case of actual conscious living systems, it is difficult to
Re:"Consciousness is finite?" (Score:2)
Err - Freudian slip there ... as suck -> as such ... :P
Re:"Consciousness is finite?" (Score:2, Funny)
In other words, they completely fudged the issue and just made up assumptions that suit their purpose.
I'm afraid that rather puts limitations upon their computations.
KFG
Re:"Consciousness is finite?" (Score:2)
Most likely, As we get better at observing interactions at the very small level between atoms and particles, we are going to find some very unexpected behaviors, some of which will explain the interface between consciousness, and the physical world we know around us. Then things will get very interesting indeed as it will then be possible to manfacture beings/
Re:"Consciousness is finite?" (Score:5, Interesting)
Or, even more likely, an emergent byproduct of highly complex strange loops and pattern matching that, unlike any sense, does not have an explicit biological presence.
Re:"Consciousness is finite?" (Score:4, Funny)
Re:"Consciousness is finite?" (Score:3, Interesting)
The brain is a (hugely complex) collection of mental tools. For instance some of the tools that are fairly unique to the human brain:
- sense of self
- language
- short-term memory
- long-term memory
- social feedback
- empathy
- self of justice
- sense of time
- abstract sense of danger
- sense of opportunity
All these mental tools (and many more I can't even name) will eventually be mapped and understood.
Conscious
assumptions (Score:2, Interesting)
Re:assumptions (Score:2)
Just out of interest, where else do you suggest?
Re:assumptions (Score:3, Insightful)
I consider the vast amount of storage required for the average human being to function. Try making a list of all the faces you would instantly recognise, celebrities, friends, family, work-mates etc. The storage for that alone would be immense.
Perhaps the brain is just the processor, acting on some transdimensional storage area. Its not totally crazy right?
To take this a step further, perhaps our entire consciousness is stored externally a
Re:assumptions (Score:3, Interesting)
A Fire Upon The Deep by Vernor Vinge (Score:4, Informative)
Strongly suggest you read Vernor Vinge's A Fire Upon The Deep - he develops a very interesting view of expansion of the universe and consciousness.
If you've not heard of Vinge before that isn't a big surprise, but he did write True Names as well - the very foundation of the cyberpunk/hacker genre. This is also a good read if you can actually locate it.
Moore's Law? It's not a theory, just a curve! (Score:5, Insightful)
All technologies seem to obey this general law. Software, chips, disk space, they all tend to zero.
Even a passenger jet costs a fraction of what it did 20 years ago.
Moore's Law turns this around to say that for the same price we can expect more and more capacity. Long before 600 years are passed, this capacity will effectively reach "infinite", being the point where no-one can use more capacity or power, no matter what the application. At which point Moore's Law will gently slow down.
Submitted, not accepted (Score:2, Informative)
Limits to pr0n? (Score:2, Funny)
Re:Limits to pr0n? (Score:3, Insightful)
Eye glaze (Score:2, Funny)
Infinity (Score:2, Funny)
Would anyone like some toast?
Re:Infinity (Score:2)
Good god, man! You've solved the problem already! Now it's just a matter of engineering [everything2.com]...
Allow me to point out a huge assumption (Score:5, Interesting)
"...consciousness must be finite."
This assumes that consciousness is based solely on computation. Not proven yet.
And for that matter, even if consciousness is nothing more than computation, how can we put a limit on an activity in space-time when we don't even know how space-time functions, or even how many dimensions it has?
Weaselmancer
The Question (Score:2)
well, duh (Score:4, Insightful)
so
Useful analogy (Score:4, Funny)
It's like at the bar -- the later in the night you attempt to pick up chicks, the fewer of them are still available.
Therefore God doesn't exist (Score:2, Interesting)
Re:Therefore God doesn't exist (Score:4, Insightful)
Re:Therefore God doesn't exist (Score:3, Insightful)
The Last Question (Score:5, Interesting)
There's also another theory about that if a couple particles collide with enough energy they can create a more perfect vacuum that would essentially "take over" the current universe (I suppose like an implosion). Maybe somebody knows the link for this.
I mention this as a backdrop for an interesting short story by Isaac Asimov called The Last Question [wikipedia.org]. This link is a summary and contains significant spoilers, you may want to read the story first [maddad.org] I think that it is apropos, as it deals with a powerful computer called Multivac.
This story is interesting to read, and interesting humanistic view. Good for pondering this slashdot thread/story. Good science fiction is useful.
Multiple Universes? (Score:2)
300 years is a very long time in terms of what technology can do. I'm sure in 100 years once we have AI that can think as well (or better) than we can, getting around these pesky universal limits will get easier.
This paper has not been published (Score:5, Informative)
Is computation discrete? (Score:3, Interesting)
The googleplex, a number that CANNOT be represented digitally (not enough atoms in the universe) can be easily represented by a particularly intelligent shade of the color blue.
Re:Is computation discrete? (Score:4, Interesting)
Er, no. At least, not within the known bounds of quantum mechanics. Your mileage may vary.
What we perceive as a continuously variable analog world just happens to have a *lot* of very closely-spaced discrete states. Each time you add volume, matter, or energy to a system you increase the number of available states by a large, but still finite, amount.
A wave could have a frequency of 1Hz, 1.1Hz, 1.00057Hz, 1.2399327772883786682676376627676367267Hz, etc. If "computation" is defined as "the act or process of evaluating with numerical or mathematical methods" then there is no physical limit to computation when using analog data storage...
Here we run into quantum mechanics once again. To take the measurement of the frequency of a wave, for instance--how do you resolve a difference down at the one part in a quintillion level? Essentially you run afoul of uncertainty principles. To reduce the uncertainty in your measurement of frequency to a low enough level to resolve such small differences, you have to pay a price in measurement time. Actually, you have the same problem when you write your data in the first place. See also my remarks about number of accessible states--storing an analog wave with a finite precision will require a certain amount of matter and energy, neither of which is available in infinite amounts*.
*probably...
Argh! Turn off SETI! (Score:4, Funny)
If there's only a finite amount of computation available, surely it's irresponsible to run things like SETI and the distributed.net cracking contests?! You're using up all of the sums, dammit!
All physical computers are finite state machines? (Score:3, Interesting)
My personal theory: (Score:3, Insightful)
Wham! There's you upper bound on computing (at least for "full" simulations).... now all you need to do is figure out how much mass and time is available in the universe
Note: I'm not about to propose this in earnest to the scientific community. It's just a casual musing of mine. CONSTRUCTIVE criticism is welcome.
Not True with Reversible Computing (Score:5, Interesting)
But computation doesn't have to be irreversible. There are various proposals on how to build reversible computers that don't consume this minimum energy per operation. More information about reversible computing can be found in this introduction [zyvex.com].
Encryption limits? (Score:4, Interesting)
Re:Encryption limits? (Score:5, Funny)
[TMB]
Re:On a similar note... (Score:2)
Re:A limit on computation? (Score:2)
I think it remains in use, because it keeps everyone (semiconductor fab plants, manufacturers, board designers) going at the same speed. If anyone falls behind current technology they make al oss. If they invest a considerable amount of money to develop a component that is over-engineered and can't work with anything else, they make a loss.
Look at the evolution of desktop/workstations. In o
Re:A limit on computation? (Score:3, Interesting)
Think of it as "Moore's Observation" instead. It wasn't meant to be a "Law" or a "Theory".
Minge for example predicts that computing something will become instant,
Computing something instantly may be possible, but transmitting the information necessary for that computation won't be. And that fact is the basis for this paper. Even if you could instantly perform every computation on every
Re:2 special case light cones to consider (Score:2)
There is a lot of unknown in physics that could make this article a joke.