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Calculating A Theoretical Boundary To Computation 583

Posted by timothy
from the this-far-and-not-beyond dept.
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."
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Calculating A Theoretical Boundary To Computation

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  • Roger Penrose (Score:3, Informative)

    by andy666 (666062) on Wednesday April 28, 2004 @08:03AM (#8995221)
    This doesn't mention Penrose's work, which is very much like this.
    • (feel free to enlighten us then, eh? :)
      • by Flyboy Connor (741764) on Wednesday April 28, 2004 @08:13AM (#8995306)
        Penrose is a mathematician who attempts to be philospohical and fails miserably, because he can't distinguish his intuition from fact. You don't need a link. Just remember that he wrote "The Emperor's New Mind", and coil away in horror.
    • Re:Roger Penrose (Score:5, Informative)

      by julesh (229690) on Wednesday April 28, 2004 @08:39AM (#8995526)
      I haven't seen anything by Penrose which is like this. In fact, this article states an assumption ("consciousness is fundamentally computational in nature") that directly contradicts Penrose's most well known result, a rather dubious pseudo-mathematical "proof" that consciousness _cannot_ be computational as a consequence of Godel's Incompleteness Theorem.

      So, no, it isn't really like Penrose's work.
      • Re:Roger Penrose (Score:5, Interesting)

        by Yartrebo (690383) on Wednesday April 28, 2004 @10:28AM (#8996659)
        Here's Turing's counter-example to bust Penrose's theorem:

        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)

          by NonSequor (230139)
          Except it is currently unknown whether quantum mechanical systems can be simulated with even probabilistic Turing machines. In fact, if it is possible to do so in polynomial time, you can do anything a quantum computer can do in polynomial time and so you can factor numbers and solve the discrete log problem in polynomial time.

          Basically it is believed that it is very unlikely that you can do quantum simulation on a classical computer.
      • Re:Roger Penrose (Score:5, Informative)

        by wytcld (179112) on Wednesday April 28, 2004 @12:31PM (#8998110) Homepage
        Penrose in recent years isn't saying "consciousness isn't a computer." Rather in collaboration with Stuart Hameroff [arizona.edu] and a number of physicists is saying that "consciousness is a quantum computer."

        So for all you /.'ers whose first reaction is: "He says we're not computers. Uncool!" consider the contrary reaction: "He says we're quantum computers. Way cool!" Also note that, as all /.'ers should know, quantum computers don't have the same limitations as conventional computers on capacity, thus the well-known threat they pose to encryption, being able to break it (in theory) in trivially short time periods.
  • Exception (Score:5, Insightful)

    by Zweistein_42 (753978) * on Wednesday April 28, 2004 @08:05AM (#8995230) Homepage
    "consciousness must be finite"

    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?
    • by Anonymous Coward on Wednesday April 28, 2004 @08:13AM (#8995303)
      Like dude, I was totally smoking this bogus weed and it like totally occured to me that there is like only so much stuff my finger can totally interact with. I mean warp theory is like so just no way, that I've got to depend on my photons you know. Cause when you think about it, all we ever feel is electron, and all anyone sees is the light we you know ... hey you gonna eat that brownie ... uh scatter. So like there's only so much stuff that we can interact with in the visible universe. So it's like there is an edge, and only so much stuff. Which means that the whole universe can only hold so many states. So there's like a finite ability for it know and like cosmically meditate about what's going down you know??

      Dude...

      I know, I totally stayed at a holiday inn express last night.
    • Re:Exception (Score:3, Interesting)

      by RLW (662014)
      We live in a finite universe. So, all things which require a physical framework are finite. It's just that there's a lot of stuff and therefore a *lot* of possible states. From the point of view of a very limited life span we can never even come close to witnessing an even small fraction of the number of states for the universe.

      In fact there are limitations to everything. Even to our ability to determinie limitations.
  • enough! (Score:4, Insightful)

    by Anonymous Coward on Wednesday April 28, 2004 @08:05AM (#8995234)
    Moore's law was never intended to be a scientific theory. It was just a useful observation. It has never had anything other than economic incentive to keep it going. Using it to discuss the calculational ability of the universe is idiotic.
    • Re:enough! (Score:3, Informative)

      by spangineer (764167)
      Exactly - Moore's law is certainly not a real scientific law. It often approximates what actually happens, but because it's based on human activity, it's not very precise. Humans are unpredictable, and thus, cannot possibly be the basis for a scientific law (as far as I know)

      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)

      by Curunir_wolf (588405) *
      Moore's law was never intended to be a scientific theory. It was just a useful observation. It has never had anything other than economic incentive to keep it going. Using it to discuss the calculational ability of the universe is idiotic.

      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

      • Current theories say that conciousness exists in the inredibly complex electrical fields generated by the human brain.

        As such, it has no weight.

        Unless you can weigh electricity.

        Unfortunatly, I dont remember where I read this.
    • Re:enough! (Score:5, Informative)

      by jcoleman (139158) on Wednesday April 28, 2004 @08:23AM (#8995391)
      Ummm...read that again. Moore's Law is not the basis of this paper. Physicists and mathematicians using economic theories (yes, Moore's Law is economic in nature) to predict physical laws are neither published nor credentialed. The finiteness of Moore's Law is an implication of the findings of this paper.
    • Re:enough! (Score:5, Informative)

      by julesh (229690) on Wednesday April 28, 2004 @08:29AM (#8995447)
      Have you read the article? All it states is that no civilisation could possibly extend Moore's Law beyond 600 years. That's the only reference to Moore's Law in the entire article, and its a reasonable one. It puts into terms we can (just about) understand the implications of the discovery. Who knows what 1.5 * 10^220 bits of information processed is? But 600 years of development at the current rate is slightly more imaginable (although, I'll admit, only marginally so).
    • Re:enough! (Score:3, Insightful)

      by janimal (172428)
      Using it to discuss the calculational ability of the universe is idiotic.

      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
  • by ab762 (138582) on Wednesday April 28, 2004 @08:05AM (#8995238) Homepage
    We should now be able to compute the asymptotic limit of web-server bandwidth for slashdot-proofness per year for 600 years. I bet it's a constant price in street dollars.
  • Sweet (Score:5, Interesting)

    by DrLudicrous (607375) on Wednesday April 28, 2004 @08:06AM (#8995246) Homepage
    I was a physics undergrad at Case, and actually had Starkman as a professor for a mathematical physics course. I have chatted with Krauss a few times since graduation on science topics involving public education. These are good guys, glad to see them headlining slashdot this morning.
    • Re:Sweet (Score:5, Insightful)

      by Total_Wimp (564548) on Wednesday April 28, 2004 @08:33AM (#8995471)
      This is great work.

      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
      • I've always thought that life would probably require a heady number of different chemical elements, thus we would find it in areas of the universe with an abudance of heavy elements (like gold, lead, etc.). My guess would be that this solar system is probably a 2nd or 3rd generation system meeting those requirements- 7-10 billion years ago there probably was not nearly as many star systems with the abundance of transistion metals that we have here. And I think those are just as necessary for "complex" lif
  • Physics of star trek (Score:3, Informative)

    by REBloomfield (550182) on Wednesday April 28, 2004 @08:08AM (#8995258)
    Physics of star trek [amazon.com]

    It's not a referer link, don't worry...

  • by jelle (14827) on Wednesday April 28, 2004 @08:08AM (#8995259) Homepage
    "and that consciousness must be finite."

    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?

  • And in other news... (Score:5, Interesting)

    by Glock27 (446276) on Wednesday April 28, 2004 @08:11AM (#8995282)
    Skewe's number of angels can dance on the head of a pin...

    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... ;-)

    • by mjh (57755)

      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... ;-)

      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

      • The GP was pointing out what those of a scientific bent already know - nothing is true, everything is permitted. We only know what we have seen. When we say something is impossible, what we mean is that it's impossible under any conditions we currently know how to bring about, or can repeatedly observe from natural phenomena.

        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

      • by WormholeFiend (674934) on Wednesday April 28, 2004 @09:10AM (#8995815)
        and everyone knows that when you make an assumption, you make an "ass" out of "U" and.. uh, "mption".

      • by Glock27 (446276)
        Considering that we have Einstein with a proof that faster than light is impossible,

        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

    • 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... ;-)

      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'
  • by mattjb0010 (724744) on Wednesday April 28, 2004 @08:15AM (#8995315) Homepage
    Please use the mirrors. In Australia, the closest one is here [arxiv.org].
  • by heironymouscoward (683461) <heironymouscoward@NoSpAm.yahoo.com> on Wednesday April 28, 2004 @08:18AM (#8995338) Journal
    Can you even define consciousness?

    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.
    • 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

    • Consciousness is most likely related to some laws of physics that we haven't discovered/fully understood yet,. possibly related to quantum mechanics.

      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/
    • by AllUsernamesAreGone (688381) on Wednesday April 28, 2004 @08:38AM (#8995518)
      Most likely, consciousness is a sense, like sight or sound.

      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.
  • assumptions (Score:2, Interesting)

    "consciousness must be finite"
    ... this only works if consciousness is based on computation, surely? It's getting into the area of metaphysics, but so far there's nothing to suggest that consciousness is based within the brain, let alone is computational.
    • there's nothing to suggest that consciousness is based within the brain

      Just out of interest, where else do you suggest?

      • Re:assumptions (Score:3, Insightful)

        by D-Cypell (446534)
        This is a question that I tend to think about from time to time...

        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)

      by phfpht (654492)
      but so far there's nothing to suggest that consciousness is based within the brain, let alone is computational. Sure there's evidence to suggest that consciousness is based within the brain. If the brain is damaged, consciousness can be removed or reduced (level of). That alone is a strong ling that consciousness is brain based. Enough brain damage and one can die or be reduced to a vegitable, metaphorically speaking. That seems to be a distinct reduction in consciousness. Smaller brains seem to have var
  • by puzzled (12525) on Wednesday April 28, 2004 @08:24AM (#8995398) Journal


    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 is not a physical theory, it is the observation of a common phenomenon, namely the curve that technology goes through as it becomes cheaper and eventually free.

    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.
  • by caek (571864)
    This article hasn't (yet) been accepted for publication. Caveat lector!
  • by phil reed (626)
    The obvious conclusion of this paper is that there is a finite limit to the amount of pr0n in the universe. That's good to know -- I can now relax, knowing that I won't have to keep buying bigger hard drives forever.
  • Eye glaze (Score:2, Funny)

    by Woogiemonger (628172)
    Yep, even the title of this paper is designed to make one stare blankly and nod. With such eloquent scientific lingo wrapped around such an outlandish subject matter, the end effect is comparable to drinking a Pan-Galactic Gargle Blaster.
  • Infinity (Score:2, Funny)

    by MrDoh! (71235)
    So, if the universe has a limit, and the Mind isn't infinite, and we're all constrained by the entropy of the ever so slowly expanding universe, I have just one question.

    Would anyone like some toast?
    • So, if the universe has a limit, and the Mind isn't infinite, and we're all constrained by the entropy of the ever so slowly expanding universe, I have just one question.

      Would anyone like some toast?

      Good god, man! You've solved the problem already! Now it's just a matter of engineering [everything2.com]...
  • by Weaselmancer (533834) on Wednesday April 28, 2004 @08:32AM (#8995467)

    "...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

  • Time to get cracking on the calculation of how to reverse entropy (per Asimov).
  • well, duh (Score:4, Insightful)

    by Too Much Noise (755847) on Wednesday April 28, 2004 @08:35AM (#8995494) Journal
    (note of caution - let's see whether this gets accepted, looks more like a Science article than a Phys. Rev. Lett. one to me)

    so ... duh. This is more or less a geometrical analysis (finite causal volume) + basic information theory. No questions asked about physics of inflation and how would that affect the result. So you end up with a trivial result, too - a finite volume can only hold a finite amount of information. If a lot of other assumptions hold - such as whether the available energy in this volume is really finite (how does one sustain an infinitely accelerating model this way?)
  • by gkuz (706134) on Wednesday April 28, 2004 @08:36AM (#8995500)
    the later the observer attempts to collect energy within the accessible volume, the less of it there is.

    It's like at the bar -- the later in the night you attempt to pick up chicks, the fewer of them are still available.

  • If this paper is true and there is a limit on consciousness, wouldn't it make the existence of an omnipotent being an impossibility?
  • The Last Question (Score:5, Interesting)

    by eclectro (227083) on Wednesday April 28, 2004 @08:43AM (#8995568)
    There are some intersting ideas [wikipedia.org] as to what the end of the universe could be.

    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.

  • On power up the new chips would create several parallel universes (one for each execution unit) and destroy them when you turn the machine off -- this way it can continue to grow beyond the 600 year theoretical limit.

    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.
  • by jludwig (691215) * on Wednesday April 28, 2004 @09:07AM (#8995784) Homepage
    Not commenting on the paper itself, but it has been submitted to PRL, not accepted. It hasn't gone through that wonderful process of peer-review that is the very heart of the scientific method (that and falsifiability but thats another topic). NASA has been setting a particularily bad example here with science by "press release". PRL is not an easy journal to publish in, lets wait until other experts have a look and not cheat the scientific method like this. PRL should not be mentioned in connection with this paper until this get published - Anyone can submit a paper to PRL...
  • by clambake (37702) on Wednesday April 28, 2004 @09:12AM (#8995835) Homepage
    Any analog variable has an INFINITE number of states. 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...

    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.
    • by Idarubicin (579475) <allsquiet@hotm a i l .com> on Wednesday April 28, 2004 @10:43AM (#8996847) Journal
      Any analog variable has an INFINITE number of states.

      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...

  • by troon (724114) on Wednesday April 28, 2004 @09:29AM (#8996015)

    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!

  • by Jagasian (129329) on Wednesday April 28, 2004 @09:34AM (#8996065)
    So does this imply that all physical computers are finite state machines? Even when connected to the internet, their total number of computational states are finite, though extremely large, and therefore Universal Turing Machines are only a mathematical construct.
  • by theLOUDroom (556455) on Wednesday April 28, 2004 @09:45AM (#8996179)
    1. A computer to fully simulate the actions of a given piece of mass, MUST be of equal size or larger than the mass it's trying to do a computer simulation of.
    2. This computer MUST simulate the actions of this mass at the same speed or slower than the actions it's actually trying to simulate.


    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.
  • by hweimer (709734) on Wednesday April 28, 2004 @10:02AM (#8996390) Homepage
    They claim that every computation step requires at minimum energy of ln 2 k_B T (k_B is Boltzmann's constant, T is the temperature of the system). This is only true for irreversible operations such as setting or erasing a bit.

    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)

    by harlows_monkeys (106428) on Wednesday April 28, 2004 @10:07AM (#8996447) Homepage
    So, how many bits do I need for a symmetric cipher key in order to push a brute force search past the computational limit of the universe?

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