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Wolfram's 2,3 Turing Machine Is Universal! 288

Posted by CmdrTaco
from the theory-flashbacks dept.
Rik702 writes "Wolframscience.com have announced that an undergraduate from Birmingham, UK has proved Wolfram's 2,3 Turing Machine is universal." You can read a pdf of the proof as well as some related coverage.
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Wolfram's 2,3 Turing Machine Is Universal!

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  • by CRCulver (715279) <crculver@christopherculver.com> on Wednesday October 24, 2007 @11:19AM (#21100283) Homepage
    I remember the discussion here five years ago when Wolfram released his book A New Kind of Science [amazon.com] . Many claimed that it was hogwash and (as it was apparently not peer-reviewed) irresponsible, but at least the movement he has tried to spark is finally showing some results.
    • Re: (Score:3, Informative)

      by r3m0t (626466)
      No, it isn't. Certainly they have some results but he's still an arrogant bastard who hasn't brought anything new to the table.

      Nor is this 2,3 machine going to revolutionise science.
      • by nagora (177841) on Wednesday October 24, 2007 @11:32AM (#21100501)
        No, it isn't. Certainly they have some results but he's still an arrogant bastard who hasn't brought anything new to the table.

        Very true. He is a clever guy, but not as clever as he thinks he is, and the book was just regurgitated from other sources. There was very little new or really much science in it. Basically he enumerates a lot of possible combinations of rules and says "some of these will turn out to be slightly interesting, you mark my words.". Well, some of them did. I'm SO impressed.

        Look at a map of the world. Some of those countries are going to end up going to war, you mark my words. See? I'm a visionary too.

        TWW

      • by meburke (736645) on Wednesday October 24, 2007 @12:10PM (#21101055)
        Sometimes the changes in Science come from just thinking about things differently. Whether Stephen is arrogant or not is irrelevant to the ideas or claims being evaluated. I doubt anything is invented in a vacuum, but rather a product of all the little discoveries and thoughts finally coalescing into something tangible.

        The main point here is that we are reaching limits in machine technology, and jumping to a different scale will require a new way of thinking about what we've already learned.

        Let me recommend three books: "The Structure of Scientific Revolutions" by Kuhn, "Bionics" by Salsburg, and "How Inventions Begin" by Leinhard. Three different thinkers; three different descriptions of the progress of technology.

        I have heard a number of criticisms on NKS, but most of the critics I've met have not actually read the book. (OK, it's a big book... I've found the same problems with people criticizing "Science and Sanity" by Korzybski, "Synergetics" by Fuller, and "Democracy in America" by de Tocqueville.) If you are going to criticize a book, please read it and understand it first.

        Recently William Gibson mentioned the problems with writing Science Fiction due to the unpredictability of the future and rapid technological change. As our technology becomes more abstract, more materials will be "intelligent" in new ways. For instance, imagine concrete with the intelligence to repair itself when a pothole is in imminent probability of forming. This type of "Turing Machine" computational ability at the molecular level may be the key to inventing this product.
        • Re: (Score:3, Insightful)

          by machinelou (1119861)
          It's probably more common that criticism comes from people who have NOT read the source material than from people who have. And, for reasons I do not completely understand, such criticism tends to be believed.

          Case in point? In a recent interview, Noam Chomsky admitted to writing his famous criticism of B. F. Skinner's book "Verbal Behavior" BEFORE it had been published and, in fact, based his review on much older material including notes taken by students of lectures presented YEARS before the final versi
        • "This type of "Turing Machine" computational ability at the molecular level may be the key to inventing this product."

          What I find so hilarious is that it already exists in biological systems, I really think if we want to go to the next level we really have to get our minds around what already exists and use that as a base, since so many things are already figured out we just don't understand them.
        • Re: (Score:3, Informative)

          by nwbvt (768631)

          Here are a few reviews from people who did read the book (or at least who claim to have read it, I admit I did not sit down and watch them each read it from cover to cover):

          • http://www.kurzweilai.net/articles/art0464.html?printable=1
          • http://www.goertzel.org/dynapsyc/2002/WolframReview.htm
          • http://www.cscs.umich.edu/~crshalizi/reviews/wolfram/
          • http://www.nybooks.com/articles/15762
          • http://www.maa.org/reviews/wolfram.html

          Now I did not go and seek out reviews that were critical of the book, these were just

      • by Quadraginta (902985) on Wednesday October 24, 2007 @12:56PM (#21101689)
        Boy you're telling me. I remember when Wolfram was the wunderkind of the 1980s and 1990s, who was going to Change The World(TM) with his brilliance. Ha. Jeff Bezos or Linus Torvalds have done far more with computers to Change The World in interesting and useful ways.
    • by UbuntuDupe (970646) * on Wednesday October 24, 2007 @11:28AM (#21100427) Journal
      Clarification:

      Wolfram's claim in NKS that he had discovered some fundamentally new way to approach science that couldn't be handled by existing peer review processes was hogwash. Others had done that kind of thing long before, and little in NKS helped advance the state of the art.

      Wolfram's proof in NKS that his Rule 110 cellular automaton was a Universal Turing Machine, was not hogwash. (That UTM was different from the one described in the story, obviously.)
    • by pohl (872) on Wednesday October 24, 2007 @11:32AM (#21100503) Homepage

      I recall that to, and was curious enough to see if the criticism was summarized well in Wikipedia. (It is.) [wikipedia.org] Personally, I loved the book, and read it knowing that it was standing on the shoulders of other's work. I remember first reading about the idea of modeling the world as cellular automata in a 1988 issue of The Atlantic, in an article called Did the Universe Just Happen? [digitalphysics.org] (search for "Wolfram" in that page) and I thought his book was a terrific work on the subject.

      I can understand how people's nerves got a little tender by having their contributions not been properly attributed, footnoted, etc. It didn't ruin my enjoyment though.

    • by Anonymous Coward on Wednesday October 24, 2007 @11:56AM (#21100847)
      Wolfram never actually proved Rule 110. The actual work for that was done by Matthew Cook - who presented the paper at a conference while he was working in Wolfram's employ - and eventually got himself and the conference sued. Apparently, working for Wolfram means you sign over any and all papers, ideas, and patents over to him, without receiving any real credit for them.

      Another little-known fact is that Wolfram was just one of several people who initially coded Mathematica. He decided one day to take all the code, form a company on his own, and engage in expensive lawsuits with all of his former collaborators to gain ownership of the code.

      As far as I'm concerned, the man at this point is wasted intelligence. He may come out with another non-trivial result or two over the course of the rest of his life, but his best contributions to the science may yet come from his wallet - like sponsoring prizes like this one.

      http://www.cscs.umich.edu/~crshalizi/reviews/wolfram/ [umich.edu]

      The above link is worthwhile, entertaining, and should help bring back anyone who drank the Wolfram Kool-Aid.

      (go blue)
      • Re: (Score:2, Interesting)

        by Anonymous Coward

        Another little-known fact is that Wolfram was just one of several people who initially coded Mathematica. He decided one day to take all the code, form a company on his own, and engage in expensive lawsuits with all of his former collaborators to gain ownership of the code.
        It (Originally SMP) was also written on Caltech equipment and Caltech sued Wolfram; hence Caltech does not pay for any copy of Mathematica they run.
  • by ebolaZaireRules (987875) on Wednesday October 24, 2007 @11:22AM (#21100331) Journal
    I hated this subject... If only my lecturer had told me I could win US$25k, it might have been more interesting.
  • Damn (Score:5, Funny)

    by 3.5 stripes (578410) on Wednesday October 24, 2007 @11:23AM (#21100345)
    That was on my To Do list for next week.
  • But... (Score:5, Funny)

    by Hatta (162192) on Wednesday October 24, 2007 @11:28AM (#21100429) Journal
    ...does it run Linux?
    • Re:But... (Score:5, Funny)

      by Palshife (60519) on Wednesday October 24, 2007 @11:31AM (#21100469) Homepage
      It runs everything.
      • Re:But... (Score:5, Funny)

        by johnw (3725) on Wednesday October 24, 2007 @11:42AM (#21100641)

        It runs everything.
        But it needs a memory upgrade and a new video card to run Vista.
        • Re:But... (Score:5, Funny)

          by sapone (152094) on Wednesday October 24, 2007 @11:59AM (#21100891)

          But it needs a memory upgrade and a new video card to run Vista.
          Well, next time you buy a Turing machine, you should go right for the Aleph brand memory tape. Plain old infinity obviously doesn't cut it.
        • Re: (Score:3, Funny)

          by Anonymous Coward

          But it needs a memory upgrade and a new video card to run Vista.
          Huh... I go through the trouble of finding a new card and an infinite tape that's twice as long, and the damn thing still displays the same three colors as before the upgrade.
          • by iabervon (1971)
            An infinite tape that's twice as long isn't any bigger. You need a continuous tape. Or if you prefer, you can get a tape that has a position for each possible state of the entirety of a regular infinite tape, although those tend to get torn too easily.
      • by eclectro (227083)

        It runs everything.
        The real problem is not compatibility. The problem is that when the machine goes to 65536+1 position on the tape, one of the colors turns to blue and the machine halts.
    • by johnw (3725)

      ...does it run Linux?
      Surely what he's proved is that it does?
      • ...does it run Linux?
        Surely what he's proved is that it does?
        Yes, in some sense at least. FTFA:

        This result ends a half-century quest to find the simplest universal Turing machine.

        It demonstrates that a remarkably simple system can perform any computation that can be done by any computer.

        And STOP CALLING ME SHIRLEY!
    • Well Linux definitely runs it. ;-)
      • by ultranova (717540)

        Well Linux definitely runs it. ;-)

        Maybe, but only a limited version at best. Do not forget that one of the assumptions for a Turing machine is infinite memory space; AFAIK the largest memory space Linux currently runs in has 2^64 memory locations. As such Linux doesn't run all the programs a theoretical Turing machine runs.

    • by Tibor the Hun (143056) on Wednesday October 24, 2007 @11:42AM (#21100645)
      Without even reading the article, I feel confident to say that, no it does not. All that universal applications allow you to do is run them on both PPC and Intel chips, as opposed to having 2 separate forks for each platform.

      (If anyone bothers to take this seriously, and feel they need to correct me, they need to step outside for a while and get some fresh air.)
      • Re: (Score:2, Funny)

        by Anonymous Coward

        (If anyone bothers to take this seriously, and feel they need to correct me, they need to step outside for a while and get some fresh air.)
        I work next to a waste treatment plant you insensitive clod!
  • 2,3. 2+3=5 (Score:4, Funny)

    by ettlz (639203) on Wednesday October 24, 2007 @11:41AM (#21100629) Journal
    The Law of Fives is never wrong.
  • by arsheive (609065) on Wednesday October 24, 2007 @11:58AM (#21100883)
    I for one now admit that all previously welcomed overlords can be emulated by this 2 state, 3 color overlord.
  • Slashdotted.. (Score:5, Informative)

    by Seismologist (617169) on Wednesday October 24, 2007 @12:05PM (#21100973)
    The wolfram site is slashdotted but this link for the article in nature [nature.com] is not.
  • if you like this... (Score:5, Interesting)

    by kisrael (134664) on Wednesday October 24, 2007 @12:17PM (#21101131) Homepage
    yeah, I know I'm pimping my own site, sosumi...
    Anyway, I was thinking about 1D CA the other week, and realized one of the attractions was that you plot time and make it 2D... but there's no particular reason you can't do the same thing to a 2D CA, like Life...
    http://kisrael.com/2007/10/21/ [kisrael.com] is the result, ethereal blue sculptures made by plotting 2D Life with Time as a physical dimension.
    I'm not sure if I learned a lot or proved anything, but it *is* pretty...
  • Uhh, what? (Score:4, Insightful)

    by Webz (210489) on Wednesday October 24, 2007 @12:17PM (#21101143)
    Could anyone take a stab at explaining what this discovery actually means, in layman's terms? What is the significance of a univeral turing machine? What if the turing machine wasn't universal?

    What's the significance of 2,3? (Bit states, color?)

    What did this discovery actually teach us? How is it useful?
    • Re:Uhh, what? (Score:5, Informative)

      by spikenerd (642677) on Wednesday October 24, 2007 @12:56PM (#21101705)
      Alan Turing is usually considered to be the father of computers. He invented a theoretical machine that he conjectured could solve any problem that could be solved by a machine. It consisted of a an infinitely long tape (memory) and a small finite set of operations that could be performed infinitely fast. Modern computers are *very* similar to his theoretical machine, except they're only very fast (as opposed to infinitely fast) and they only have a lot of memory (as opposed to an infinite amount of memory). No one has ever found a problem that could be solved by a more complex machine and could show that it couldn't be solved by a Turing machine. (BTW, Turing eventually killed himself by eating a poisoned apple after most of the scientific community shunned his work due to some personal habits. This was the inspiration for Apple computers' logo.) So Wolfram proposed an even simpler machine and conjectured that it could solve anything that a Turing machine could solve. Now this guy proved that Wolfram was right. I should mention for completeness that two other guys (Church, and dang, I forgot the other one) also proposed systems that were provably equivalent to Turing's machine around the same time, but Turing's was the easiest one to turn into an actual machine.
      • Re:Uhh, what? (Score:5, Informative)

        by Budenny (888916) on Wednesday October 24, 2007 @04:42PM (#21104941)
        "Turing eventually killed himself by eating a poisoned apple after most of the scientific community shunned his work due to some personal habits."

        Much, much sadder. He was prosecuted for homosexuality, sentenced to a course of estrogen, became depressed, and killed himself. This was how a grateful nation treated a man who had played a large, perhaps the leading role, in decoding Enigma signals - the decisive element in the Battle of the Atlantic. Tragic and wicked.
        • Re:Uhh, what? (Score:4, Interesting)

          by Eponymous Bastard (1143615) on Wednesday October 24, 2007 @06:43PM (#21106353)
          Became depressed because of:
          - Estrogen treatment: gained weight, physical changes, possibly psychological changes too.
          - Lost his security clearance so couldn't work on any of the crypto/high security work he did. (spies usually tried to subvert homosexual and/or prosecuted people who were dissatisfied with their government). Half of that work couldn't be published either which left him in a bad position as an academic.
          - "most of the scientific community shunned his work due to some personal habits." as the GP said. Guess which habits this means

          Probably caused a lot of rifts in his personal life too.

          BTW, the inspiration for Apple computers' logo was actually Newton's apple. Older logos have a picture of Newton sitting under a tree with a glowing apple above him. It is unknown how much Turing influenced it. People often mention the rainbow apple in this regard.
    • Re: (Score:3, Informative)

      by hansraj (458504)
      Turing machines are abstractions of what it means to compute.

      Consider the problem of sorting n numbers. In terms of computing, one is interested in computing a sequence of integers formed by rearranging given numbers that satisfy some (here monotonicity) property. Now, if one is given 5 numbers to sort one could do one thing to sort them and given 10 numbers to sort, something else. Existence of a Turing machine for sorting means that one could follow only one set of rules and sort any number of input numbe
    • by jgoemat (565882) on Wednesday October 24, 2007 @01:01PM (#21101795)
      Universal Turing Machines [wikipedia.org] are "[...] extremely basic abstract symbol-manipulating devices which, despite their simplicity, can be adapted to simulate the logic of any computer that could possibly be constructed."

      From the article [wolfram.com]:

      Here it is. Just two states and three colors. And able to do any computation that can be done.
      [...]
      There were some simpler universal Turing machines constructed in the mid-1900s--the record being a 7-state, 4-color machine from 1962.

      So basically there's a machine that has two states, each of which can be three colors, and that machine can perform ANY computation that an x86 cpu can perform. The code to add two 32 bit numbers in an x86 processor might be just a few bytes and the code to do the same thing with this 2,3 Turing machine might be thousands of bytes, but it can do it. It will be horribly inefficient and slow, but it can be done. They've proved that a 2,2 machine is impossible so a 2,3 machine was the simplest possible theoretical Turing machine. This paper proved that one exists. It doesn't have a practical application right now, but the article mentions possible molecular computers that can use this simple machines to do calculations on strands of molecules like DNA.

      • Wolfram's hype about molecular computers is hogwash. Yes, any molecular computer would need to be simple and follow simple rules like a Turing machine. But there's no way you'd actually build one using the Turing machine structure, moving left and right along a linear tape. The simplest algorithms taking linear time on an ordinary general-purpose computer can take quadratic time or worse when implemented on a one-dimensional Turing machine. A TM is a universal computer in that it can imitate any other,
      • by Relic of the Future (118669) <dales@nOsPaM.digitalfreaks.org> on Wednesday October 24, 2007 @01:44PM (#21102437)
        I believe it's "the machine can be in one of two states, and each point on the tape can have one of three colors"; not "a machine that has two states, each of which can be three colors," which is gibberish.
    • Re:Uhh, what? (Score:5, Informative)

      by johnnyb (4816) <jonathan@bartlettpublishing.com> on Wednesday October 24, 2007 @01:38PM (#21102337) Homepage
      A system is "Universal" if it can, given infinite memory and an appropriate program, compute any computable function. A previous system even more simple than this one, Rule 110, was proven to be Universal by one of Wolfram's associates (Wolfram had the idea that it might be, Matthew Cook discovered the proof). However, a Universal Turing machine has some extra requirements with regards to the implementation. So this is the simplest Universal Turing Machine.

      If you already know programming, then here's how to think of it:

      A Turing machine is a programming language.
      A Universal Turing machine is a language that is sufficiently flexible enough to perform any computation (including emulate any other Turing machine - i.e. language)
      A Non-Universal Turing machine is a language that is built to do a specific purpose well, but does not have enough flexibility to perform any computation.

      For computer programmers, nearly every general-purpose language we deal with on a daily basis is Turing-complete. An example of a non-Turing Complete language might be a configuration file. It has a language, you may even be able to do some basic scripting in it, but unless it is built out of a general-purpose language you cannot perform any computation in it.

      What they think it is useful for is to help us to make nanomachines easier. If we can construct a Turing machine with simpler parts, we can have a compiler that can pick up the slack in making the program.

      On another note, Wolfram takes these tiny Turing machines as reason to not believe that people have souls, while I on the other hand take these to indicate that life must be designed, but that has to do more with the general properties of Turing machines rather than the size of them.
      • So Rule 110 is universal but not a turing machine?

        I may have to correct the phrasing of a previous post...
  • Does this mean it runs native on the new Intel based Macs? :)
  • Automat size (Score:2, Interesting)

    by ledvinap (412654)
    Can anyone estimate how long tape is needed to do some real-world computation? Like adding/multiplying two (n-bit) integers?
    • by zx75 (304335)
      It is irrelevant to the problem. Turing Machines in general, and simple Universal Turing machines especially require exponential space/time to solve problems that have quicker solutions on more advanced machines.

      Usually no consideration is paid to how efficiently a Turing Machine works, as long as it is finite. There is no optimization that goes into the Machine to solve real-world problems, only proof that they could be solved. In your example a plausible answer for the amount of tape necessary could range
  • by Tim2 (151713) <twegner@swbe l l .net> on Wednesday October 24, 2007 @01:07PM (#21101861)
    As an undergraduate in 1968, I did an independent study that estimated the size of the universal turing machine described in: Davis, Martin (1958), Computability and Unsolvability, New York NY: McGraw-Hill Book Company. This was tedious but not hard. For any slashdotters ready to rush out and implement a working universal Turing machine, be forwarned that your parts list needs to include an infinitely long tape. Worse, when calculating the output of an arbitrary recursive function on your universal Turing machine, you won't know in advance how long the tape needs to be, in case you were cheap and bought a tape with finite length rather than the more expensive infinite one.

    The universal Turing machine itself consists of a large but quite finite set of quadruples. The problem is the longish tape.
  • by BoRegardless (721219) on Wednesday October 24, 2007 @01:19PM (#21102029)
    Lots of nitpicking of the solution and Wolfram and such have been posted. Let the nitpickers contribute!

    It takes a push from various people, and communication and conflicts of opinions to wind up exciting someone to sit down and solve some excruciating problem.

    I don't care whether it is math, mechanics, biology or physics, someone has to do the HARD work, and Wolfram contributed in his own promotional way, and Alex Smith solved the SOB of the smallest Turing problem, with a significant set of input from the judging panel requesting addtional work.

    A community of interested people wound up involved in getting an advanced solution. Then others said "but what good is it in requiring an infinite memory/tape". Similar things were said about past inventions, until other inventors figured out how to make the prior/first invention practical.

    I love math, but am not a mathemetician, so I have to contribute with the mundane discoveries and designs I do in my arena of medical product design, and they too will live on beyond me.

    The complainers should leave something that outlives them. That is what makes for a great society.
  • by deuterium (96874)
    Wolfram probably could have chosen a more modest title for his book, and put his particular ocntributions to CAs in better perspective, but I do think he's right in his emphasis on CAs being a relatively new and important tool for future scientific modeling. It's an attempt to describe the world at a more fundamental level than we currently do, using computers and rules instead of equations. Equations aren't old hat, but they are old school. They often describe things in a macroscopic way - you don't get mo
  • by matt me (850665)
    $proof =~ /perl/
  • At the wolfram site, he shows the rules for this 2,3 machine (two state, three color).
    The two states being up and down, and the colors being white, yellow and orange.

    Is there an equivalent 3,2 machine - {up, down, charm} and {white, black}?
    His machine:
    {S,C) -> {S,C,O}
    {
    {D, O} -> {D, Y, L},
    {D, Y} -> {D, O, L},
    {D, W} -> {U, Y, R},
    {U, O} -> {D, W, R},
    {U, Y} -> {U, O, R},
    {U, W} -> {D, O, L}
    }

    3,2 machine
    {
    {c, w} -> {u, w, L},
    {u, w} -> {c, w, L},
    {d, w} -> {u, b, R},
    {c, b} -> {d, w, R
    • by Anonymous Coward
      of this, but it doesn't pass the /. lameness filter ;^)
    • Re: (Score:3, Informative)

      by Iwanowitch (993961)
      No. Color and state play very different roles. Colors ("symbols" are also used) decide the input and output strings that can be generated (the colors make up the alphabet), states are internal parts of the machine.

      As such, you're using a different alphabet to write on your tape. Which means that the strings that will be accepted/rejected by one machine will definitely be different from those accepted by the other machine, and as such they are not equivalent. By definition, machines are equivalent if they
  • Something doesnt sound right here.
    • Re: (Score:2, Insightful)

      by Ant P. (974313)
      unsigned int x,y,z,n;
      arbritrary_values(&x, &y, &z, &n);
      assert(n>2);
      assert(pow(x,n)+pow(y,n)!=pow(z,n));


      The proof for that was something like 200 pages and took approximately 300 years, IIRC.
  • But will it blend?

    Imagine a Beowulf cluster of these.

    But will it run Linux?

    In Soviet Russia, universal are turning machines.

    • by oatworm (969674)

      But will it blend?

      Yes, but you'll need an infinitely large blender to blend the infinitely long tape.

      Imagine a Beowulf cluster of these.

      Impossible. Following the Wikipedia article [wikipedia.org] on Beowulf clusters: "It does not contain any custom hardware components and is trivially reproducible." The entire machine (not including the infamous infinite tape) is a custom hardware component. You could make a cluster of these, though. It just wouldn't be a Beowulf cluster.

      One question to think about, though: Since Beowulf clusters are predominately open-source dri

  • by ortholattice (175065) on Wednesday October 24, 2007 @01:55PM (#21102625)
    I had thought that the Game of Life was one of the simplest possible, although someone else will have to clarify how it compares. I think Life was proved universal a number of years ago, and an actual implementation is here: Life Universal Computer [igblan.co.uk].

    There are also universal machines possible with S-K combinators, which in a sense is also one of the simplest if not the simplest, with only two possible commands: S and K. (I guess it depends on how simplicity is measured.) Amazingly, the shortest universal machine [slashdot.org] found so far with SK combinators has 272 bits, compared to 5495 bits for Roger Penrose's universal Turning machine built from the original Turing machine and 268,096 cells for the Life version.

    I couldn't quite glean the size of a universal machine implemented with Wolfram's 2,3 cellular automaton, but I would imagine it would be very large.

  • So, this is apparently a 2 bit machine, with each bit representing 3 states. That's 9 possible states total.

    If this is correct, then I - or any other hobbyist - should be able to build this Turing machine with a few flip flops and LEDs. And supposedly, could use such a machine to emulate any other computer. What I'm wondering, though, is how practical such a machine would be.

  • by viking80 (697716) on Wednesday October 24, 2007 @03:48PM (#21104157) Journal
    The original Universal Turing machine was defined to end with the *HALT* instruction. The 2,3 Turing Machine can not halt, and is therefore not universal. It appears that Wolfram conceded that computers today dont really halt, they just keep ticking after the program is complete, so they accepted the 2,3 machine as universal, and the proof as completed.

    Maybe someone should submit the same proof, concluding that it is *not* a universal Turing machine, and claim the $25k.
  • Not What It Seems? (Score:3, Interesting)

    by Jane Q. Public (1010737) on Thursday October 25, 2007 @12:52AM (#21109619)
    TFA (the first link) claimed that this was a proof of the "simplest" Turing machine. However, as I understand what was written, and without reading the whole proof, it appears that this is not actually the case. What he proved was that his chosen candidate Turing machine, among "2,985,984 possible 2,3 machines", was universal. That does not mean that his machine is the simplest possible, only that it is among the simplest known CATEGORY of such machines (2,3). There may well be some kind of example of one that is, in a significant way, simpler.

Simplicity does not precede complexity, but follows it.

Working...