Wolfram's 2,3 Turing Machine Not Universal

Fishbat writes "In a cutting message to the Foundations of Mathematics mailing list, Stanford's Vaughan Pratt has pointed out an elementary mistake in the recently announced proof that Wolfram's (2,3) machine is universal." Update: 10/30 04:18 GMT by KD : Ed Pegg Jr. from Wolfram Research points to this response to Dr. Pratt's note, which has been submitted to the FoM mailing list but has not yet appeared there due to moderation.

The Filter

[eldavojohn (898314)]

The author of this e-mail, which is incredibly insightful and articulate asked:

How did an argument containing such an elementary fallacy get through the filter?

To be fair, I don't consider this that elementary a fallacy but when you break it down to if x + y = infinity then both x & y are infinity it does sound like a rather obvious pitfall. And, in response to his comment on catching it, it had not yet been through "the filter" as the original story stated:

Smith's proof will be published in the journal Complex Systems.

Meaning it had not yet been peer reviewed. But I agree with the author however abrasive he came off (although this is far higher than an elementary mistake) when he said:

Had I pushed my luck my second question would have been, who has verified this proof that has taught an automata theory course at a suitably accredited institution?

I can forgive an undergrad prof for missing this, his response was probably just to look at it and encourage the kid to submit it. Afterall, what motive would a professor have to read it?

But what should really be noted is what Wolfram himself is quoted as saying from the initial publishing of this proof:

"I had no idea how long it would take for the prize to be won," said Stephen Wolfram. "It could have taken a year, a decade, or a century. I'm thrilled it was so quick. It's an impressive piece of work."

Alright, that last sentence there is pretty damning. I have heard time and time again on Slashdot that Wolfram just took other people's work, that he had people working underneath him & that he didn't actually know what he was talking about in his book. This is some corroborating evidence, in my opinion.

I don't know a lot about finite automata but this whole display has shown that Alex Smith is talented but not the winner of the prize, it's best to accept and seek out all criticisms from your community before publishing & Wolfram is not the genius he makes himself out to be. I don't believe I will ever read "A New Kind of Science" as I have many other books in front of that one on my list.

Sounds like just another step in the learning process for Alex--too bad about the cash but he is only 20 and from the looks of it has a bright and promising future. Quite the embarrassment for Wolfram, however.

The real kicker would be if Wolfram had asked his staff to review the proof and they knew it had an elementary mistake and had told him it was golden. Now that would be poetic justice.

Re:The Filter

[by Anonymous Coward]

You misread the post. He said that if x + y = infinity and y is finite, then x must be infinity. This is TRUE for numbers. You cannot apply this by analogy to automata and think it is still true. It is not.

Re:The Filter

[jnana (519059)]

Indeed. A prior email in that thread -- by the same author, Pratt -- makes it very clear by giving the example of 2 pushdown automata [wikipedia.org] (PDA). A single PDA by itself is not universal, but the system comprised of 2 PDAs is universal, since each stack can represent one side of the Turing machine tape.

As Pratt states, the fallacy is of the following form: a system comprised of 2 PDAs, PDA A and PDA B, is universal. PDA A alone is not universal. Therefore, PDA B must be universal (because the system as a whole is universal). QED.

Of course, in the actual proof, it was not 2 PDAs, but a 2,3 machine and an encoder (i.e.,"PDA A" == "encoder" and "PDA B" == "2,3 machine").

Re:The Filter

[by Anonymous Coward]

Incidentally, for anyone who wants to learn something about automata and theory of computation and doesn't know where to start, I highly recommend the following book by Michael Sipser: Introduction to the Theory of Computation [amazon.com].

It's quite pricey for such a small book, but it's worth its weight in gold (i.e., the time you save by reading this little masterpiece instead of something else that's less well written). You can find the 1st edition used for much cheaper than the 2nd edition, and the differences between the two editions are pretty minor.

p.s. I have no connection to the book or the author. I'm just a very happy customer.

Re:The Filter

[by Anonymous Coward]

"p.s. I have no connection to the book or the author. I'm just a very happy customer."

Hey Michael, long time no post.

Re:

[Workaphobia (931620)]

Am I correct in thinking that the issue is that the given proof proves that the wrong system is universal? I read part of the proof but I'm unfamiliar with some of their terminology and models. From what I can tell, the question was whether a particular Turing machine was universal, and the answer was "yes, with some minor modifications", and the prize panel decided that this adaptation fit within the definition of the problem.

Re:

[ais523 (1172701)]

This argument and Pratt's argument that the universality proof are incorrect both share the same flaw; see the response linked in the update of the original story for more details. Yes, a system consisting of two push-down automata that can communicate back and forth is universal. However, the proof of the universality of the 2,3 Turing machine in question doesn't set up a system where two systems can communicate back and forth; instead, the output of one is used as the input of the other, and there is no

Re:

[EveLibertine (847955)]

Regardless, the guy that wrote that email is a prat.

Re:The Filter

[HeadlessNotAHorseman (823040)]

Wolfram's 2,3 Turing Machine Not Universal
HOLLYWOOD - In a shock move, MGM has undercut Universal in its bid for the movie rights to Wolfram's 2,3 Turing Machine. Insiders had predicted that Universal would make the deal to build on the phenomenal success of Wolfram's 2,2 Turing Machine, but it has since become apparent that Universal failed to include an option for all sequels in the original contract. The exact figure offered by MGM is unknown, but is believed to be approximately x + y, and we can confirm that y is a finite number. More details will follow.

Give Wolfram some credit

[Starky (236203)]

I have heard time and time again on Slashdot that Wolfram just took other people's work, that he had people working underneath him & that he didn't actually know what he was talking about in his book. This is some corroborating evidence, in my opinion.

Hardly. Wolfram disappeared for a decade to produce A New Kind of Science. Was he picking his toes while his team of crack Mathematica techies were developing the ideas for the book? I find that hard to believe. In fact, the way I heard it, he did all of his own editing on the book, much to the dismay of some who found it in need of editing.

He probably did have staffers assisting in running simulations (and with his bankroll, I would certainly entertain that notion myself), but name me one prominent professor who hasn't stood on the shoulders of graduate students.

Whether you consider him a genius or a crackpot (and he certainly gives reason to entertain both opinions), Wolfram is undoubtedly brilliant and seems to be dedicated to the advancement of mathematical ideas that he considers to be important. It hardly seems that a lack of academic integrity would be consistent with his actions to date.

Whether history will ultimately judge him a genius or a crackpot, I would guess that he has done more to advance mathematics than all the posters to this article combined, myself included. So give the man some credit.

Ouch

[JoeShmoe950 (605274)]

I'd hate to be involved in either the submission or the review of that proof. I was rather intriuged when the proof was first posted, but I must say, this is something of an embarrassment

Is not universal?

[MarsDefenseMinister (738128)]

What does that mean? Does this mean that it hasn't been proven to be universal (which is the case if there was just a bug in the proof) or is there another proof that shows the machine is not universal?

kdawson, not proven to be universal and proven to be not universal are two different things.

No, just not proven

[billstewart (78916)]

As others [slashdot.org] have pointed out [slashdot.org], no, it doesn't mean anybody's proven that it's not universal, it just means that this alleged proof of universality was incorrect.

Back to the drawing board

[tepples (727027)]

Does this mean that it hasn't been proven to be universal (which is the case if there was just a bug in the proof)

It appears to mean the proof has a defect in the same way that message passing in Windows has a defect [wikipedia.org]. As far as I can tell based on the article, Vaughan Pratt alleges that so much of the proof relies on the defect that the proof would need a complete rewrite.

Bad Headline

[EvanED (569694)]

Wolfram's 2,3 Turing Machine Not Universal

That's not, from my reading, what is true. What is true is that the proof is wrong, which means that it may not be universal, but reverts back to the unknown state.

I rate this proof

[MillionthMonkey (240664)]

(Score: 2,3 Possibly Informative)

Re:

[Alsee (515537)]

(Score: 2,3 Possibly Universal)

-

"from the whoa-not-so-fast-there-big-fella dept"

[ZorbaTHut (126196)]

Yeah, speaking of that . . .

"Wolfram's 2,3 Turing Machine Not Universal". What? Where'd you get that? This issue doesn't prove anything of the sort - it merely shows that this proof is invalid. It may be universal, it may not, but we still don't know.

So, ironically - whoa, not so fast there big editor.

Bad romantic consequence

[Hao Wu (652581)]

I hope that this young man's girlfriend stands by him. So many women would be driven away from an otherwise suitable boyfriend who is skilled at the maths.

Peer Review Rules

[tjstork (137384)]

This sort of thing is science when it works at its best. Someone throws something out there, and another scientist checks it, and bam, we learn something.

Re:

[Reaperducer (871695)]

This sort of thing is science when it works at its best. Someone throws something out there, and another scientist checks it, and bam, we learn something.

Did we learn something? As far as I can tell we're back where we started.

Re:Peer Review Rules

[ispeters (621097)]

Did we learn something? As far as I can tell we're back where we started.

Well, to paraphrase someone famous (perhaps Edison?), we've learned another way that doesn't work. It sounds like the author of the proof has used a faulty syllogism. Perhaps the syllogism can be patched up such that the rest of the proof plus the patched syllogism equals a correct proof.

Ian

Re:

[tjstork (137384)]

Yep. We learn that "never hold the press conference until after peer review and acceptance of publication".

Well, in all likelihood, we really didn't learn -that-. :-)

Every now and then I take a crack at P=NP, and sometimes, I feel like I've really got a good proof - a program idea, that, when implemented, could FACTOR fairly quickly. I'll be practicing my "move over Al Gore, here's what the Nobel Prize is really about" speech as I'm typing my breakthrough in, and there will be some implementation detail th

Re:

[rbarreira (836272)]

Every now and then I take a crack at P=NP, and sometimes, I feel like I've really got a good proof - a program idea, that, when implemented, could FACTOR fairly quickly.

Do you mean factor numbers? Even though it would be impressive to have an algorithm which factors numbers quickly, it wouldn't prove anything about the P=NP? problem. Factoring numbers is not known to be an NP-complete problem, so solving it in polynomial time doesn't automatically imply that P=NP.

Re:

[tjstork (137384)]

Factoring numbers is not known to be an NP-complete problem, so solving it in polynomial time doesn't automatically imply that P=NP

I thought FACTOR was NP-complete... if not, I think you could probably show it to be at least NP-complete by taking the selection of prime numbers as a sort of a napsack problem against a set of "special numbers" or, numbers that one could generate primes with.

Indeed, the one insight that I tried out but failed at was trying to see if I could arrive at what those special numbers

Re:

[rbarreira (836272)]

How can generating a prime number be NP-complete? Do you mean testing if a number is prime or not? That problem is in P:

http://en.wikipedia.org/wiki/AKS_primality_test [wikipedia.org]

Regarding whether FACTOR is NP-complete or not, most computer scientists don't think so.

Where are my meds?

[synthespian (563437)]

Jesus Christ, is Wolfram out of Lithium again?

part of Wolfram's reply cracks me up

[Danny Rathjens (8471)]

I don't think that the details of which particular Turing machine is or is not universal are all that significant.

http://cs.nyu.edu/pipermail/fom/2007-October/012149.html [nyu.edu]

I must admit that NKS is a bit over my head at the moment, though. So I could be reading something into it not meant. ;)

Title Proven To Be Misleading

[yerdaddie (313155)]

The Wolfram's 2,3 Turing Machine proof of universality was found to be flawed. This does *not mean* the 2,3 Turing Machine isn't universal. It just means it has not been proven to be universal. That would require another proof. Subtle distinction, I know; but in any case, the title of this article is fallacious.

Can I get a refund

[stinkfish (675397)]

for the Wolfram's 2,3 Turing Machine I bought from amazon?

Re:Can I get a refund

[rubycodez (864176)]

only if the infinite tape is mostly unused

Elmentary Mistake in Title

[Cyberllama (113628)]

This doesn't mean it's not universal, just that it's not PROVEN that it is. Not at all the same thing.

Vaughn Pratt is confused

[evgalois (1181567)]

No, Vaughn Pratt is confused. There is a post from on the FOM mailing list that explains the confusion. There are subtle issues concerning the nature of computational universality in the presence of infinite initial conditions. Vaughn Pratt is probably remembering work from the 1950s on computational universality, which does not address these issues. There are different definitions that could be given for computational universality with infinite initial conditions. Alex Smith's proof was verified with a particular, natural, definition that was chosen for the prize. So all is well. The 2,3 Turing machine's universality has not been toppled with one email and the personal opinion of Vaughn Pratt. What has happened, though, is that questions that have not been discussed since the 1950s are (this week) back in vogue again.

Re:Vaughn Pratt is confused

[Ed Pegg (613755)]

I'm posting from Wolfram Research. Basically, a message from Vaughan Pratt was posted to the correct spot, the FOM list. Dr. Pratt likely didn't expect his message to get a late night SlashDot level exposure. A response to his message has already been sent to the FOM list, but it is a moderated list, and the response is not visible yet. Here is a copy of the FOM posting from Todd Rowland, from the Wolfram prize committee. http://forum.wolframscience.com/showthread.php?s=&threadid=1472 [wolframscience.com] This is how math is done ... trying to poke holes in proofs.

Re:

[by Anonymous Coward]

Since you are posting from the heart of the Wolfram Hype Machine (TM), perhaps you could comment on why the prize was announced by Wolfram Himself as being successfully awarded, when Martin Davis on the FOM list states that the committee members were not polled [nyu.edu].

This appears to be a flagrant violation of the rules for the prize [wolframscience.com], which state that "For the purposes of this prize, the treatment of universality in any particular submission must be considered acceptable by the Prize Committee."

To make my ques

Re:Vaughn Pratt is confused

[evgalois (1181567)]

Here is the explanation about the suposed "flaw", which of course it is not: http://forum.wolframscience.com/showthread.php?s=&threadid=1472 [wolframscience.com] And my comments on the guy that thinks that the universality definition was changed for the prize benefit (I am surprised about how many people write about this without knowing a bit of the subject and trying to sound technical talking about "AND gates" completely nonsense): Concerning the definition of universality, a halting state or halting instruction wasn't a requirement. This is a common usage nowadays in the field of small universal Turing machines, which is a generalization of previous definitions. If there is no clear definition is because there is no clear-cut, established procedure to determine when an initial condition is computationally simple enough to be acceptable. Some would wish universal computation stick to a finite initial condition with an unbounded tape filled with "blanks", because that's the only case where the theory is entirely clear. However, others accept generalizations such as periodic "blank" words as long as they remain computationally simple enough (possibly generated in the same fashion as an unbounded "blank" tape). So Alex Smith's use of a non-periodic but still sufficiently computationally simple background is a natural generalization of this sort. The key point is that the background can be set up without doing universal computation, so the 2,3 machine itself actually carries out the computation. We are glad that this is making a contribution to the discussion on universality. We expect that others will further clarify what Alex Smith has done. We particularly hope that his methods can be extended to other similar proofs.

Re:Vaughn Pratt is confused

[evgalois (1181567)]

Well, that's a good start: you accepting you are not an expert in the field but pretending to judge something that of course experts reviewed. It is nice that people like you think on this problem and possible flaws, though. Of course nobody is changing the well stablished definition of a Turing machine. But it is accepted even by the experts that there is no clear definition on universality. You should follow all the FOM posts and not only those that you think are better to critic others. The definition of universality has been modify several times, the first to make a generalization exactly as the used by Alex Smith was Watanabe as early as in the 60's. Alex Smith contribution is a generalization of the same sort. If you want some references feel free to check these and then reply again: [1] Manfred Kudlek. Small deterministic Turing machines. Theoretical Computer Science, 168(2):241-255, November 1996. [2] Manfred Kudlek and Yurii Rogozhin. A universal Turing machine with 3 states and 9 symbols. In Werner Kuich, Grzegorz Rozenberg, and Arto Salomaa, editors, Developments in Language Theory (DLT) 2001, vol. 2295 of LNCS, pp. 311-318, Vienna, May 2002. Springer. [3] Maurice Margenstern and Liudmila Pavlotskaya. On the optimal number of instructions for universality of Turing machines connected with a finite automaton. International Journal of Algebra and Computation, 13(2):133-202, April 2003. [4] Claudio Baiocchi. Three small universal Turing machines. In Maurice Margenstern and Yurii Rogozhin, editors, Machines, Computations, and Universality (MCU), volume 2055 of LNCS, pp. 1-10, Chisinau Moldavia, May 2001. Springer. [5] Turlough Neary and Damien Woods. Four small universal Turing machines. Machines, Computations, and Universality (MCU), volume 4664 of LNCS, pp. 242-254, Orleans, France, September 2007. Springer. [6] Yurii Rogozhin. Small universal Turing machines. Theoretical Computer Science, 168(2):215-240, November 1996. [7] Shigeru Watanabe. 5-symbol 8-state and 5-symbol 6-state universal Turing machines. Journal of the ACM, 8(4):476-483, October 1961. [8] Shigeru Watanabe. 4-symbol 5-state universal Turing machines. Journal of Information Processing Society of Japan, 13(9):588-592, 1972. That is why many serious authors as Vaughn Pratt can be confused. Because they are not updated on the subtles of the field. We are glad that this is making a contribution to the discussion on universality. We expect that others will further clarify what Alex Smith has done.

Serious authority

[Emnar (116467)]

I don't understand the math behind this argument and counter-argument, but Vaughan Pratt is a CS legend [wikipedia.org] and one of the early cofounders of Sun, to boot. You also might have run across his name in a cite or two in The Art of Computer Programming series by Donald Knuth. And if you don't care who Knuth is, then you probably don't care about this post at all.

I knew Pratt's daughter in college -- nice woman. Wrote her term papers in LaTeX, on a Linux workstation, in 1996 :P

The really damning thing from the FOM list...

[Bob Hearn (61879)]

... is this:

http://www.wolframscience.com/prizes/tm23/technicaldetails.html [wolframscience.com]

It says there that for the prize, the notion of universality is to be judged
acceptable by the Prize Committee.

I clicked on Prize Committee:

http://www.wolframscience.com/prizes/tm23/committee.html [wolframscience.com]

And found these members:

Lenore Blum
Greg Chaitin
Martin Davis
Ron Graham
Yuri Matiyasevich
Marvin Minsky
Dana Scott
Stephen Wolfram

Since the prize was awarded, what definition of universality was used during
the deliberations?

In particular, Martin Davis, Ron Graham, and Dana Scott are subscribers to
the FOM list. What definition of universality are they using?

Harvey Friedman

followed by:

...
But, as I said in an earlier message, although the committee was kept
informed, we were never polled.

Martin

                                                      Martin Davis
                                        Visiting Scholar UC Berkeley
                                            Professor Emeritus, NYU

Let's see Wolfram explain that.

glass houses

[m2943 (1140797)]

Pratt asks: "How did an argument containing such an elementary fallacy get through
the filter?"

Proofs containing elementary errors are published all the time. Peer review is, and always has been, only a way of weeding out some percentage of bad submissions. Peer review that is strict enough to ensure that only correct papers get published would also result in the rejection of many good papers. In fact, some good papers that have advanced science have contained elementary errors.

And people who sit in glass houses shouldn't throw stones. Looking through Pratt's publication list, the first two papers I came across on a topic that I know a lot about should never have passed peer review.

Everybody who publishes makes elementary mistakes and makes a fool of himself sometimes; one should respond kindly and gracefully.

Re:duh

[schon (31600)]

the universality of Turing machines is... gosh, what the heck does this mean?

It means you need to study more for your CS exams.

Re:duh

[ispeters (621097)]

You sound like a troll since you're so belligerent, but, in case anyone else here is legitimately wondering what it means for a Turing machine to be universal, I'll try to answer.

Basically, a Turing Machine [wikipedia.org] is an abstract "computer"--it's a tape (a skinny piece of paper) that has a start but no end (it's infinitely long, but it has a start), and a read/write head that can zip up and down the tape writing, reading, and erasing symbols on the tape. The Church-Turing Thesis [wikipedia.org] postulates that a computable algorithm is any algorithm that can be computed in a finite number of steps by a Turing Machine. There are some things that look like algorithms and seem like they should be computable but are in fact impossible. The classic example is the Halting Problem [wikipedia.org].

Anyway, a regular Turing Machine only computes one function--it's a single-purpose machine. A Universal Turing Machine [wikipedia.org] is a Turing Machine that can simulate any other Turing Machine by interpreting a codified description of the other machine. Since every computable function is isomorphic to some Turing Machine and every Turing Machine can be simulated by a Universal Turing Machine, every computable function can be computed by a Universal Turing Machine. The computer you're using to read this is an approximation to a Universal Turing Machine (the RAM would have to be infinite in size to be a proper Turing Machine), and the codified descriptions that it interprets are the binary executables that you run on it.

Hope that helps,

Ian

Re:

[chgros (690878)]

No, not at all, for a Turing machine to be universal means that it can simulate any other Turing machine.
Gödel's incompleteness theorem (although I don't remember it so well) is more akin to the impossibility of the halting problem.

Re:

[by Anonymous Coward]

Apparently some Slashdot editors have still to defeat the gargantuan misteries of propositional calculus...

Re:Proven?

[EvanED (569694)]

Apparently some Slashdot editors have still to defeat the gargantuan misteries of propositional calculus...

Or, you know, English.

Re:

[mrpeebles (853978)]

I agree he could have been more diplomatic. Still, it is pretty crappy to let a student (an undergrad if I recall) publish a now embarrassing proof with an error that is apparently pretty obvious to an expert. That is sort of how I read this comment.

math the religion

[slew (2918)]

Sadly, it seems to be the case that there is only a small fraction of the math and science worshiper that have fallen from the "purer" faith and dare question the high priestess of truth promulgated by the science and math establishment.

Euler was probably one of the people responsible for some the old theorems that are the foundations of mathematics. Euler had some famous flaws in his early proofs (most notably his polyhedra formula and radical product proofs). These proofs were fortunatly repaired along

Summary for Average Joe Geek (formatting fixes)

[Otheus (897195)]

Sorry about the lack of spacing. I forgot I was posting HTML. Re-do here:

After following the links, I surmised the following:

Undergraduate Alex Smith submits a proof claiming that Wolfram's 2,3 Turing Machine is Universal. Next, Wolfram's staff review the proof for several *months*. Then, an announcement is made. Next, we have a notable computer scientist, Dr. Vaughn Pratt, who by the way, received his PhD under Donald Knuth, point out an "elementary" flaw in the proof. Wolfram's researchers and Wol

Re:wolfram inc. sinking deeper and deeper

[tlord (703093)]

Yes, exploited. It's a professional embarrasment at the start of his career. I'm not sure $25K is a fair price.

It's a basic logic problem. The original challenge problem could be restated: "Prove either that there exists a non-universal machine which emulates the 2-3 machine OR that the 2-3 machine can only be emulated by a universal machine." The proof does neither. Wolfram Inc. reps have come back with "Well, perhaps we should change the definition of universality!" Only, they aren't very concrete in offering an alternative with any rigor and the vague suggestions they are making don't add up (e.g., don't answer Vaughn Pratt's counter-example of paired push-down automata).

What the student proved here may turn out to be an interesting and useful result (not the universality of 2-3 but the universality of this interesting combination of machines). Students should be encouraged to work on such problems. Students should be encouraged to write as well as this student is learning to do -- it's a nicely presented paper (trivial formatting bugs aside). But students shouldn't be encouraged to go in those directions on false premises.

At least two interesting questions come from the students work. To Wolfram Inc.'s credit, they are pointing in the general direction of these new questions (even while not yet acknowledging their mistake). The new questions: Do there exist simple machines whose universality is undecidable (and might 2-3 be an example)? If 2-3's universality is either false or undecidable (and especially in the latter case) can we find any useful structure to what combinations of it with other machines clearly are universal?

I'll leave it as an exercise to figure out how raising those questions relate to the "Priciple of Computational Equivalence" in NKS but, meanwhile, leave the student out of it!

We'll see. With an additional step that "finitizes" the student's construction the proof is rescued and raised to the status of an important lemma -- but if that step isn't very quickly forthcoming, the prize -- in no small part an advertising vehicle -- was administered in a pedagogically misleading way.

-t

Re:

[jpop32 (596022)]

Jeez, Can someone please give me the short version explanation about why everyone is bagging on Wolfram?

Well, the very short version is that he is not a very nice person. And, basically, he said 'fuck you' to the traditional scientific community and went his separate way, turning from a prodigy to an outlaw. Which the traditional scientific community didn't appreciate much. And now they are, more or less, openly out to get him.

Which is, IMHO, unfortunate, because his scientific ideas should be judged on the