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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.
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.
My work here is dung.
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 weapon in the arsenals of the world is so formidable as the will and moral courage of free men.-Ronald Reagan
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.
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.
Breaking Into the Industry - A development log about starting a game studio.
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.
I suggest you read Slashdot
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.
This is my sig.
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.
Apparently some Slashdot editors have still to defeat the gargantuan misteries of propositional calculus...
Or, you know, English.
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 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 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.
Anyway, a regular Turing Machine only computes one function--it's a single-purpose machine. A Universal Turing Machine 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
(Score: 2,3 Possibly Informative)
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.
only if the infinite tape is mostly unused
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