User blog:Vel!/Hilbert ITTMs

Within hours of setting up our IRC channel ##googology, an interesting discussion occurred:

[12:32:55]  Anyways, let's talk math [12:33:15]  some dude is raving about "infinity-dimensional Turing machine" http://googology.wikia.com/wiki/User:65.26.80.144 [12:33:32]  I didn't even waste my time reading that [12:33:46]  However it is actually an interesting idea if we put a TM in Hilbert space [12:33:55]  Why not, that's fun [12:33:56]  and use ITTM instead of TM [12:33:57]  It is [12:34:18]  However we would need to revise the rules a bit [12:34:26]  But how would ruleset work [12:34:41]  We would have infinitely many ways to move [12:34:46]  How to choose one? [12:35:19]  We need a way to specify an arbitrary natural number, and have the TM move in that direction [12:35:46]  the important thing is that this number must be specified at runtime [12:36:08]  I don't really see how this is supposed to work [12:36:20]  You could do something very, very clunky [12:36:59]  In addition to L and R, the motion M reads the 1D tape along (x, 0, 0, ...) [12:37:21] <FB100Z> and adds that vector to the TM's current position [12:37:55] <FB100Z> it's a start, although it basically leaves the TM stranded after moving outside of the line :/ [12:38:16] <Wojowu> Hilbert space is uncountable, right? [12:38:26] <FB100Z> I didn't use the right term [12:38:54] <FB100Z> it's a countably infinite set of natural-number coordinate positions [12:39:08] <FB100Z> equivalent to Bowers' X^X space [12:39:08] <Wojowu> Okay [12:39:15] <Wojowu> I see [12:39:42] <Wojowu> So only finitely many coordinates can be nonzero [12:39:45] <Wojowu> At a time [12:39:51] <FB100Z> Right. [12:40:07] <FB100Z> But if we use ITTM instead of TM... [12:42:33] <Wojowu> ...then we have no control over how many coordinates are finite [12:42:53] <Wojowu> *are nonzero [12:43:09] <FB100Z> How come? [12:43:30] <Wojowu> Well, using your idea of adding vectors [12:43:42] <Wojowu> Machine can write infinitely many ones [12:43:45] <Wojowu> And boom [12:44:33] <FB100Z> what are the consequences of that? [12:44:56] <Wojowu> Then we are working in uncountable Hilbert space [12:45:13] <FB100Z> Sure. [12:45:26] <Wojowu> Okay, so it's not a problem [12:45:33] <FB100Z> However it seems we can only access countably many positions in the Hilbert space [12:45:54] <FB100Z> because there are countably many rules that can lead up to the M motion [12:46:48] <FB100Z> In other words, at all points the machine's position vector must be somehow computable with ITTMs, I believe :(

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[13:10:05] <FB100Z> Regarding Hilbert ITTMs [13:10:27] <FB100Z> although exploring higher-order spaces is interesting, I hypothesize that it ultimately gets us nowhere [13:10:46] <Deedlit11> Hilbert ITTMs? [13:10:47] <Wojowu> Yeah [13:11:05] <Wojowu> The idea is this [13:11:27] <Wojowu> We have an ITTM with one tape head and other head inside a Hilbert space [13:11:52] <Wojowu> We now use the tape to specify where we move Hilbert space head [13:12:11] <Wojowu> I think it's clear where this is going [13:12:22] <FB100Z> so the colors in the 1D tape encode a vector that gets added to the Hilbert head's position [13:13:23] <FB100Z> it's important that the machine is infinite-time, so we can access positions with infinitely many nonzero coordinates [13:13:37] <Deedlit11> so we are talking about Z^Z rather than a Hilbert space really [13:13:45] <FB100Z> yeah, I didn't quite know what to call it [13:13:48] <Wojowu> Actually, yeah [13:13:58] <Deedlit11> or 2^Z [13:14:02] <Wojowu> It's a discrete subset of Hilbert space [13:14:48] <Deedlit11> sounds interesting [13:15:08] <FB100Z> despite the fact that the larger space is uncountable, I think that we can only access countably many positions [13:15:38] <FB100Z> due to ITTM rules being countable [13:15:43] <Deedlit11> right [13:15:54] <Wojowu> But it'll be hard to keep track of which positions have been visited and which haven't... [13:16:10] <FB100Z> which leads me to suppose that ultimately this is computationally equivalent to ITTMs [13:16:32] <Wojowu> I think that's correct, although it's not that obvious

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[13:24:22] <Wojowu> Guys! [13:24:26] <Wojowu> We were wrong! [13:24:35] <Wojowu> This model actually is stronger! [13:24:44] <FB100Z> Wait whaaaaat [13:24:52] <Wojowu> Let me modify it a bit: [13:24:53] <Deedlit11> waaat [13:25:00] <Wojowu> We have two tapes, scratch and control [13:25:18] <Wojowu> Scratch does what it does, control tape controls head in Hilbert space [13:25:33] <Wojowu> So now suppose we do the following: [13:25:44] <Wojowu> We simulate some machine M on scratch tape [13:26:12] <Wojowu> On control tape we write instantaneous description of M at every step [13:26:22] <Wojowu> And we move to that position in Hilbert space [13:26:26] <Wojowu> And flash that cell [13:26:48] <Wojowu> If configuration repeats infinitely many times, this flag will flash infinitely many times [13:26:59] <Wojowu> And will ultimately end up being on [13:27:07] <Wojowu> So if this configuration happens one more time [13:27:11] <Wojowu> We can see that [13:27:15] <Wojowu> This way [13:27:21] <Wojowu> We can solve halting problem! [13:27:30] <FB100Z> Huh. Let me process that a bit. [13:27:41] <Wojowu> Because machine doesn't halt iff some it's configuration repeats w times [13:27:53] <Wojowu> *its [13:28:17] <FB100Z> Scratch is the Z^Z head, right? [13:28:34] <Wojowu> No, scratch is another linear tape [13:28:39] <Wojowu> I modified model a bit [13:28:47] <Wojowu> So we have to linear tapes and Hilbert space [13:28:54] <Wojowu> But this can also work on one tape

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[13:29:29] <FB100Z> If you're right that it is stronger, then the following extension should also work: [13:29:43] <FB100Z> using the Hilbert heads to control vectors in even larger Hilbert spaces [13:30:10] <Wojowu> It's hard for me to imagine this [13:30:17] <Wojowu> But I guess it might work too [13:30:27] <FB100Z> Yeah, I can't really wrap my had around anything of cardinality Beta_2 either :( [13:30:31] <FB100Z> *head [13:30:34] <Wojowu> Okay guys, I'm really sleepy [13:30:42] <Wojowu> So I'm leaving for good for today [13:30:51] <Wojowu> Have good time thinking about this! [13:30:53] <Wojowu> Cya o/