So we use a 15 state 2 symbol UTM. To compile that to the waterfall model we use 30 clocks that each encode one of the TM transitions. Everytime one of those empties it corresponds to the TM taking a step with that transition. 1 clock encodes the left half tape in base 2. We need 7 clocks to help manipulating the tape: multiply the half tape by 2 when we move right and write the correct symbol onto the tape afterwards. Or divide with rest when we move left and read the current symbol. Those 8 clocks are also needed for the right half tape. We get a total of 46 clocks. In general we need 8+4m+nm clocks for n state m symbol TMs.
Then the conversion to the flooding waterfall model happens. There are comments in ais523's compiler that explain how it works. To be honest I don't fully understand the magic that is happening in there. It looks like for the bulk of the simulation each TWM clock is represented by 4 FWM clocks, split into two clusters that alternate emptying. Both clusters also have a fixed number of helper clocks. When those clocks can possibly empty is strictly managed and the clocks balance each other out in a delicate construction to manage useful comparisions despite the inherently exponential growth of the numbers in the model. We also need additional flooding clocks to get the FWM program into the correct starting position. Depending on which clocks need input this might double the number of required clocks. (Maybe one of the clusters can always start empty? That would give the roughly 6 flooding clocks per original clock I see in practice)
Running the TWM program for the 15x2 UTM through the compiler gives me a required 300 clocks. This is different from the 296 I got before, since I included a sizeable input to the right half tape clock. It looks like without that input some startup clocks could be skipped, but for our operations we definitely need that input.
In MTG terms we need 1 creature type per flooding clock. Wall is not usable. We also need 1 additional type that always stays alive so we don't fizzle our swaps. This is best accomplished by using 1 more type to add a heartbeat. It looks like the creature type count has grown to 315 by now, so using a couple types more than we though won't be a problem. The number of bishops required stays at just under 14 million (13944268)
Apparently the 15x2 UTM simulates bi-tag systems, so I can't say how we would encode a normal TM into input for that. I don't remember any bounds for comparing MTG(n) with BB(n). I just assume the input conversions are doable with a few exponentiations, and are outscaled by the repetitions anyway.
Edit: Minor correction for the writeup: We don't need the opponent to control a creature for the last Soulblast before the final precursor trigger. We can just target any of our creatures when declaring the targets of the spell. Only after that do we pay the additional cost and sacrifice all creatures. We sacrifice everything, Soulblast goes on the stack, we get all our golems. Then Soulblast fizzles because we sacrificed the target, but we don't care about that.
Also the paragraph about us controling a dragon is wrong now after you changed the types of our initial sacrificed bishop.
Gotcha, thanks. That conversion is the big thing I want to get a better grasp of, then.
And yeah, the writeup is currently halfway through some changes I need to untangle at some point.
In the writeup, you have "Spend the last mana to have Exalted Sunborn activate Crypt Rats’ ability, dealing 1 damage to each of the other 29 creatures and both players, and putting us at 31 life.", but that ability does not self-exclude.
What would be especially fun to include is strategies involving a traditional stage and maybe even a hyperstage, but I'm not entirely sure where to start on trying to fit those into a relatively small number of cards. Are there any existing ones on the more compact end?
Well, we had the previous incarnation of the writeup, which went to 20+ cards and and went up to a bunch of stages (I know you guys already know this, just bringing it up to talk about it ), but I presume we can improve on this, maybe a bunch? Well, I guess the 12 card hyperstage deck will pretty much obselete anything we've done before. We could probably build off of that efficient hyperstage to get a new record megastage, and even a gigastage that we never got quite working? And we could also address the w^w construction that plopfill described earlier, and try to come up with both the smallest deck that can make it work, and also the highest damage 60 card deck that uses it.
In terms of the category where these would become relevant, the obvious one would be to use the same rules and just forbid anything Busy Beaver-like. For more natural rules that don't just forbid Busy Beaver specifically, we could require not just the deck but also the specific play sequence (well, we could still do this by precisely describing the conversion from a Turing machine to an exact monster creature setup, then referring to a specific Turing machine to convert - I think there's like a 45 state, 2 symbol machine that gets up to epsilon_0, which is gonna be very hard to beat with our current computable methods, and also 150-ish machine that can run the full Bashicu Matrix System, which is supposed to go incredibly far, although it hasn't been proven exactly how strong it is. So maybe not this one.) We could also say the deck must be "computable" in nature, although this isn't formally defined; computability refers to functions not numbers, so while we can say the entire setup of having a Flooding Waterfall Machine implemented and considering all possible starting permanents is uncomputable, a deck that can only run the FWM under a finite collection of starting permanents, well that well just be finitely many numbers assigned to a finite collection of starting permanents, so uncomputability doesn't enter into it. There is the obvious thing where we don't actually know what the Busy Beaver setups are, and there is no general method for figuring out what those are... but that depends on our state of knowledge, and not the actual deck, and I don't think there is a way to limit what Busy Beaver numbers we can figure out, other than each arithmetically sound axiom system will have their own maximum Busy Beaver number that they can prove, so for instance, there is a maximum n for which ZFC can decide the value of BB(n)... however, #1: we'll probably never learn that number, #2 ZFC is just one axiom system of infinitely many possibilities, #3 dealing with the formal system of mathematics that we assume seems going too far away from the initial challenge :D. Dang, maybe the best way to define this challenge is just to rule out Busy Beaver methods.
Whoops, I kinda just rambled off topic there! But I'll leave it since it's about the non-Busy Beaver challenge which I find interesting, and is now more interesting with plopfill's w^w construction. Also still interested in the possibility of higher-order Busy Beavers, if we're ready to tackle that one!
Yeah, definitely the w^w one. I think that was the biggest driver in the idea, actually.
Using the Saw hyperstage as a base for a megastage and gigastage certainly sounds exciting, but is it viable? I was under the impression that Saw doesn't tend to play well with other w^N methods.
For the secondary writeups, I wasn't thinking it had to be particularly rigorous formal challenges so much as just going, like, "here's something we can do in N cards without Saw in Half", "here's something we can do in N cards without Arcbond", etc.
That does look like a useful start. What sort of things did you have in mind for supporting it?
I'm not sure. I was thinking of using an instant or sorcery card to feed into the stage (by adding life or putting permanents onto the battlefield) that is limited in how many times it can be cast (perhaps by having an additional cost), but haven't found a good choice yet.
That does look like a useful start. What sort of things did you have in mind for supporting it?
I'm not sure. I was thinking of using an instant or sorcery card to feed into the stage (by adding life or putting permanents onto the battlefield) that is limited in how many times it can be cast (perhaps by having an additional cost), but haven't found a good choice yet.
Thanks, fixed properly now.
The only thing I can think of to round it out in two cards is Dosan's Oldest Chant and Gravitic Punch, but those numbers aren't particularly high. For a spell to feed into the stage, Kuldotha Rebirth looks appealing, but nothing quite fits the bill to round out the list.
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And yeah, the writeup is currently halfway through some changes I need to untangle at some point.
Whoops, fixed.
That does look like a useful start. What sort of things did you have in mind for supporting it?
Yeah, definitely the w^w one. I think that was the biggest driver in the idea, actually.
Using the Saw hyperstage as a base for a megastage and gigastage certainly sounds exciting, but is it viable? I was under the impression that Saw doesn't tend to play well with other w^N methods.
For the secondary writeups, I wasn't thinking it had to be particularly rigorous formal challenges so much as just going, like, "here's something we can do in N cards without Saw in Half", "here's something we can do in N cards without Arcbond", etc.
I'm not sure. I was thinking of using an instant or sorcery card to feed into the stage (by adding life or putting permanents onto the battlefield) that is limited in how many times it can be cast (perhaps by having an additional cost), but haven't found a good choice yet.
The only thing I can think of to round it out in two cards is Dosan's Oldest Chant and Gravitic Punch, but those numbers aren't particularly high. For a spell to feed into the stage, Kuldotha Rebirth looks appealing, but nothing quite fits the bill to round out the list.