Yeah and if dependencies don't look all the way ahead, and just apply one by one until a loop happens, there is probably a construction that has the infinite even with a version of DC that doesn't have the +1/+1 effect.
(Edit: the rule that implies this interpretation is
"613.8c After each effect is applied, the order of remaining effects is reevaluated and may change if an effect that has not yet been applied becomes dependent on or independent of one or more other effects that have not yet been applied.")
As for Auspicious Ancestor the reason I didn't go with that was its types: Human Cleric
Both of which we have earmarked for the computation. Centaur Advisor does not collide with anything.
Yeah that's probably better to keep panharmonicon, but Gandalf the White seems really good anyway so it might not matter. Though the cowardice issue is still a problem.
Edit:
Ooh, what about Attendant of vraska as a 3 life / tick creature that we can't gain life from? It does force us to include a pretty useless vraska but maybe getting life back as a useable resource is worth it?
So dependency loops only look forward one 'step' of effects at a time?
That's even worse than the way I just thought it worked! (which was look forward through all of them and if a loop is found, break it by going in timestamp order)
I think this means that even a hypothetical DC without the +1/+1 part still has these problems, just from coat of arms!
Edit: and yes any creature decreasing in toughness from another being created is sufficient for there to be an infinite.
Issues with getting flash again, getting a boom keg again, getting the coat of arms on the opponent's side, etc etc.
yeah its tricky.
But profane memento doesnt work anyway, as we'd need three creatures not overlapping the 6 clocks, so they gain 3 per tick, but that's enough to make a simple infinite 3 clock machine, and gain 6 life a tick forever before swapping off of soul collector.
Yeah I think Profane Memento would be better but it is not legendary so we can given them a dozen of them and they live through the extra arcbonds easily. Having it be attached to the actual creature card saves that.
Maybe we can make it hard to copy artifacts? We don't actually need multiple copies of anything though having extra boom kegs is very convenient.
Hmm, maybe we can keep the opponent alive with something like a Lagonna-Band Elder that dies and is soul collectored every tick to keep them at constant life? That way if there is another arcbond creature they will die to the extra damage.
It is unfortunately easy lifegain for us and prevents panharmonicon and any lifelink creatures, so it's not great. But I think it at least solves the multiple arcbond problem?
ok the proof of flooding not being TC is a bit more complex I thought. Proving logclocks (maxclocks?) not TC via bounded states is easy and the flooding clocks can behave like them but that's not sufficient. Taking snapshots of the flooding computation's state at every power of 2 steps reveals the similarities. No flooding clock can trip more than once between powers of two, unless it is tripping every single step (or part of a cycle of clocks that trip every step, these clocks have an essentially constant effect on the computation and can't be perturbed by other clocks). So flooding computations still seem to be somewhat 'bounded' or 'approximated' by them but its not a perfect approximation.
This seems to still be a death knell for flooding computation, even if not a full fatal proof.
Edit:
I think there may be a way to make it so that using a different creature than soul collector's arcbond triggers ends up killing the opponent. Something like Profane Memento where if they stop getting soul collector triggers, they stop gaining life. (and die to all the extra arcbond triggers on the stack)
This doesn't fully work because we can give them a ton of them and they can gain enough life to unravel the computation and all that extra damage.
Ah! That cuts to the heart of it!
Yeah this isn't TC because after every non halting clock has tripped once, the max value for those clocks is bounded by the program (say by multiplying all the nonzero values, no tripped clock will have a value greater than that)
Thus the number of states becomes finite and this is not TC.
Darn.
Because I was gonna do a trick and point out that the computation works the exact same way **without** the multiplier and viola flooding waterfall is TC.
Instead we prove both are not TC.
!!
Idea for possible flooding style computation!! Not proven but looks promising!
So the problem with flooding clocks has always been that it quickly takes way too many ticks for clocks to trip and that makes the behavior explode in a pretty terrible way.
But what if the tick damage scaled up too?
There are two ways, 1 by 1 with something like Mechanized warfare but those cards can let us pick a black creature sometimes and mess things up. Rankle and torbran would be good for this but they need to hit the opponent, and Tor Wauki the Younger can trigger unwanted computations. Even if there was a good way to do this, this strategy makes things tricky and its not clear we can use it to simulate computation.
But we also have effects like furnace of rath to continuously double the damage the tick does. (or triple with city on fire)
What does this do? Well if we look at the clocks in base 2(or 3), instead of just doing a bunch of -1s we are also bit shifting the clock right.
This means that the number of ticks to trip these clocks is more like the log of its value instead.
Which starts to look a LOT more like normal clocks.
For example, a heartbeat clock for some output can double(triple) itself each tick, and also add that to an output clock that is always one more than the tick value.
I’m not sure this is TC but this looks so, so very much more likely to be TC than any other nearly flooding computation.
If this works this also has the minor benefit of flooding computations’ enormous outputs. (though it does lose the nice stairstep output from before allowing for easy arbitrary inputs for repeated computation.)
The main problem with this seems to be that manipulating clocks with large values doesn’t affect how long they take to trip. I’m not sure how big of a problem it is, but it prevents this from behaving exactly like normal waterclocks. We do have access to 250+ of them though so that’s probably enough to work with??
I think these logclocks (needs a better name?) have serious potential to be TC and save us with the simple change of dralnu’s crusade -> city on fire (or furnace of rath if base 3 doesn’t work for some reason)
However I am not adept enough to prove these TC, they seem capable of non-trivial non-exploding behavior, but do still have some of the quirks that make flooding clocks difficult to work with.
Looking at ways to keep the opponent alive during computation, a ravenous daggertooth style card shows it can be done with one slot, the other similar cards generaly aren't more useful, though metropolis reformer is kind of interesting, I there had been a desire for an ivory mask effect that ended up with a different workaround.
I'd still like some way to force only soul collector to be arcbonded, even if we have TYS or w/e way to copy spells. But I haven't found anything to prevent the computation just switching to another arcbonded creature to halt.
Yeah flooding clocks making multiplication cheap is nice, X does need to satisfy some divisibility properties, but we can force 9X/10 to be a square, or at least factor nicely. It basically just can't be a prime, or a small multiple of a prime.
Obviously with a larger clock we need to divide by more than 10, but that example is for the 7 clock spiral rise implementation. X doesn't even need to be all that large. Because the massive N is just a number of input zombies, and d and m are much smaller.
The real issue with cards like broodmoth and Mikaeus, the Unhallowed is that they can let us keep a bishop/necromancer on our side after kaervek's spite to mess with computation. That's why I went with soul collector/dread slaver, as any damage on things is from the opponent's side, and so they get anything that dies due to damage.
Hmm, good point, 4 seems unlikely. 5 seems like it should be possible, though I remember that being tricky. I'll think on it, now that things are a bit different maybe there's some way to fix it.
Each of the others have tradeoffs, so its not clear which of those options would be best. (I think plague for what we have right now but that could easily change.)
Edit: 5 is possible using temp flooding clock to multiply
rough proof of 5:
Convert clock to all positive values
Add large number X to each entry in the desired clock
So rows look like X+a, X+b , X+c, etc.
Replace x with X/10 in all the non diagonal entries and add new temporary flooding clocks filled at less than X/20. Now the diagonal entry is largest in the row (6X/10+6X/20=9X/10<X)
These new helper clocks are each less than X/20 necromancers making their nondiagonal entry
This takes at most X/20+X/20 ticks to go off = X/10 which is before any of the “real” clocks can resume
Makes up to X^2/400 tokens of the real type.
We actually just need exactly 9X/10, so X>360
Soul collector doesnt grant a type, and so can't keep an animated coat of arms alive forever.
edit: To be more clear, we only have coat of arms card that can return, if the boom keg is animated so is coat of arms. Once coat of arms dies, soul collector will die too, breaking the arcbond cycle. This does seem to puncture the Sliver legion version, but coat of arms is just generally better anyway.
the 6 dual typed creature cards can't share any types naturally too. There are a few overlaps in creatures we want (IE dread slaver is a zombie horror, which overlaps with rotlung being a zombie and with yarok being a horror).
How does comeuppance work with multiple arcbonds again? It doesnt say instead, so the damage half is a trigger? If it triggers and we can resolve another arcbond first, can't we just ignore it and do the same multiple arcbond tricks from before?
The idea is that dread slaver can only do damage via arcbond, arcbond can only be triggered initially with boom keg, and boomkeg can only be killed with kaervek's spite, so the Dread slaver must me on the opponent's side, and thus we can't get a bishop/necromancer/reanimator onto our side during computation.
what makes dread slaver work better than the other options (ie broodmoth) is that it can keep returning the actual cards forever.
(Soul collector works as well, morph is typeless and dies quickly)
(Edit: the rule that implies this interpretation is
"613.8c After each effect is applied, the order of remaining effects is reevaluated and may change if an effect that has not yet been applied becomes dependent on or independent of one or more other effects that have not yet been applied.")
As for Auspicious Ancestor the reason I didn't go with that was its types: Human Cleric
Both of which we have earmarked for the computation. Centaur Advisor does not collide with anything.
Yeah that's probably better to keep panharmonicon, but Gandalf the White seems really good anyway so it might not matter. Though the cowardice issue is still a problem.
Edit:
Ooh, what about Attendant of vraska as a 3 life / tick creature that we can't gain life from? It does force us to include a pretty useless vraska but maybe getting life back as a useable resource is worth it?
That's even worse than the way I just thought it worked! (which was look forward through all of them and if a loop is found, break it by going in timestamp order)
I think this means that even a hypothetical DC without the +1/+1 part still has these problems, just from coat of arms!
Edit: and yes any creature decreasing in toughness from another being created is sufficient for there to be an infinite.
Issues with getting flash again, getting a boom keg again, getting the coat of arms on the opponent's side, etc etc.
yeah its tricky.
But profane memento doesnt work anyway, as we'd need three creatures not overlapping the 6 clocks, so they gain 3 per tick, but that's enough to make a simple infinite 3 clock machine, and gain 6 life a tick forever before swapping off of soul collector.
Maybe we can make it hard to copy artifacts? We don't actually need multiple copies of anything though having extra boom kegs is very convenient.
It is unfortunately easy lifegain for us and prevents panharmonicon and any lifelink creatures, so it's not great. But I think it at least solves the multiple arcbond problem?
This seems to still be a death knell for flooding computation, even if not a full fatal proof.
Edit:
I think there may be a way to make it so that using a different creature than soul collector's arcbond triggers ends up killing the opponent. Something like Profane Memento where if they stop getting soul collector triggers, they stop gaining life. (and die to all the extra arcbond triggers on the stack)
This doesn't fully work because we can give them a ton of them and they can gain enough life to unravel the computation and all that extra damage.
Yeah this isn't TC because after every non halting clock has tripped once, the max value for those clocks is bounded by the program (say by multiplying all the nonzero values, no tripped clock will have a value greater than that)
Thus the number of states becomes finite and this is not TC.
Darn.
Because I was gonna do a trick and point out that the computation works the exact same way **without** the multiplier and viola flooding waterfall is TC.
Instead we prove both are not TC.
Idea for possible flooding style computation!! Not proven but looks promising!
So the problem with flooding clocks has always been that it quickly takes way too many ticks for clocks to trip and that makes the behavior explode in a pretty terrible way.
But what if the tick damage scaled up too?
There are two ways, 1 by 1 with something like Mechanized warfare but those cards can let us pick a black creature sometimes and mess things up. Rankle and torbran would be good for this but they need to hit the opponent, and Tor Wauki the Younger can trigger unwanted computations. Even if there was a good way to do this, this strategy makes things tricky and its not clear we can use it to simulate computation.
But we also have effects like furnace of rath to continuously double the damage the tick does. (or triple with city on fire)
What does this do? Well if we look at the clocks in base 2(or 3), instead of just doing a bunch of -1s we are also bit shifting the clock right.
This means that the number of ticks to trip these clocks is more like the log of its value instead.
Which starts to look a LOT more like normal clocks.
For example, a heartbeat clock for some output can double(triple) itself each tick, and also add that to an output clock that is always one more than the tick value.
I’m not sure this is TC but this looks so, so very much more likely to be TC than any other nearly flooding computation.
If this works this also has the minor benefit of flooding computations’ enormous outputs. (though it does lose the nice stairstep output from before allowing for easy arbitrary inputs for repeated computation.)
The main problem with this seems to be that manipulating clocks with large values doesn’t affect how long they take to trip. I’m not sure how big of a problem it is, but it prevents this from behaving exactly like normal waterclocks. We do have access to 250+ of them though so that’s probably enough to work with??
I think these logclocks (needs a better name?) have serious potential to be TC and save us with the simple change of dralnu’s crusade -> city on fire (or furnace of rath if base 3 doesn’t work for some reason)
However I am not adept enough to prove these TC, they seem capable of non-trivial non-exploding behavior, but do still have some of the quirks that make flooding clocks difficult to work with.
I'd still like some way to force only soul collector to be arcbonded, even if we have TYS or w/e way to copy spells. But I haven't found anything to prevent the computation just switching to another arcbonded creature to halt.
Obviously with a larger clock we need to divide by more than 10, but that example is for the 7 clock spiral rise implementation. X doesn't even need to be all that large. Because the massive N is just a number of input zombies, and d and m are much smaller.
The real issue with cards like broodmoth and Mikaeus, the Unhallowed is that they can let us keep a bishop/necromancer on our side after kaervek's spite to mess with computation. That's why I went with soul collector/dread slaver, as any damage on things is from the opponent's side, and so they get anything that dies due to damage.
Each of the others have tradeoffs, so its not clear which of those options would be best. (I think plague for what we have right now but that could easily change.)
Edit: 5 is possible using temp flooding clock to multiply
rough proof of 5:
Convert clock to all positive values
Add large number X to each entry in the desired clock
So rows look like X+a, X+b , X+c, etc.
Replace x with X/10 in all the non diagonal entries and add new temporary flooding clocks filled at less than X/20. Now the diagonal entry is largest in the row (6X/10+6X/20=9X/10<X)
These new helper clocks are each less than X/20 necromancers making their nondiagonal entry
This takes at most X/20+X/20 ticks to go off = X/10 which is before any of the “real” clocks can resume
Makes up to X^2/400 tokens of the real type.
We actually just need exactly 9X/10, so X>360
edit: To be more clear, we only have coat of arms card that can return, if the boom keg is animated so is coat of arms. Once coat of arms dies, soul collector will die too, breaking the arcbond cycle. This does seem to puncture the Sliver legion version, but coat of arms is just generally better anyway.
How does comeuppance work with multiple arcbonds again? It doesnt say instead, so the damage half is a trigger? If it triggers and we can resolve another arcbond first, can't we just ignore it and do the same multiple arcbond tricks from before?
what makes dread slaver work better than the other options (ie broodmoth) is that it can keep returning the actual cards forever.
(Soul collector works as well, morph is typeless and dies quickly)