Does the pigeonhole principle rule out the possibility of losslessly simulating a universe the size of our own?

Say you had a very powerful computer and wanted to run a completely lossless simulation of a universe approximately the same size as our own: $$10^{80}$$ particles.

Each particle in the simulation has properties like velocity, mass, charge, etc. Assuming that your program didn’t use any tricks (like compressing this simulated universe by storing groups of 1000 particles as if they were one), does the pigeonhole principle mean that you would need a computer made out of at least $$n$$ particles to losslessly simulate a universe of $$n$$ particles?

I say this because I don’t see how it’s possible to store all of the physical properties of a particle on a piece of hardware without using at least one actual, physical particle.

Am I right about this? Does this mean it would be impossible to ever hope we could create a high resolution, lossless simulation with a number of particles similar to the actual number of particles in our universe?

Is there a guide/reference for character hairstyle possible in D&D Forgotten Realms universe?

I was playing a D&D session as a female cleric and, at some point, the party encountered some kind of wind monster. As I tried to cast focused spells, the DM argued that I couldnt because my hairs kept getting in my face and made me unable to focus.

Later on, I told the DM my character would ponytail her hair but he refused stating that I had nothing to tie my hair with nor have any knowledge of hair-tying as female characters usually do not tie their hairs in D&D universe.

Regardless of the “hairs in the face break your focus” situation, is there a reference for hairstyle for DnD universe ? Or can I trust Google about medieval tied hairs?

Resource/Methods for “leveling” spells from the Harry Potter Universe?

I’m playing a homebrew version of “Broomstix” with my nephews and am looking for a resource that doesn’t just list all the Harry Potter universe spells and their functions but also lists the approximate “level” of the spell (age at which it is learned at Hogwarts for the average student). I don’t have time to read the books again, but does anyone know of a resource (website) that does this? Alternatively, can you suggest a rule of thumb for deciding how advanced a spell is?

How Secure is the Universe Repository?

I’m wondering about the security of downloading things from Universe. The Ubuntu help page seems to say that the community can make updates to pieces of software in that repo. Does that mean that just anybody can make changes, or that the maintainers of the original software can make changes?

For example, there’s an instance of Discord maintained in Universe, so can just anybody edit that instance, or only the Discord developers? Or do I misunderstand the whole thing?

How would you have avoided the Miss Universe 2015 mistake?

In case you missed it, over a week ago Steve Harvey incorrectly announced Miss Colombia as winner of the Miss Universe 2015 pageant. He later apologized and corrected himself; Miss Philippines won.

Harvey essentially blamed his cue card for being confusing, and I don’t blame him for blaming said cue card:

Assume that it’s not a publicity stunt (I don’t think it is):

If you’re a program director of the pageant, how would you prevent this mistake? [Added Jan 1] In other words, how do you ensure the announces the correct winner and then immediately announces the first runner-up?

I’m assuming most would re-design the cue card (I would), but that doesn’t have to be the solution.

Here’s some requirements and restrictions, though:

1. The host may not have a teleprompter or large cue cards held by a stage hand (because the finalists are onstage with the host).
2. The host must refer to the third-place contestant as “Second Runner-Up” and the second-place contestant as “First Runner-Up” (though they don’t necessarily need to be referred to as such on paper).
3. Once the final three contestants are determined, the host announces the Second Runner-Up who then leaves center-stage. Once the final two contestants are determined, the host announces [edit: the winner, referring to her as as Miss Universe YYYY, and then immediately names the First Runner-Up.]

Assume that

1. Any printed information that’s seen by the host will only be seen by the host until The End of Time.
2. [Edit again] Over the previous three pageants (2012-2014), there has been no commercial break between the announcement of the Second Runner-Up and the winner. In other words, there’s usually not sufficient time to hand a separate cue card to host when only two contestants remain.

I have my solution (I’ll post it) but I’m curious to see what others plan.

Data Structure for applying transformations upon universe of tuples

Problem

I want a data structure which describes a universe of tuples all of length $$n$$, $$U$$. In $$U$$ we associate each $$tuple$$ with a non-negative value, which we will denote as $$U.count(tuple)$$. If $$U.count(tuple)=0$$, then we consider $$tuple$$ to not be in $$U$$. I want to be efficiently update these $$U$$ with five operations, some of which involve two fixed constants $$maxVal$$ and $$minVal$$. (to clarify, $$maxVal$$ and $$minVal$$ should not change throughout the process, and should a tuple in $$U$$ should never have a value which exceeds $$maxVal$$)

1) append(U,c): yields an updated universe $$U’$$, $$c$$ should be below $$maxVal$$. For each tuple $$t$$ in $$U$$, make a copy $$t’$$, and append $$c$$ to $$t’$$. For each $$t’$$, $$U’.count(t’) = U.count(t)$$.

2) add(U,c,i): yields an updated universe $$U’$$. For each tuple $$t$$ in $$U$$, we create a copy $$t’$$, and add $$c$$ to $$t'[i]$$. If $$t'[i], then then $$t’$$ is in $$U’$$ and $$U’.count(t’) = U.count(t)$$. Otherwise $$U’.count(t’)=0$$ and for our purposes we no longer need to keep track of it, and can exclude it from $$U’$$.

3) merge(U,i,j): yields an updated universe $$U’$$. For each tuple $$t$$ in $$U$$, we create a copy $$t’$$, we pop $$t'[j]$$, and add its value to $$t'[i]$$. In this case, there can be multiple tuples $$t$$ which correspond to $$t’$$ in $$U’$$, thus let us initially say $$U’.count(t’)=0$$ and then add to this. Now, if $$t'[i] < maxVal$$, $$U’.count(t’) = U’.count(t’) + U.count(t)$$.

4) finish(U,i): yields an updated universe $$U’$$. For each tuple $$t$$ in $$U$$, we make a copy $$t’$$ and delete $$t'[i]$$. Again, for the same reasons as in 2, let us initially say $$U’.count(t’) = 0$$. If $$t[i] > minVal$$, then $$t’.count = t’.count + t.count$$. At the end, all $$t’$$ with non-zero count are included in $$U’$$.

5) combine(U1,U2): yields an updated universe $$U’$$, $$U_1$$ and $$U_2$$ should have same length tuples. For each $$t$$ in $$U_1 \cup U_2$$, $$U’.count(t) = U_1.count(t) + U_2.count(t)$$

My thoughts so far (messy)

If we did this naively, and represented $$U$$ as a list of the tuples with non-zero, this would be horribly inefficient. Let’s assume $$U.count(t) \neq 0$$ for all $$t$$ that don’t exceed $$maxVal$$. Then, with $$maxVal = 10$$, a universe for tuples of length $$4$$, that’s 10,000 if statements for the add operation…

I think one optimization would to have a tree, where the first level has nodes for each value of $$t[0]$$, and the child of a node are the next element in the tuple. I’d imagine you’d do this by having a list of all your first level nodes. Each node itself, is represented as a tuple of its value, and a list of its valid children, or $$U.count(t)$$ for leaves. With add at lower levels, you have much less updates to do, and by ordering children in a fixed way according to their value, and keeping childless nodes as placeholders, your only need to do the validity check once too. Furthermore, to make append efficient, instead of having the levels be consecutive indices, we could make the order arbitrary, describe the tree-level to tuple index conversion with another list. For stuff like combine I am less sure what’s a good way to do things, maybe it would be beneficial to actually sum the counts of shared tuples, but maybe there are a clever cheats or work arounds…

My explicit questions

1a) Is there an efficient data-structure which performs these operations?

1b) Otherwise, are there some general rules of thumb for data structure which are relevant for this problem or my proposed solution?

2) Same as above but for the case where combine is applied to a very large number of universes, instead of just 2.

Datalog Program Equivalence with Fixed Universe Size

It is well known that to compute equivalence of two Datalog programs (or equivalently of two first order formulas with the least fixed point operator) wrt all possible inputs and universes, is undecidable. If we fix the universe size to be N, it is clearly decidable in time exponential wrt N. My question is whether one can do better, maybe even in time logarithmic in N (and maybe much worse wrt the length of the programs). Note that equivalence of two nonrecursive Datalog rules can be done in constant time for all inputs, let alone for fixed N, but is NP-Complete wrt the rules’ size.

Number of nodes in Van-Emde-Boas tree of universe size u?

The universe size $$u$$ in vEB trees is square rooted at each level until $$u = 2$$. So, unlike search trees, the branching factor is different at each level.

The height of the tree is $$h = \lg \lg(u)$$ and $$u$$ is an even power of 2, $$u = 2^{2^k}$$. I tried to calculate the count using the sum:

$$\sum_{i=1}^{h-1} 2^i\,2^{i + 1}$$

But it doesn’t work. Any idea on how to do the math?

[ Comics & Animation ] Open Question : Who is your top 5 favorite actor/actress who played the villian role of Marvel Cinematic Universe?

Movies and TV Shows. Here’s my top 5: 1. Vincent D’Onofrio (Kingpin) 2. Josh Brolin (Thanos) 3. Michael B. Jordan (Erik Killmonger) 4. Tom Hiddleston (Loki) 5. Michael Keaton (Vulture) HM: Andy Serkis (Ulysses Klaue), David Tennant (Killgrave), Kurt Russell (Ego), Hugo Weaving (Red Skull), Mickey Rourke (Whiplash), James Spader (Ultron), Sigourney Weaver (Alexandria), Jeff Bridges (Obadiah Stane), Ben Barnes (Billy Russo), and Cate Blanchett (Hela)

Do Stephen Wolfram and Gerard ‘t Hooft propose that literally every Cellular Automata is a universe?

Both Stephen Wolfram and nobel laureate Gerard ‘t Hooft think that the universe is a Cellular Automata.

As far as I know, ‘t Hooft developped a series of frameworks to build different models of Cellular Automata and Wolfram also proposed a framework where network nodes could produce different Cellular Automata universes.

Both of them proposed specific models of Cellular Automata to describe our universe (or rather they are working in a Cellular Automata description that can be applied entirely to our universe), but since they proposed a framework to create different Cellular Automata models, this makes me think that these authors think that literally every type of Cellular Automata correspond to a universe.

For example, apparently Stephen Wolfram in his book “A New Kind Of Science”, chapter 9, he defines a series of network nodes that could be capable of reproducing all models of cellular automata. But I am not sure about this and I would need someone that knows better this field to confirm this to me

So, does anyone here know of this is correct? Does anyone here know these works well enough to tell me if these physicists think that every Cellular Automata model corresponds to a universe?