Blaise Agüera y Arcas: evaluation

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…unning process to mix more water in some cases, but, right, it's it's it's it's not, it's not part of the code, let's say. Viral is the case in which the the code and the thing that is copied overlap. So in other words, some of the code that does the copying is actually some of the stuff that gets copied, but the, the code is not fully contained by what gets copied. So this is an incomplete replicator that would need to cooperate with another replicator in order to reproduce. So that's what I mean by viral. In the beginning of BFF, all of the replicators are inanimate and viral. The great majority are inanimate and a few of them are viral. A few of them happen to copy, you know, 1 of those, bytes that is actually an instruction that is doing the copying. But as you move toward, the time of jolation, which I've normalized to 1 here, you can see that cellular replicators suddenly emerge. So they can't emerge before, you know, about halfway through the run and they and they shoot upward at the end. And that's really interesting because that tells you that that the moment of a cellular replicator where the machinery for copying yourself is part of the thing that is copied, emerges through the symbiosis or the symbiogenesis of inanimate and viral, replicators. Okay. So a full equation would have 2 terms. It would have this reproduction and, you know, Laca Volterra type type term and it would have a merger or Smolchowski type term. 1 on the left is normal population dynamics, that's normal Darwinism, and the 1 on the right, you could think about the left as evolution and the right as revolution. Right? Those are the moments when things come together. Now, the population dynamics part for BFF looks like this. It's a little bit more complicated, but it has the same basic form as Lotka Volterra. There's a linear part on the left. I'm just writing that as a matrix, r I j, operating on the on the whole thing. And on the right, the reason that looks a little bit a little bit different from Lotka Volterra is that when something gets copied, it overwrites other stuff. So, so now we have to say, well, how does how does that suppress the populations of everything else in the soup? In order to figure that out, you have to look at niches. What are the bytes where something gets copied? And the overlap between the niches of 2 replicators tells you, you know, how much 1 thing getting copied, you know, how likely it is that that will overwrite, something else that shares its niche. Okay. The symbiogenesis part is a bit of a mess, so I'm not gonna go through it. I hope that's okay. But it looks just like Smolchowski, just gnarlier. The the reason that it's gnarlier is because Smolchowski has only binary fusion, between 2 parts. And in BFF, sometimes a bunch of things come together, so you have to take into account these kernels that have more than 2, parameters in them. Also, when things come together, they don't necessarily look like the sum, of of the things that came together. o you have to take into account these kernels that have more than 2, parameters in them. Also, when things come together, they don't necessarily look like the sum, of of the things that came together. You could have something that is 3 bytes long or something 5 bytes long come together and the result that copies itself is only 2 bytes, 1 byte from each 1 or or anything, right along those lines. So to to account for those complexities, you end up with a a much more complicated k term, but it's essentially the same as as a small hovske coagulation. To prove that this kind of symbiogenesis is needed in order to get, these complex programs, you you can do a very simple intervention, which is, when you're interacting 2 tapes, you can sort of do it in a sandbox before committing. And in the sandbox you see whether, whether a new replicator arises, and if so, what replicators is it made out of. In other words, you know, when you look at the source, you can see what, you know, whether whether any of those, source bytes were actually the output the outputs of copies of some previous replicator. And if so, then you have a tree. You have an ancestry tree for that for that replicator. That means that you can think about the depth of such a tree, you know, how many things have come together, and you can limit the depth of that tree. You can say, if the tree depth exceeds 10 for a new replicator, then I'm gonna I'm gonna actually not not do this interaction. I'm gonna take them back apart, pretend it never happened, put it back in the soup, and try again. If you if you limit the depth of the tree to say 24, then the number of operations that you have to block, the number of interactions you have to block is actually very small. You only have to block 1 in 1000 operations, but that 1 in 1000 operations is really important as it turns out. If you block those, no gelation will happen. You need, at least 3 depths of 20 or so in order to get these complex, programs. So this is very nice proof that symbiogenesis is what what is needed in order to get to these complex tapes. When you do that blocking, you end up with sort of logistic curves for the populations of all the replicators in that soup. They they should they go up and then they saturate and stabilize. That's fun because it lets you do a little bit of math. So as you can see, you know, not only do things go up and saturate, but then there's some random, oscillations and those oscillations can be correlated. Sometimes you can see the 2 of those populations go up and down together, that so means that they they may be collaborating with each other, and sometimes they go in opposite directions, they're anti correlated, and that means that they're competing with each other because they're, you know, 1 is overwriting the other for instance. So that's what 1 would expect from, from off diagonal production and competition from those those equations I wrote earlier. And if you linearize the dynamics around that steady state, then you can sample the correlations in those population fluctuations and you can reconstruct the matrix r. those equations I wrote earlier. And if you linearize the dynamics around that steady state, then you can sample the correlations in those population fluctuations and you can reconstruct the matrix r. I'll skip the details of how 1 does this, but this is a classic fluctuation analysis. You solve the Lyapunov equation and you get a Jacobian, and from that you get the matrix R. And the matrices R look really cool. First of all, they have a strong diagonal that tells you that by and large, things replicate themselves, just as you would expect from Lotka Volterra. But, there's some other stuff going on here as well. Aside from that dominant diagonal of self replication, there is some negative stuff off the diagonal and some positive stuff off the diagonal. The negative stuff off the diagonal you can see looks largely symmetric about the diagonal and that's as you would expect too. Basically, if a competes with b, then b competes with a. 2 things that are that are fighting for the same niche are are in a kind of 0 sum relationship with each other. But the cooperation part where where, where something helps something else is not symmetric, and that's as you would expect too. Just because a helps b or enables b doesn't mean that b enables a or at least not directly. Right? So there are complex cycles in this in this graph on the right of of of co dependency or enablement. So, negative component is symmetric, positive component is is asymmetric and there's this big diagonal. Do the submatrices that are about to undergo symbiogenesis have any special properties? They do. So in other words, if it's these, let's say, 4 rows and columns that are about to about to undergo symbiogenesis, you can ask, what are the what are the eigenvalues of that matrix, of that sub matrix? And it turns out that that they are generally cooperative. So essentially, if you if you were to pick random rows and columns from this matrix, then you get high dimensional picture of the rank of the matrix. But when you look at the ones that actually combine, it's much lower rank. They're already, working together. So in other words, there's a relationship between the r and k parts of this equation. Symbiogenesis happens among guys who are already working together. Not all the same, not independent, cooperative. Here's another really interesting thing. If you look not at the r matrix but at the Jacobian itself, then you can find the signs of imminent instability in it, of when it's about to pop, when it's about to go run away and gelate, or gel. You don't say gelate, say gel. Right? So in particular, if you block the depth of the possible trees to a low number, then the eigenvector the eigenvalues of the Jacobian are always negative, meaning that the system is stable. But as you look at larger depth ceilings, you find that more and more of this leading eigenvectors, or the real parts of those leading eigenvectors, pop positive and that means that the system is about to blow.…

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