Cellular Automata (Cont.)

The last thread filled up quite quickly and I didn't want to see discussion of the subject die needlessly. Below is a neat little rule I found for the Generations algorithm which uses only one cell to create a fractal-like structure. Pic related is the successor at around 120 generations or so. Rule: 2578/123/23

>>9558 changing the number of states to 8 yields a wholly new kind of vaporwave album art.

>>9559 That looks like a Mandala.

Is this >>5399 physicist still here? I have been reading about the spin glasses and the Ising model lately.

What happened to the old thread? I can't find it with dashchan.

>>10859 I archived it. You may find it here: >>>/archive/17

Reposting the spin glass question.

>>10872 Stephen Wolfram conjectured that a very simple one-dimensional CA is Turing complete, and in 2004 Matthew Cook proved this. See the link below: http://wpmedia.wolfram.com/uploads/sites/13/2018/02/15-1-1.pdf

>>10859 I was surprised to see it gone too. Good thing it got archived. Unfortunately it seems as if the CA discussion has died needlessly in direct contravention of the OP's wishes. Maybe we ought to fix that. There's a lot of stuff we haven't yet covered and many potential advancements out there waiting to be found. https://hackernoon.com/cellular-automata-and-their-elegant-complexities-musings-about-blockchain-quantum-computing-and-c8d2c2d1cf70?gi=43d022201aa9 This article is a fairly recent one. It was posted at the tail end of the last thread's lifetime. It doesn't have that much in the way of hardcore technical information, but since we have a whole new thread here, it may be a good idea to post an intro piece of this sort to grab attention since it has a lot to do with many topics that many people here have an interest in. Also, since I had a lot of fun posting about CA art near the end of the last thread, here's more. Source is here: https://www.deviantart.com/redcurry79/art/Hexagonal-Game-of-Life-743349659

>>10874 Here's something from the same artist in a very similar style but with different coloration that makes all the difference. https://www.deviantart.com/redcurry79/art/Hex-Pattern-736029321 Now where should this thread truly begin? One thing missing from the last thread was discussion of other types of automata. CAs aren't the only game in town for automata. Far from it. There are automata of every imaginable configuration. An example of this that piqued my curiosity is the continuous CA. Continuous CAs can have a range of values for each of their cells instead of discrete sets of values. The concept extends even further if you decide to do away with discrete cells as well and have the entire automaton be continuous, which is called the continuous spatial automaton. I'd like to get somebody in here who has a good understanding of all these other automaton types and compare the uses for them with CAs. In some situations, it may be a good deal easier to code one of these alternative automata, but I think this needs a more expert opinion than mine. We also had a discussion about CAs with Darwinistic traits under the possible name of genetic or memetic CAs in the last thread, but that was never really followed up on to any great extent. Substantial discussion relating to neural networks will probably feature in this thread as well.

I really should get back to that image feedback program I started...

>>10884 >myopic related

>>10884 elaborate? what was your approach?

Those look pretty sweet, makes me quite interested in this but lack of time kinda kills it to get into this for me.

>>10932 Check out Golly, it's FLOSS and available for desktop and Android. Comes with some preloaded algorithms for you to play around with. http://golly.sourceforge.net

I've been reading about Edward Fredkin's Finite Nature argument, that a cellular automata is behind the universe. It's pretty impressive. Turns out many famous computer scientists are of the same opinion. Fredkin was a close associate of Feynman too. https://www.forbes.com/sites/startswithabang/2017/09/26/it-from-bit-is-the-universe-a-cellular-automaton/ https://www.theatlantic.com/past/docs/issues/88apr/wright.htm http://www.bottomlayer.com/bottom/finite-all.html

Is there anyone following this thread? I'm the CA developer from the last thread and made significant progress. Willing to share if anyone is interested. Especially the physics poster who was asking about spin glasses. I will blow your mind if you're still around.

I'm a nonprogrammer but this stuff really interests me. I have some mathematical maturity . Is there an introductory enough book I can consult to make a project and learn to program with CA?

>>12502 im also interested in wanting to do some cellular automata. in R . is there an introductory resource

>>12502 >>12505 I don't know about R, but in Python I would recommend Sayama's work. Scroll down to the cellular automata section. He covers Game of Life and more. To use his CA code, download PyCX package from Sourceforge. In there you'll find Game of Life and many more snippets.

>>12424 Are you the guy who was developing the ergodic CA/AI stuff? I was interested to hear more about that. I just got busy and haven't been to Lainchan in a while. I haven't really gone to many other boards either. I think we should get back to doing something here. You might not actually come back since this is almost a month and a half old message, but we'll lose what I thought was the most compelling part of the thread.

>>11537 Konrad Zuse, the inventor of the computer, he wrote about the universe being cellular autamata as well. It was in German though so if course American pop scientists would be remembered.

>>12853 I'm checking here regularly, so all is well. Are you the spin glass physicist? Perhaps there could be ergodic implications that could be uncovered in the future, I don't know. What I do know is that the code I've written in the past thread is a model of many theoretical and real world systems. I haven't spent any time improving the code. The code gave results and I had to make sense of them. So far I know: 1. It has both first order and second order transitions 2. Upper and lower bound first order transitions are distributional conjugates, or complementing each to 1. 3. The system acts like an axial Ising model. 4. Second order transition point is extremely close to the exactly calculated result. 5. These transition points are trigonometric, thus the system can be represented in hyperbolic space. 6. Some liquid crystal models behave in this manner: crystallization and quenching Ising models go hand in hand. I could go on and on. I don't think these are relevant to the thread.

>>12855 >Are you the spin glass physicist? I was the guy who developed a bit of a recursion obsession and wanted to focus on building neural networks and other forms of bioinspired programming with CAs, as well as the generalized use of CAs as a programming paradigm all their own. I actually figured out how this can be useful since then, so it's somewhat less of a noobish trait than before. I did research a while back towards the end of the last thread based on what one of the posters said about needing a recursive structure to code a CA recursively, and found that a project had already been done where a CA was implemented on a tree structure, which grants it the recursive properties that you can never get by putting it on a grid. I was thinking of using a recursive CA of this style to create a genetic program simply and easily, since one of the most popular ways of writing a genetic program is by using recursion, and then using the program to generate neural networks. You might try to write a recursive neural network directly as well. This was discussed at the very beginning of the first thread and intermittently throughout, but it was never seriously followed up on until now. Unfortunately I have very little actual knowledge of machine learning as opposed to interest, so anyone who has that knowledge would be very much appreciated for posting some of it here.

>>12856 In my opinion, the extension of my work to neural networks is still somewhat speculative. I believe the binary neural network as presented in the attached papers might be of relevance. The neural network quine, as they have called it, exports the node of a neural network as its output, thus duplicating itself. The output node is very alike to the initial random distribution of my model. What my model does is flattening the noise in one direction, allowing me to calculate relations through the number of bottlenecks each distribution generates. This is speculative of course, but gives some neat insights. Self-Replicating Neural Network #Quine - A neural network that learns to output its own weights: https://arxiv.org/abs/1803.05859 ("introduces the possibility of continual improvement through natural selection.") And the information bottleneck of deep learning is a theory by Naftali Tishby. https://www.quantamagazine.org/new-theory-cracks-open-the-black-box-of-deep-learning-20170921/

>>10872 >>12424 >>10556 Ah, that's me. I'm back and ready to have my mind blown. I'm also the Julia raytracer guy.

>>12865 Nice, I thought you were long gone. Alright then: I'm referring to this code >>277 I was talking about some majority condition back then, I had no clue. That condition turns out to be the axiality. The neighborhood function I ran, first forced the system into a next-nearest neighbor formation (Moore), then frustrated it with a nearest neighbor formation (Neumann). Pic related. I've been trying to fine tune the system changing these neighborhood parameters and and thought it was the central element of the code. The code turns out to be a variant of Per Bak's Axial Ising Next Nearest Model (ANNNI). The frustration between nearest and next-nearest neighbors leads the system into arrest at some point, but until then, the system is ferromagnetic along the axis, and paramagnetic on the normal axis. Here's where things start to get insane. This dual magnetic property brings a two-fold phase transition with it. Usually this happens at the Curie temperature, but the system is so bizarre that the ferromagnetic-paramagnetic transition happens as a 1st order transition, at both lower and upper bounds of the system. After these bounds are passed, the system remains in constant criticality, until the fluctuations reach a global minimum at the Curie point, when a second order phase transition happens, and I can get an approximation to Onsager's 2D Ising critical point from my model much more accurately than fuarrrking deep neural networks that got published in top journals. The neighborhood frustration in an axial direction leads to quenching of the Ising model. The reverse process of renormalization is taking place. Instead of decimation (zooming out), system is zooming in, and that's why I believe I can read what deep learning nodes have in it. This is crystallization at its best.

>>12867 >>>/archive/277

>>12867 Let's see if I can follow you. You are considering a 3D cellular automata with randomized initial conditions, you identify the on/off states with spins in various extensions of the Ising model, and the CA rules allow you to find some easy and reasonably accurate way to sample states from its Gibbs distribution at any given temperature? And in particular, it predicts the phase transition temperatures more accurately than other methods?

>>12874 2D cellular automata, but yes. There is no exact solution for the 3D Ising model. In less than 100 lines of code. The upper 1st order boundary generates a Lennard-Jones potential, meaning we are not only talking about A phase transition, but phase transitionS, due to the correlated nature of the spin states. These become degenerate at the Curie point.

>>12867 I forgot the pic related: First is the Moore and Neumann neighborhoods for cellular automata. Second is the next-nearest and nearest neighbor points for Ising models. The last one is from a recently published pre-print on short-range interacting particles. In summary, they suggest a diagonal formation to minimize total potential energy (degenerating the Lennard-Jones potential), which is exactly a decimation process, which in turn is exactly a frustrated cellular automata neighborhood. Which allowed me to plug these boundaries into hyperbolic space and solve for the geometries behind this quenching.

>>12867 I have yet another speculation about what you've created even though I don't understand much of this post or spin glasses. I know of another system similar to the Ising model called the cellular Potts model. I found it a while ago without really having a clue what to do with it. I think I have a somewhat better understanding of it now. The cellular Potts model is a model of muscle contraction. Muscle contraction is a stochastic process just like other human biological processes. This means that the Fokker-Planck equation can be used to describe it. One of the strangest mysteries in all of science to me is that the Fokker-Planck equation, the heat equation and the Schrodinger equation are all equivalent. I wonder if the CA you have here is related to this. It should be a very interesting line of research to see how far this CA's versatility extends. I also find the NN quine relevant to my interests due to the possibility of writing a genetic program as a recursive CA and using it to quickly and easily train this NN to do basically whatever the user wants. You could write a GP where the population consists of NN quines and have them mate with each other instead of merely self-replicating to get results at a much faster pace, and maybe implement the NN quine as a CA as well. Sadly I'm not an expert on machine learning, so I may not have anything worth investigating here. Sorry it took so long to get back to you. Things have been hectic lately. But I at the very least want to avoid taking a multi-week break from this.

>>12948 >One of the strangest mysteries in all of science to me is that the Fokker-Planck equation, the heat equation and the Schrodinger equation are all equivalent. I wonder if the CA you have here is related to this. It absolutely is. The Potts model is related to the XY model: the molecules are in rotation. The square-lattice Ising model, which my axial model is based on, is directional. Only from these distinctions one can derive a whole lot of analogies. For instance, if you'd want to look into the Nobel Prize in Physics 2016, it's a variation of the XY model, the Kosterlitz-Thouless (KT) transition. My model is an analogy of liquid crystal transitions of a specific type. This type is the equivalent of superconductor phase transitions between Type-I and II. Pics related The generality of spin-glass systems is not a secret. There are articles out there speculating that the universal model of physics is one giant Ising model. It's also the reason why the Ising model is jokingly referred as the model that has fed the most physicists: when academics run out of publishing material, they quickly make up a model that behaves like an Ising model and voila: published in a respectable journal. Don't worry about it, work tempo is also getting in my way lately. I'll be interested to keep in touch.

>>12952 Can you make a 1D CA or an otherwise somewhat more lightweight CA that acts similarly? Or how about a NN quine version of it? It sounds like a model with numerous applications.

>>12856 >>12864 ML grad student here, with background in deep learning + some graph theory + I've always loved CAs although I never did much with them. I want to help, but I need a bit of spoonfeeding to get me up to speed. I have read >>277 and I'm trying to follow the stuff you're posting here, but I have to admit I'm a bit lost. The concepts I am not getting are: - recursive cellular automata - how do trees fit into this - how does hyperbolic geometry fit into this, although I'm guessing it's related to the fact that you can embed trees in hyperbolic space - most of the physics you posted, although >>12874 cleared it up a bit (?) - why are you so fascinated with the NN quine?

>>12956 Good to hear from a ML specialist. I'll try to explain this the best I can. >recursive cellular automata This refers to at least two different concepts. The confusion is a result of the field being niche enough that nomenclature isn't as set in stone as it is elsewhere. I use it to refer to a CA defined on a recursive data structure such as a tree instead of being defined on a grid as CAs usually are. I've seen another project using this term to describe a CA where each new state depends on every previous state. I have yet to decide which one is more entitled to the name. >how do trees fit into this See above. >how does hyperbolic geometry fit into this, although I'm guessing it's related to the fact that you can embed trees in hyperbolic space I didn't know this was possible. Guess you learn something new every day. >most of the physics you posted, although >>12874 cleared it up a bit (?) That post was another guy. This stuff about the Ising model and spin glasses that some people are talking about here is over my head. >why are you so fascinated with the NN quine? Besides the novelty of the NN quine, which really should have been investigated a long time ago, it seems like a great way to develop a well-rounded ML regimen which is accessible to anyone because of its autodidactic properties. I like seeing a concept like quines which was commonly held to be useless be discovered to have more to it than most people thought. My interest in the ML side of this is about learning how to use CAs as more general implements of programming and particularly as part of control software for embedded systems.

>>12957 I see. So the idea would be to build some sort of programming paradigm based on CAs, and then use it to build neural networks? How would you go on about doing that, tho?

>>12958 >So the idea would be to build some sort of programming paradigm based on CAs, and then use it to build neural networks? This is exactly what I want to do with CAs, but I want to have other uses for CAs besides just building NNs. One example of the sort of thing I'm going for is how you can build a blinking light pattern either by programming it like a conventional digital clock or by using a two-state CA as the main element of your control program. You have to have a way to distinguish between situations that preclude the effective use of CAs and situations that don't. The problem is that there isn't really a developed doctrine for this. Another good example of CAs getting some practical programming use is when Wolfram used Rule 30 as a RNG, though this is a deprecated method now. I suspect that the way to tell how and when to use CAs in a program is based on the Wolfram typology. You can theoretically render any program as a CA, but some programs have a natural inclination to be written that way and not others. >How would you go on about doing that, tho? I hoped you had the answer to this. I only get to work on this intermittently for the most part. I've attempted to get this off the ground for a while now to little effect.

>>12955 >Can you make a 1D CA or an otherwise somewhat more lightweight CA I don't think so. The entire model depends on neighborhood frustration as shown in >>12881

>>12956 >>12957 >>12958 >>12959 Programmer of the model here. I need to emphasize that I've built the model first and then spent a significant time trying to understand what the model is doing. My advice would be to leave the terms used in the last thread out of discussion. They are not as relatable compared to exact descriptions I will be providing. Quickly to summarize this evolution: >recursive CA is now a periodic boundary axial neighborhood frustration >ergodic CA is now a spin-glass or Ising model > how does hyperbolic geometry fit into this, although I'm guessing it's related to the fact that you can embed trees in hyperbolic space > why are you so fascinated with the NN quine? These two are related. The numeric solutions found in the model are trigonometric and the model's evolution is similar to a geometric figure called "the Gabriel's Horn". https://en.wikipedia.org/wiki/Gabriel%27s_Horn The figure's geometric name is a pseudosphere and to calculate its trajectories, hyperbolic transformations are required. Pic related, some neat triangles start showing up. These hyperbolic definitions are the hot soykaf in cosmology right now. The quine: 1. outputs the node of a deep learning neural network, which is awfully similar to the random initialized state of cellular automata. This is called the black box of deep learning, and there are articles comparing the deep learning process to an "Information Bottleneck". If you haven't noticed, the pseudosphere is by definition, a bottleneck. Hyperbolic transformations are used to project the global shape of the earth on a 2D map (Mercator projection) and I believe the actual process behind the deep learning could be read in the hyperbolic space. What my model does is frustrating this distribution enough so a human can see the bottlenecks. It's a dictionary between human and neural networks. 2. If you go and read von Neumann's first released works about cellular automata, he addressed the replicability of the model as a must. Now, I'm not sure a NN quine is suited for this, but it's a step in the right direction.

>>12956 >>12957 >most of the physics you posted, although >>12874 cleared it up a bit (?) >That post was another guy. This stuff about the Ising model and spin glasses that some people are talking about here is over my head. It's pretty straightforward. It's a model of a collection of atoms, or more properly, a chain in 1D and a lattice in 2D and 3D. The spin in the spin glass is similar to the 0-1 boolean state, the spin is either up and down. There are more complicated definitions, such as the -1 - 0 - 1 spins, but the 0-1 boolean spins are called the half-spin type. Glass is the positionality disorder that happens when the system starts to evolve. Within a window glass, you would see a high level of disorder in its structure, unlike a crystal. What we are trying to accomplish is have these spins dictate terms on their neighbors. That's where the cellular automata comes in. The well-known nature of cellular automata is having a set of rules that allow the neighbors to decide the state of a cell. What I've done in the previous thread is the exact opposite: I wanted a cell to dictate the terms of its neighbors. Now we have a half-spin, or a cell with 0 or 1 state. The frustration arises when one cell tells to its right neighbor "you better spin 1 my friend" and the cell on the other side telling the same neighbor at its left "don't listen to that schmuck, spin 0!". I've made the necessary adjustments for this based on Moore and Neumann neighborhood types of cellular automata. When the system evolves enough that these frustrations can't be solved by directionally shifting cells (move them somewhere so they can't fight over their neighbors), the system locks itself down. It's called an arrested state. Arrested states are profoundly important from a computer science point of view, because Turing's definition of a halting problem specifically states a computer can't decide on its own to halt without an instruction. Well, I beg to differ Alan.

>>12965 >boolean Binary* What I’m trying to do is use the binary neural network as shown in the pic >>12864 To describe what is called a “binary system” https://icerm.brown.edu/materials/Slides/sp-s18-w1/Crystal_problems_for_binary_systems_%5D_Laurent_B%C3%A9termin,_University_of_Copenhagen.pdf This is where I am right now.

>>12959 Off the top of my head, I see only a significant path that one might take to build a CA-powered NN: 1. have the weights be a continuous CA 2. let the CA evolve 3. give an input to the NN and see what is produced as output by the input + CA system By tuning the rules of the CA, you could control the response of the whole network. What I just described is VERY akin to reservoir computing, although tuning the CA may constitute a step forward w.r.t. echo state networks and similar models. One cool thing that I'm thinking is to use a convolutional layer to compute the CA (I should have some code lying around to do just that) and then learn the weights of the convolutional layer by backprop. This would give you a dense NN whose weights are the state of the CA, and the CA evolves thanks to another set of weights. Probably impossible to train properly, but you'd have reduced the number of weights from input_size * output_size to 9. If we can get genetic algorithms in the mix we have achieved the holy trinity of AI, too. >>12964 >>12965 >>12966 Let me preface by saying that I've spent the first year of my phd building NNs with non-Euclidean latent spaces, but I'm still having a hard time following you, Lain. You seem to be collating a lot of stuff together and I can't really see where you're aiming. >If you haven't noticed, the pseudosphere is by definition, a bottleneck. but what does that have to do with the information bottleneck of NNs? They seem to me like two substantially different concepts. >some neat triangles start showing up yes, it's pretty known property of constant-curvature manifolds. How does that relate? >I wanted a cell to dictate the terms of its neighbors. How is that different that changing the update rule? The next state is still going to be a function of the 8 neighbours. >Turing's definition of a halting problem specifically states a computer can't decide on its own to halt without an instruction Not really, it says that's not possible to decide a priori whether a computation will halt. >Well, I beg to differ Alan. the halting problem's been proved logically. And finally, I still don't see any real link between the CA, the quine, and the binary network. I feel that you might actually be up to something here, but you should try to explain this better.

>>12967 >but what does that have to do with the information bottleneck of NNs? They seem to me like two substantially different concepts. What is information? It's the resolution of uncertainty as defined by Claude Shannon. Information is measured by entropy, which is the outcome of a random process. In my example, the initial cellular automata distribution is completely randomized. You could imagine I'm calculating its entropy. >yes, it's pretty known property of constant-curvature manifolds. How does that relate? The model is producing ideal triangles, which are later popping up in real-world applications of golden ratio :) >How is that different that changing the update rule? The next state is still going to be a function of the 8 neighbours. Hence, the recursiveness. This is why I don't want to use the term recursive CA. Recursiveness is a property of the system, not the characterization. But when a cell determining its neighbors is determined by its previous neighbors, that's pretty recursive to me. >Not really, it says that's not possible to decide a priori whether a computation will halt. which by definition means that you can't construct a computer program on these premises without adding a halting instruction, knowing that the halting problem is undecidable. >the halting problem's been proved logically. That's exactly what I'm challenging. By whose logic? Do you really think if we advance to supercomputing computational logic as we know it will still be a thing? >And finally, I still don't see any real link between the CA, the quine, and the binary network. I feel that you might actually be up to something here, but you should try to explain this better. This much I concede. Brainstorming could help.

>>12969 >The model is producing ideal triangles, which are later popping up in real-world applications of golden ratio :) You can observe this effect in the pic related. Left is the accident-vehicle pattern that keeps up popping all over the world. When you zoom down to a daily basis, that's where the phi keeps popping up.

>>12969 >That's exactly what I'm challenging. By whose logic? Do you really think if we advance to supercomputing computational logic as we know it will still be a thing? Suppose you had a magical oracle H, such that for any program P H(P) will return true if P halts, false if P doesn't halt, and H always halts. P here is assumed to be bundled with its inputs, that is P = P'(x) for whatever input x you want. Construct a program H', which does the following: if H(H') == true, then run forever. else, halt. Does H' run forever, or halt? If H' runs forever, then H(H') = false, so H' halts. If H' halts, then H(H') = true, so H' runs forever. this is a paradox, which means that the introduction of H was logically inconsistent. Doesn't really matter what computational or logical model you're using, as you can see we made no assumptions except that you can encode programs as inputs to other programs, the disjointedness of true and false and "halting" or "running forever", and the existence of branching primitives. All of these are true of anything that could reasonably be called a computer. We didn't even assume H is a program: we assumed it's just something that a computer can do. Interestingly godel's incompleteness theorems follow pretty similar lines. If you want to challenge this, be my guest, if you can prove it you'll be considered the greatest genius to ever walk the earth.

>>12971 >if H(H') == true, then run forever. H' is a program that runs: if H'(H(H')) == true, then halt. ECQ, or from contradiction, anything follows. I've just placed a paradox in your paradox, so you can contradict while you contradict. Enjoy your troll science. And it doesn't matter if you place another paradox, because after contradiction, anything follows, so it all works out for me. You silly Lain, I'm not a genius. I just went down the rabbit hole a long time ago.

>>12972 The H' I constructed doesn't take inputs or return anything(we can just assume that it always returns true when it halts), so I don't really understand what you're talking about at all. >I've just placed a paradox in your paradox that doesn't resolve the paradox in the slightest. >and it doesn't matter if you place another paradox, because after contradiction, anything follows, so it all works out for me it is true that the principle of explosion means that you can prove anything by assuming something false, but that's irrelevant because the thing you're assuming is false.

>>12973 er rather, you can prove anything by making contradictory assumptions. You're free to work with contradictory logic, but don't expect anybody to listen to you when you say you've proved something with the principle of explosion.

>>12974 Working with contradiction is exactly what Godel did. His work is liar's paradox in mathematical terms.

>>12973 Let's try something else. >All unicorns are red. This statement is true. How do you contradict it?

>>12977 >How do you contradict it? I don't, the statement you have made is vacuously true.

>>12978 Here's what I do. >All unicorns are red. >All unicorns are blue. >All unicorns are green. >All unicorns are apple. >All unicorns are banana. >All unicorns are soykaf. . . . . . .

>>12979 yes, all of those are vacuously true. Do you have a point?

>>12979 >>12980 Then I get to stretch on the meaning of red and green and blue and work on their relativity. The reality itself is based upon such descriptions that we take for granted. It is these implicit agreements we make with each other that allows us to set up logic. My point is, this logic allows me to stretch anything I want to anywhere I want. If one can prove a contradiction, I can double contradict that. If there is a vacuous statement, I can infinitely expand it. Don't hate the player, hate the game.

>>12981 the only reason "all unicorns are _" is true for any _ is because there are no unicorns. If there were a unicorn, only some of those statements would be true. This is assuming that unicorns are well-defined mathematical objects, we're not doing linguistics here. >If there is a vacuous statement, I can infinitely expand it. indeed you can say a lot of vacuous statements, but they're vacuous. Since there actually are computer programs, when we talk about them we're not making vacuous statements. You made no substantiative refutation to my proof whatsoever, if you say you can contradict it you should do so. I'm not gonna accept "well I can do whatever I want because reasons!" because as you have said, there are implicit agreements about what is logical and what is not logical, and if you break those agreements you're not doing logic.

>>12982 Hold on, I'm trying to find a link. I'm not trying to irritate you, but I'm afraid if I keep on talking without substantiating anything, I'll start looking like an asshole.

>>12982 Okay I've got it. This will stray a bit off topic, but here it goes: Stuart Kauffman is a theoretical biologist at the Santa Fe Institute. He is working on the mechanisms that entail development of biospheres, and his specific interest is in self-organizing structures that create their own constraints, in which they develop. Kind of similar to how humans build their own logic and then define their lives as a product of their logic. https://arxiv.org/ftp/arxiv/papers/1303/1303.5684.pdf https://arxiv.org/pdf/0907.2492.pdf Read these carefully and think about what the cause-and-effect structures mean to a logical structure, especially thinking from a time-arrow perspective. In my opinion, if a model like Stu's is possible, then I can define a space in which a program can first set up its constraints (or axioms) and then law its rules on its own that allows the program to halt at a specific setting. That is what I have in mind with arrested states. Its main property is topologic, it's not driven by logic.

>>12982 >>12984 It's called the Work-Constraint Cycle. Work is defined as "constrained release of energy into relatively few degrees of freedom". Constraint refers to "a reduction on the degrees of freedom of the elements of a system exerted by some collection of elements, or a limitation or bias on the variability or possibilities of change in the kind of such elements." In Kauffman’s terms, “Work begets constraints beget work”. You have a chicken-egg situation in your hands. Life, Lain, is a giant feedback loop. Your logic is built on cause-and-effect. https://halshs.archives-ouvertes.fr/halshs-00792440/file/Constraint_2013_web.pdf

>>12985 In other words, "the cycle is inevitably a thermodynamic irreversible process, which dissipates energy and requires couplings between exergonic (spontaneous, which release energy) reactions and endergonic (non spontaneous, which require energy) ones, such that exergonic processes are constrained in a specific way to produce a work that may be used to generate endergonic processes, which in turn generate those constraints canalizing exergonic processes."

>>12984 these papers are pseudoscientific gibberish. We're talking about formal logic, not physics or biology. We can simulate formal logic with physical systems, like brains, but formal logic is something that exists only in the abstract, like an ideal form. If you want to change the rules, *you're talking about something else*, plain and simple. If you assert something paradoxical, you've left logic behind and you can no longer claim logical certitude of your results. That doesn't make them useless, but it does make them no longer logical. Call them something else. Yes, you can always write a program that halts. You can even make a programming language such that every program you write with it halts(look into Coq). A program running on a physical computer will eventually stop when it runs out of power. But you can't know ahead of time whether *any* program will halt, you can't write every program in Coq, and if a computer runs out of power before it can finish computation that says only something about the system you're using to simulate a program, not the program itself. Please understand lain, this is not something that mathematicians or logicians or philosophers are happy to say, it dashed many hopes and made a lot of people very angry when Godel came out with his incompleteness theorems. But we have to say it, because it's incontrovertibly true.

>>12988 Yeesh, what Godel did was to prove that the idea of a mathematical proof being infallible, is in fact fallible. Get a hold of yourself Lain. It has shown we can't have an axiomated set theory that can bypass Russell's paradoxes, and what we have now in Zermelo-Fraenkel is a bastardized fragment of Frege's and Hilbert's idealized fantasy. Physics and biology did not spring out of mathematics like computer science did, however, and their dependence on formal logic is far less than you believe. Physics is more intimately tied to mathematical foundations, but it has always been biologists leading the way in areas where mathematicians and logicians shunned like you are doing now. We owe chaos theory to biologists like May, a field that was discredited for decades. Synchronization is a booming research area. It is disappointing that you are discrediting the work of an accomplished scientist like Kauffman because it is incompatible with your worldview, but let me remind you something: mathematics is an idealized version of the world you are living in, and your logic is a simulation of human activity in a formalized structure. We owe language to logic, yes, but there are many more ways of communicating than putting things into words. Does synchronized behavior adhere to formalized inferences? (I'm withholding the use of the word 'logic' since it gets you ticked off) Anyway I won't let this thread spin off into a flamewar, so I'm ending my contributions here. Feel free to tag along.

>>12989 >Yeesh, what Godel did was to prove that the idea of a mathematical proof being infallible, is in fact fallible. this is not true. Godel proved that for any axiomatic system that can formalize arithmetic, it is either incomplete(there are true-but-unprovable statements) or inconsistent(there are statements which can be proven both true and false). It showed a very particular thing, and in fact it is equivalent to the proof of the halting problem I gave above(computation <=> proof). You can't say that actually you can write a program which will determine whether every program halts, and then turn around and use godel's incompleteness theorems. They're the same thing. >and their dependence on formal logic is far less than you believe. I believe nothing in particular about their dependence on formal logic, I'm saying physical/biological conclusions are in fact irrelevant to logical considerations. >and your logic is a simulation of human activity in a formalized structure. I think it's the other way around, when we do logic we are simulating the ideal formalized structure. >We owe language to logic, yes I disagree

>>12967 >have the weights be a continuous CA Why this type of CA? There are just so many different kinds of NNs that it's hard for someone who isn't a specialist in ML to know how even a respectable portion of them work. >What I just described is VERY akin to reservoir computing Is it now? >One cool thing that I'm thinking is to use a convolutional layer to compute the CA (I should have some code lying around to do just that) and then learn the weights of the convolutional layer by backprop. This would give you a dense NN whose weights are the state of the CA, and the CA evolves thanks to another set of weights. I understood little of this, but can we have this with only one set of weights? >Probably impossible to train properly, but you'd have reduced the number of weights from input_size * output_size to 9. We can't have a NN that can't be trained properly. >If we can get genetic algorithms in the mix we have achieved the holy trinity of AI, too. Interesting, but after seeing the NN quine I don't know if anyone should bother with this since they can now train themselves.

>>12984 I'm also a computer scientist. I'm looking to expand what we know as learnable or computable (different domain spaces). I am having trouble understanding how this bio-logic can do more than turing machines. Additionally, the quantum turing machine can solve problems in bqp space-- where at least some nonempty subset is unsolvable by classic turing machines. https://www.quantamagazine.org/finally-a-problem-that-only-quantum-computers-will-ever-be-able-to-solve-20180621/ https://en.m.wikipedia.org/wiki/Quantum_Turing_machine Between the quantum turing machine and the classical turing machine, a ton of space is covered. What space could this bio-automata cover?

>>13128 The Turing machine, and the Haltung problem that we’ve covered ITT relies on certain axioms for the problem definition. The self-construction of constraints is akin to a self-construction of axioms. The Kauffman bit I’ve shared can be rephrased (just for informative purposes, nothing concrete) in a mathematical manner like this: To know a number, you need a number theory. To create a number theory, you need the knowledge of a number. This is similar to the work/constraint definition of Kauffman. They define each other.

>>13131 >the Haltung problem that we’ve covered ITT relies on certain axioms for the problem definition. sure, but so does this bio-evolutionary whatever. The really neat thing about the halting problem(and godel's incompleteness theorems, which I stress are the same thing) is that they apply to *anything* which can express peano arithmetic, independent of other assumptions. >>13128 I don't think they know anything about computability theory.

>To know a number, you need a number theory. The naturals can be bootstrapped as the free monoid of one generator.

>>13139 >I don't think they know anything about computability theory. Whatever pal. >>13142 Don’t take number theory literally. It’s any number generation scheme (hence the disclaimer).

>>13139 Godels incompleteness theorems rely on self contradiction such as the language ALAN (all Turing machine encodings that are not accepted by the Turing machines they encode or that define invalid Turing machines). If the ALAN Turing machine encoding is accepted by this turing machine then it's not in the language ALAN and is falsely accepted. Else it's not accepted when it should be. This is fundamentally different from the halting problem.

>>13144 I think the Lain is confused with the Halting problem and Hilbert’s Entscheidungsproblem.

>>13139 >I don't think they know anything about computability theory. There is no "they" here. Only one poster has been actively researching this aspect of CAs as far as I can tell. I've been focused on trying to understand CA applications for machine learning. I don't know enough about computability theory to have much of an opinion on what people seem to be arguing about here. I also know relatively little about ML, but I'm attempting to improve my knowledge in this area. But my post that asked about how to determine which CAs are suitable for the purpose of ML at >>13007 got drowned out by the talk about Turing machines. It seems like the ML guy left the discussion. I only know that some computer scientists have theorized without any proof so far that random Turing machines have fundamentally more computational power than normal TMs.

>>13144 suppose you have some proposition about the natural numbers, with of course only a finite number of quantifiers. Call it P. Write a program V(P) which checks whether the given statement is true for each set of inputs (n1,n2,...nk). V(P) halts when it finds a counterexample, else it continues forever. So: if you have a program H that can determine when other programs halt, you can decide any statement P about the natural numbers by running H(V(P)), including the Godel sentence. Flipside: if you can decide any statement about the natural numbers, then you can encode every program as a natural number and write a proposition P such that P(n) = true iff the program encoded by n halts. P can be encoded as a program, so it qualifies as an oracle. Thus the halting problem is decideable iff every statement about the natural numbers is decideable. I left out all the boilerplate about encoding constructions and stuff because it's tedious. >>13151 ML probably has a lot more interesting applications, but unfortunately it is nowhere near by areas of expertise. It would be interesting to see if there are specific CA's which lend themselves to rapid evolutionary convergence for particular problem areas or something

This thread could use more discussion. It's bereft of actual implementations, unlike the last thread. I've been playing with cellular automata in APL and some other environments, recently: >>15664 Since the APL is so concise, I'll add it to this post as well; this is Rule 30 in one line of APL:N←(1 1,~N)=(0,N,0)∨N,0 0 Here's some functions that make it easier to work with: ∇ R←R30 N R←(1 1,~N)=(0,N,0)∨N,0 0 ∇ ∇ L R30I R ⍝ The L is the repetition count and R the pattern. R◊→0↓⍨1≠1⌈L◊L←L-1◊R←R30 R◊R←(-~¯1↑R)↓(~↑R)↓R◊→1 ∇ I have other implementations as well and I may branch into other cellular automata later. What implementations have you been up to writing lately?

these threads were promising, i was expecting one of you guys to actually create a small intelligent agent (or at least something akin to small ai) but it's just masturbatory talk in the end shame

>>12855 >5. These transition points are trigonometric, thus the system can be represented in hyperbolic space. Turns out the automaton is pumping out the roots of an equation (approx. 0.8535 and 0.1465) as phase transition critical points. See the code below, the equation is at the bottom: import matplotlib matplotlib.use('TkAgg') from pylab import * import numpy as np L = 100 # size of 2D lattice: LxL p = 5 / 9 # Approximation to upper bound of 2nd order phase transition # p = 4/9 #Approximation to lower bound of 2nd order phase transition # p = float(0.5 + (1 / (2 * sqrt(2)))) #upper bound of 1st order phase transition # p = 0.8536 # p = float(0.5 - (1 / (2 * sqrt(2)))) #lower bound of 1st order phase transition # p = 0.1464466 # p = (cos(π/8)+ cos^2(π/8)) /4 # best approximation to Onsager's 2D Ising Model Curie point ## # initializing randomly assigned states with probability p def init(): global c, nc c = zeros([L, L]) for x in xrange(L): for y in xrange(L): c[x, y] = 1 if random() < p else 0 nc = zeros([L, L]) ## # visualizing the content of an array def draw(): cla() imshow(c) ## # count of upper neighbors of a live cell's Moore neighborhood def number_of_upper_neighbors(x, y): upper_count = 0 for dx in range(-1, 2): upper_count += c[(x + dx) % L, (y + 1) % L] # print upper_count return upper_count ## # count of lower neighbors of a live cell's Moore neighborhood def number_of_lower_neighbors(x, y): lower_count = 0 for dx in range(-1, 2): lower_count += c[(x + dx) % L, (y - 1) % L] # print lower_count return lower_count ## # count of right neighbors of a live cell's Moore neighborhood def number_of_right_neighbors(x, y): right_count = 0 for dy in range(-1, 2): right_count += c[(x + 1) % L, (y + dy) % L] # print right_count return right_count ## # count of left neighbors of a live cell's Moore neighborhood def number_of_left_neighbors(x, y): left_count = 0 for dy in range(-1, 2): left_count += c[(x - 1) % L, (y + dy) % L] # print left_count return left_count ## # count of von Neumann neighbors of a live cell def number_of_Neumann_neighbors(x, y): Vertical_count = 0 Horizontal_count = 0 for dy in range(-1, 2): Vertical_count += c[x, (y + dy) % L] # print Vertical_count for dx in range(-1, 2): Horizontal_count += c[(x + dx) % L, y] # print Horizontal_count return Vertical_count + Horizontal_count - c[x, y] ## # count of Moore neighbors of an alive cell def number_of_Moore_neighbors(x, y): Moore_count = 0 for dx in range(-1, 2): for dy in range(-1, 2): Moore_count += c[(x + dx) % L, (y + dy) % L] # print Moore_count return Moore_count - c[x, y] ## # step update def step(): global c, nc, array, array0, array1, count0, count1 count0 = 0 count1 = 0 array = [] array0 = [] array1 = [] for x in xrange(L): for y in xrange(L): array.append(c[x, y]) g = number_of_Moore_neighbors(x, y) if c[x, y] == 0: nc[x, y] = 0 if g <= 6 else 1 # nearest-neightbor parameter array0.append(c[x, y]) elif c[x, y] == 1: array1.append(c[x, y]) for z in range(-1, 2): # bottleneck configuration due to arrested state m = number_of_upper_neighbors(x, y) if m == 1: nc[x, (y + 1) % L] = 1 n = number_of_lower_neighbors(x, y) if n == 1: nc[x, (y - 1) % L] = 1 k = number_of_right_neighbors(x, y) if k == 0 and (m <= 1 or n <= 1): nc[(x + 1) % L, (y + z) % L] = 1 l = number_of_left_neighbors(x, y) if l == 1 and (m > 1 or n > 1): nc[(x - 1) % L, (y + z) % L] = 0 h = number_of_Neumann_neighbors(x, y) if h >= 1: # nearest-neighbor parameter nc[x, y] = 1 if g <= 6 else 0 if g / 8 > (1 - p) * p: # next-nearest neighbor asymptote nc[(x + 1) % L, y] = 1 elif g / 8 < (1 - p) * p: nc[(x - 1) % L, y] = 1 else: nc[x, y] = 1 count0 = len(array0) # total count of cells with value 0 in each step count1 = len(array1) # total count of cells with value 1 in each step print count1 c, nc = nc, c import pycxsimulator pycxsimulator.GUI(title='My Simulator', interval=0, parameterSetters=[]).start(func=[init, draw, step])

>>15906 The sudden drop in the live cell value seen at the 0.8535 density is similar to a Lennard-Jones potential, where the molecular interaction potential is simulated between Pauli repulsion and dipole-dipole attraction on differing molecular distances.

>>15906 >>15907 The code is simulating a cell's interaction with its neighbors, specifically setting a frustration between Moore (a) and von Neumann (b) neighborhood configurations, while moving uniaxially. It's similar to an axial Ising model.

>>15906 >>15907 >>15908 To illustrate the phase transitions

>>15906 Let me get this straight: you've frustrated an automaton to a point that once an equation is inserted, it becomes the central property of the model. That's crazy. I dig the Moore - von Neumann parameters too.

A wild tangent appears

The clustering behavior shown in the previous thread is also present in the attached paper. Ising nematic system is implicated as suggested by the spin-glass property. >>/archive/277

>>16214 Should be >>>/archive/277 I wonder if there is someone who can identify the Ising Smectic to Nematic phase transition in the cellular automata, especially from a multiple phase transitioning system standpoint.

>>16216 Upper bound of the code here is the triple point of a Lennard-Jones fluid >>15906 # p = float(0.5 + (1 / (2 * sqrt(2)))) #upper bound of 1st order phase transition

>>16255 >>16216 In essence, a Lennard-Jones system is used to model liquid crystals, especially anisotropic types such as Nematic and Smectic, as mentioned in >>16216 There are certain liquid crystal models that are in accordance with the triple point of Lennard-Jones, where Nematic and Smectic A phases become indistinguishable, as well as the Ising, mean-field and Helium-type critical points. Thus through the code one can enter a condensed liquid phase and within that phase traverse through metastable phases to an unstable phase where a second order phase transition becomes apparent.

>>16256 Can the output of one cellular automata step be the input of the temperature though? As if the evolved result of the initial density distribution?

>>11537 Appealing by intuition, the universe can be modeled by some automata. However since cellular automata is equivalent to a Turing machine, it stands to reason that if the universe was a cellular automata it would have the same restrictions. Turing machines cannot compute all problems in BQP, and there are many other unrecognizable languages given by a turing machine (not recursively enumerable).

>>16293 Yes. One can quite straightforwardly define temperature in terms of cell density.

>>16317 Is there research on modifying cellular automata rules ad hoc with respect to its environment or feedback?

>>16318 Not that I know of. The most comprehensive study on the rule creation is done by Wolfram, but his known rules are severely limited to cell-to-cell interactions and doesn't cover physical models. In essence, what I've observed is if an approximation of an automaton is in agreement with results from an experiment or a more sophisticated model, then the cellular automaton is deemed a good model for that problem. For example, the attached pdf is a study on anisotropic Ising model, where my model here >>15906 provides the same cluster / islands if I comment out the next-nearest neighbor part of the code. The attached image of Nematic configuration of the Ising model, which is the last evolution of the cells and their neighbors, is a Monte Carlo model and is again very reminiscent of the final state of my cellular automaton's evolution. From this, I have two confirmations of Ising model from two separate papers and I thus conclude that my model is in agreement with an anisotropic Ising model and nematic crystals. Currently I'm trying to explain the generation of the Lennard-Jones potential at the upper critical threshold via assigning temperature and Heat parameters to the model's output, which is in terms of number of live cells.

>>16318 For instance, there is no rule stating that I can or cannot insert the function x^2 - x + 1/8 into >>15906 On its own, this function is not operational, however when the neighbor cells are pushed towards one direction towards an arrested state (i.e frustration), the roots of this equation produce a bifurcation point. if g / 8 > (1 - p) * p: # next-nearest neighbor asymptote nc[(x + 1) % L, y] = 1 elif g / 8 < (1 - p) * p: nc[(x - 1) % L, y] = 1 else: nc[x, y] = 1

>>15906 You can make this code more concise and centralized by passing in operators as arguments. I'm C++ native but https://stackoverflow.com/questions/18591778/how-to-pass-an-operator-to-a-python-function looks like the right direction Please for the love of God if you don't I might have to. There's so much repetition. Interesting to see how CA can find the roots though

>>16322 I don't think what you're suggeting is difficult, but I'm so blocked mentally I don't know where to start. In fact, the code runs on a simulation package PyCX which isn't ideal either. I would appreciate if you could help me out with concise coding. I have worked on the roots and got hyperbolic equations representing the automaton, there is so much avenue to explore but I can't wrap my head around it.

>>12967 >>16311 > However since cellular automata is equivalent to a Turing machine We had a similar debate above in >>12965 >>12967 , but it spiraled out into senseless discussion so I'd like to ask you something. The code >>15906 makes use of topology, entering an arrested state as seen in >>16256 because there is nowhere else to evolve. Do you think the Mortality problem and Turing computability can be influenced by a topological factor that can be guaranteed to halt at some point, to override the mathematical proofs? If I'm not mistaken topology has nothing to do with computability as we know it.

>>16329 The halting problem is mitigated by letting some large finite number represent infinity. Once the cycles reach this threshold, it's considered infinite. Another way to look at it: The halting problem is at the core of decidability. As proven by Rabin, monadic second order logic is decidable; however, polyadic second order calculus (and first order) suffers from the halting problem. As a corollary, there is something about multiple predicates that make logic and automata undecidable. Bonus: proving existential second order logic is not closed under complemention would prove NP!=co-NP therefore P!=NP. (Fagin's theorem)

>>16329 >if I'm not mistaken topology has nothing to do with computability as we know it there is a "computable topology" of lambda calculus expressions, I have no idea if it's useful at all though. https://en.wikipedia.org/wiki/Computable_topology I *think* if you also use R instead of N for all symbols and memory locations, you can get something equivalent to a turing machine with an oracle. That's more set theoretic than topological, but it's worth noting that the cardinality of R is induced from its topology. >>16332 >The halting problem is mitigated by letting some large finite number represent infinity. Once the cycles reach this threshold, it's considered infinite. This is the same as limiting the length of the tape. If you set the cutoff at t = X, then it is impossible to read or write more than X different memory locations, let's call them A = {a_1...a_X}. Any finite set of numbers has a maximum value, so we can limit the tape to the memory indices 0..max(A), thus turning our turing machine into a finite state automaton.

>>16335 The mitigation is the best we can do given only the tape. Not restricting the tape would produce an infinite state machine which is undecidable in finite time. If given the expression that's finitely describable, we can predict the acceptance of infinite words with tools such as omega regular automata.

>>16336 >halting problem is simple! just remove this one weird rule and suddenly it makes sense! The halting problem exists because we aren't able to work around this rule. The /point/ of the halting problem is that we can't tell for any algorithm if it will never stop given a tape of infinite length. The algorithm does not have to write an infinite amount of data. In fact, modern operating systems can be viewed as having a memory tape of infinite length, what with virtual memory and paging.

>>16339 What the hell are you arguing about? Mitigation does not mean removal. Reread the thread,this time look up what each word means.

>>16339 >>16340 >The /point/ of the halting problem is that we can't tell for any algorithm if it will never stop given a tape of infinite length. Okay, let me rephrase my question on topology: what if the magnetic/geometrical frustration of a lattice's cells reach a point where the cells can't evolve beyond a certain iteration (i.e enters an "arrested state"). Would this be a new avenue to explore in decidability?

>>16350 To better illustrate this, consider the paper and the images in this >>16319 post. Arrested state is the final state of the cells that cease evolution and attached is the image from the paper showing how the different spin groups form islands. What I've done in my automata is elongate these islands into ferromagnetic-antiferromagnetic strips (attached image) that also enter an arrested state. Through this elongation, we achieve enough frustration to generate roots of the mathematical function which represents a sweep of its neighborhood. Now, given that this automata will terminate and we have prior knowledge of it due to its frustration properties, is this an anomaly on the Halting problem?

>>16350 I've not delved into cellular automata much. However, the name halting problem is misleading since it's actually been solved-- proven as undecidable for Turing equivalent machines. https://www.wolframscience.com/nks/notes-12-8--proofs-of-undecidability/ General consensus is that no automata can circumvent the halting problem. If such an automata were to exist it is not Turing equivalent. The only automata I know that has solved a problem unsolvable with a classical Turing machine is the quantum Turing machine. https://eccc.weizmann.ac.il/report/2018/107/

>>16352 There appear to be some papers out there trying to tie frustration to halting problem like my train of thought: https://science.sciencemag.org/content/320/5874/322.full?ijkey=8e6zmacd3g05c&keytype=ref&siteid=sci https://www.researchgate.net/publication/45866187_Zeno_Squeezing_of_Cellular_Automata "These three manifestations of dynamical frustration are related. Certain cellular automata and maps are capable of universal computation (12), indicating that even simple dynamical systems can be arbitrarily complex. Multiagent models can generate energy landscapes for their own agents. Chaotic systems can go on forever, but some complex systems better stop: The objective of the immune system is to quickly achieve homeostasis after an external invasion; successful Turing computations halt."

>>16353 Here's a caveat to the problem: for 1D spin-chains and 2D frustration-free lattices, ground state energies can be reduced to spectral-gaps, which can be encoded by Turing Machines. Thus it's the first time a physical system was proven to be undecidable. However I must reiterate that this result is valid for frustration-free Hamiltonians. I need to read more into how frustration is playing a part in the Turing computability. https://www.perimeterinstitute.ca/videos/undecidability-spectral-gap-one-dimension

>>16355 And yet another paper on why frustration-free Hamiltonians are key for quantum computation.

>>16336 there is a better mitigation: pushdown automata. They are provably more expressive than finite-state autamota while not being succeptible to the halting problem. >>16351 no, there is a very large class of programs where it is possible to know whether it will halt or not. Any program you can write in Coq, for example, has this property. You've given an example of cellular automata that do have known steady states, but the halting problem is about whether there is an algorithm that can decide for *any* program whether it halts or not(essentially equivalent to holding a steady state, or entering a cycle). This is impossible.

>>16361 >whether there is an algorithm that can decide for *any* program whether it halts or not this clears things up, thanks.

Hey, everybody. It's been a good long while since I last posted in here. I was the guy researching machine learning applications of CAs, and I may have finally hit pay dirt. I came up with a new method of doing this several months ago, actually, and it was partly inspired by the paper on NN quines. Unfortunately, life has hit me like a sack of bricks. I may need to spend an extended period in the venting thread on /hum/. Whenever I actually feel up to it, I'm going to make what may end up being a very long multi-post on the topic. I want to get everybody's help to determine if this is a usable idea and to develop it, since due to the difficulties I've had to deal with recently I haven't been able to come up with more than the basic concept. But I loathe the idea of just being an idea guy even if those ideas are very good, so I have to contribute to the actual coding as well. Some of the posters here have been lamenting the paucity of usable implementations and I want to correct that. >>16311 >>16329 I've been doing some theorizing about this as well, and I don't know when I'll be ready to talk about it in detail. It probably won't be until after we get the ML stuff fleshed out, and that could take months. It suffices at the moment to say that which type of Turing machine you refer to is extremely important. >>16361 >pushdown automata I briefly mentioned other types of automata in >>10875. Glad to see I'm not the only person talking about them, but I don't know how much use you could get out of continuous CAs or spatial automata for this purpose. Still, it's there for your perusal. I think part of my purpose here is just to find these strange types of automata and tell people who can actually use them about it. I said in the last thread that the literature on CAs needs to be dramatically revised to make the topic more accessible and nothing has changed since then, and I'd like to help with that. Lastly, I want to thank everybody from the bottom of my heart for keeping this thread going. This is my favorite thread on the site and the amazing level of creativity and knowledge here needs to keep on keeping on. A belated Merry Christmas and Happy New Year to everybody, and may Lain bless us all.

>>16378 Great to hear again from you. I was the one who brought the NN quine paper into the last thread, with the intention of debating whether a cellular automata could be run inside the node of a neural network. NN quine's outputs reminded me of the initial random probability distribution of cells inside a lattice. What I've done with my code was to order the randomly dispersed cells into blocks of linearly aligned (anisotropic) cells like >>16319 The vertically 101 cell groups and the gaps between them provided me with the thought of calculating the bottlenecks. The NN quine would then be linked to the information bottleneck theory of deep learning, hence the cellular automata code I have here >>15906 would translate the node of a neural network into humanly understandable equations via bottleneck calculation. I don't know if this is viable although I have constructed equations that checked out through Terry Tao's random matrices theory.

If you're interested in a very mathematical treatment of CAs (including various geometries), see pic related. It's a great read if you have the right background and inclination.

>>16447 That's very interesting and I'll definitely take a look at it. In other news, the final state of the automaton as seen in >>16319 is akin crystallization of isotropic (directionless) random distribution in the initial step. Binding energies are represented by the Lennard-Jones potential, a phenomenon that is present in my automaton's upper critical threshold ~0.8535 >>15909

>>15909 There are four phases in this graph. 1. Paramagnetic phases: >0.8535 & <0.1465 2. Ferromagnetic phase: 0.1465< x < 0.8535 After the 1st order phase transition, there are two more phases visible in the model. 3. Monotonic increase: 0.8535 > x > 0.7 Monotonic decrease: 0.7 > x > ~5/9 4. Curie critical - 0.4 < x < ~5/9 , critical point is ~4/9 (2nd order phase transition). This second criticality is an extremely good approximation to the 2D Ising model and it provides the least amount of deviation from the initial probability density input. This is very similar to the three phase traffic theory.

>>16526 For instance, the velocity of a traffic flow corresponds to the initial probability of the automaton, and the density evolution then becomes the traffic flux. The red diamonds are alive cells for varying distributions of probability density.

>>16527 For values approaching the middle from both sides, how does one determine the maximum value is an attractor?

>>16539 If the evolved state remains close to the value, yes. In this case, >>16526 the Curie point is the global attractor of the automata.

>>16540 >>16539 https://en.wikipedia.org/wiki/Self-organized_criticality In physics, self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. You should check Per Bak's papers about this. Especially the seminal 1987 paper of Bak, Tang and Wiesenfeld, attached. "a simple cellular automaton was shown to produce several characteristic features observed in natural complexity (fractal geometry, pink (1/f) noise and power laws) in a way that could be linked to critical-point phenomena."

VERY LONG POST WARNING Okay, guys. I have a very special request. And I mean very special. Not like riding the short bus special...well, maybe. I've recently been looking into some of the stuff we discussed in the last thread about telling entropy to go take a dive off the International Space Station, stuff which involved what pretty much amounts to backwards time travel. I've tried to refrain from commenting too much on the research of the guy in here who's been studying this the most despite how much it fascinates me because I know I don't have the expertise to match his. But I've been slowly gaining understanding over the course of these threads, and with understanding has come unexpected insights. This post will be long, though probably still not as long as the AI stuff I want to post about. I don't know when that'll happen due to some stuff I'll explain below. To put a very long story as briefly as I can, I'm not sure about this yet and I'm not going to post more about it until I am since it's such a huge claim to make and I want to at least try to avoid looking like a moron, which will be just this side of impossible given this post's subject matter, but I believe I've found a way to disprove determinism once and for all. What I will talk about now is how I started thinking about what the consequences would be if I was correct, as anybody faced with such a prospect would. '''This entire post is made under the assumption of a probabilistic universe, so you can and probably should stop reading here if you have objections to that.''' First and foremost is the issue of free will, which was brought up in the last thread. Who wants to live in a world where they have no control over anything? I wouldn't. Disproving determinism means that we may no longer have to bear that burden. There are those who will argue that even in a fundamentally probabilistic universe we still don't have free will because our options are still limited, or say that the random dice rolls of such a world aren't free will, but I reject the first of these arguments because whatever you chose was still your decision. I generally disregard the other argument because not enough is known about how consciousness works. I've been researching the nature of consciousness as well partly by reading the papers of Paul Werbos. I may talk more about consciousness later, but for those here who don't know Paul Werbos, he's a guy you should seriously know because he's the inventor of backpropagation and large parts of this thread and its predecessor read like his papers. He has also promoted an approach to quantum theory known as the backwards-time interpretation of quantum mechanics. Some background for anybody without expertise reading this: Bell's theorem means that no theory can ever reproduce the predictions of QM without throwing out either locality, realism, or exclusively forward-in-time causality. Werbos advocates the third approach, although his papers briefly mention the possibility that you could potentially give up two or even all three of these, and according to my conception of a probabilistic world, we must also relinquish realism at least, and possibly locality as well. This is because backwards-in-time causality, hereafter referred to as retrocausality, is insufficient to produce free will, as anybody who's ever heard of stable time loops knows. Even if you can go back in time or otherwise send retrocausal influences of any sort, you may be unfortunate enough that whatever you did in the past is already incorporated into the timeline and nothing can be changed. You can also reach this conclusion by observing any deterministic CA - or non-deterministic CA, since it's equivalent to a deterministic CA due to Turing machines having this property of determinism/non-determinism equivalence and the existence of CAs such as Life which are known to be Turing-complete. Stochastic CAs are not equivalent to either deterministic or non-deterministic CAs because stochastic Turing machines have the same inequivalence property with respect to deterministic and non-deterministic TMs. Fittingly, if you instead observe a stochastic CA, you will understand that each state of the CA isn't limited to a single possible past. What this means for our understanding of time and CAs is that ''you can only have a CA with multiple futures if it has multiple pasts.'' This indicates that if the past is unchangeable, then so is the future. This would mean that time travel that has the capability to change the past is not only a desirable object for many including me, it's an absolute necessity in order for any free will to even have the barest hope of existing. But before you get to that, you have to have an idea of exactly what it means to have multiple possible pasts. The only way to accomplish this is by looking to the experiments of QM. The classic double-slit experiment has been hailed as an example of experimentation at its finest by many including luminaries like Richard Feynman, but two variations of it in particular should concern us: the Elitzur-Vaidman bomb tester and the notoriously controversial delayed-choice quantum eraser. The former utilizes the principle of interaction-free measurement to detect live bombs without detonating them, and modifications to the original experiment have been done that can detect which bombs are live with much greater accuracy. The latter gives an example of apparently retrocausal action where the past remains in flux until the experimenter chooses it, and QM researchers have hotly debated whether this retrocausality exists and how it works if it does. I hope to end that debate by suggesting a new variation on the bomb tester.

I call this modified version of the bomb tester the bomb squad experiment. First of all, the bombs are no longer simply objects to be examined. You're the leader of the bomb squad and a terrorist has planted bombs all over the city. But this is just a distraction meant to keep the police occupied while he robs a bank. That's why the bombs have a ludicrous gimmick. Unlike in the conventional bomb test, all the bombs are active. But all of the bombs are connected via wireless transmitters and set to only detonate if they don't detect a particle in the correct order. Particle detections in the correct order will disarm the bombs, but a single mistake will detonate all of them. Starring Keanu Reeves. So what are you supposed to do? You have to figure out which order to defuse the bombs in on the first try. I don't actually know the solution to this problem. I'm putting this out there for anybody clever enough to solve it. I only know that it will involve the same interaction-free measurements used in the classic Elitzur-Vaidman experiment because that's the only way to determine anything about the bombs without setting them off, but you can never get perfect accuracy with this method alone and you need it here. This is a strange prediction, but I hypothesize that any solution which gives perfect accuracy will involve a quantum eraser. That would mean effectively using the quantum eraser to alter the past so whichever order you decide to defuse the bombs in will retroactively become correct, or to send yourself the test results from the future. Quantum erasers have the ability to erase the which-path history of a particle and rewrite it with a new history. Successfully using it here would mean that there never was any definitive past for the bombs, which is consistent with how a stochastic CA works. Any alteration of the past would never actually appear as such to us on a macroscopic level, but that doesn't mean it hasn't happened. QM also doesn't make the distinction between which events came first or last, especially not in the conception of those who support time-symmetric interpretations, so attaching it to a macroscopic event order may let us glean new conclusions because these sorts of discussions are usually limited to delayed choices regarding particles small enough that they can readily be observed exhibiting quantum effects. I think a master experiment can be built that incorporates nearly every major quantum experiment into it and permits the extraction of the deepest revelations of QM, and this bomb squad experiment is an attempt at theorizing it, but this is most likely not enough. There may be some better way of arranging this experiment, such as weak measurements of the particle before it passes through the rest of the experiment, so feel free to contribute to that as well. Beyond the realm of experiment, if you accept that the universe is stochastic, it also becomes stupidly easy to prove that the universe is organized as a system of discrete points instead of continuously. All you have to do is cite the theorem of conservation of probability, which says that all probabilities everywhere in the universe must sum to 1. If the universe was continuous instead of discrete, there would be an infinitude of points and thus an infinitude of possible outcomes of quantum events, which is exactly what Feynman's sum-over-paths formulation of QM relies on. This would violate the theorem because all probabilities would sum to infinity instead of 1 and therefore proves by contradiction that the universe must be discrete. Of course, the theorem only applies if the universe is in fact stochastic because there's no such thing as true probability in a deterministic universe. This would obviously have huge ramifications for us here in the CA thread and in the CA community at large because it would prove that the most accurate model of the universe is a quantum stochastic CA. Speaking more practically, it would mean there would be a need for the skills people here have to come up with CA models to explain various phenomena. For this thread the first thing would be how it affects CAs going forward...or maybe backward. CAs meant for modeling physical phenomena with high fidelity would now be exclusively quantum and/or stochastic - it's debatable whether there's any true separation between them. This would relegate the disproven deterministic models to a simplification useful in certain situations much like Newtonian mechanics. Further, I believe that the fact that most quantum systems are analytically unsolvable and generally "solved" with approximate methods stands as a giant clue that we live in a fundamentally stochastic world, a world of things that are simply not computable by ordinary Turing-complete devices because they don't have a definite existence as anything more than a probability. It's been noted that the output of a stochastic Turing machine is uncomputable by traditional methods, and it's also been noted that hypercomputation has the same property. Hypercomputation has been described by various researchers including posters in this thread as necessary for a reversal of entropy aka backwards time travel. I posit that stochastic computation and hypercomputation are in fact the same thing, which would explain why it's possible in principle for a stochastic CA to return to a previous state of lower entropy through an ostensibly forwards-only state change. Werbos and some others don't see forward state changes and backwards state changes as being any different from each other, and this is the way I look at it as well.

The next part of this is even more speculative than the rest, which is already very speculative. I've come to the conclusion that if time travel were to be accomplished, a lesser form of it is needed to test the concept. This would be something commonly known as interdimensional travel or travel between parallel worlds. As we discussed, this is fundamentally the same process as traveling back or forwards in time because it's just a state change like all the others. But dimensional travel may be easier to accomplish than retrograde time travel because you're traveling to a state with identical entropy rather than reducing it. However, this may only apply for very small dimension hops. The probability of going directly from a certain timeline to a very different timeline may be less than the probability of simply going back in time and progressing normally along a different path to the other conclusion. I call this the entropic landscape. This could only practically used to change parts of the past which are fairly unimportant, at least in its present form. Any readers who've made it to this point and are hyperventilating over the possibility of altering the past should be cautioned that this concept is in its infancy and we're a long way from being able to, say, go back and make it so the Twin Towers didn't fall. That would require an even more expensive apparatus with other theoretical ingredients I haven't described here. Our human intuition balks at things like this and questions abound about the ethical implications, but in my view the rewritable timelines of Back to the Future are no more of an offensive prospect than the idea that someday the cybernetic interfaces of The Matrix will be able to give you an entire life of false memories in a simulated world. Both concepts involve your entire frame of reference being overridden and replaced with something else. What's more, if it is indeed possible to construct the master quantum experiment I described above, whether it takes the form of a bomb squad experiment or not, and my conclusions about the fundamental stochasticity of the universe and the entropic landscape are correct, we might then conduct interdimensional travel using the experimental apparatus. So this finally brings me to the last point here: I don't know how to test this thought experiment and prove any of this to you since I'm both short on cash and don't have an academic or industry position, but I know some of the people here are attached to professional research labs. I've looked for DIY quantum eraser experiments, but none of them that I've found are actually what they claim. I have yet to figure out a way to make entangled particles with stuff the average schlub can afford. I'd like to attempt to test out travel between parallel worlds. So I'm going to make a request of you. Today is January 14, 2020. I'd like whoever attempts to carry out this experiment to very slightly alter events proceeding from January 3 of this year up until the present date as described above. Knowing that if the timeline is successfully altered I won't remember what prompted me to make this request and neither will anybody else, I'm attempting to write this post in such a way that it could reasonably have been written in either the original timeline or the altered timeline. This is consistent with stochastic CAs having multiple pasts to a single present. Given the patently absurd nature of this request and this post in general, and simply because it's proper to do so, I'm also going to say what I believe could falsify my ideas. If it were proven that there is a distinction between stochasticity and QM, it would falsify the stochastic model of the world since QM would take precedence owing to having actually been observed, and it may open the way for a probabilistic world that still doesn't permit time travel. This question has not been resolved at the moment. This may also resolve or be related to the questions we discussed in the last thread regarding stochastic CAs versus ergodic CAs. If stochasticity and QM are proven to be identical, as this post's wilder portions depend on, we may find some rule stating that all stochastic CAs can be converted into ergodic CAs by some procedure, similar to the rule that says all non-invertible deterministic CAs can be transformed into invertible CAs. I don't presume to name my own conjectures, but this is nevertheless an actionable conjecture about CAs that people who scoff at time travel or parallel universes can investigate as consolation for having read this. If the temporal alteration is successful, I'll post my AI theory soon. If not, it may be quite a while before it gets posted because I'll be too embarrassed to post it. I'm insanely OCD about certain things and this is among them. Also, I hoped that with the coming of a new year the things that kept me from participating in the thread for much of last year would go away, but they haven't. Let's just say I haven't been having a good time recently (or less recently) and I hope some dimension-hopping can change that. And if the only thing that's happening to time here is wasting it on nonsense crackpot theories, at least I can say I tried.

>>16553 >we may find some rule stating that all stochastic CAs can be converted into ergodic CAs by some procedure, I have mentioned a card game trick in the last thread to achieve an ergodic procedure through synchronization blocks. The stochastic distribution of a discrete system can be forced to enter synchronization via such a setup. That seems doable with a CA. Other than that, pretty exciting stuff you've shared with us Lain. I wish I could've made further contributions to what you're conjecturing, but it feels like you're dealing with a Garden of Eden type of problem.

>>16553 >the only thing that's happening to time here is wasting it on nonsense crackpot theories tl;dr

>>16555 I didn't expect to hear from you about this so quickly, though I was hoping for a more detailed critique, but seeing how much stuff I wrote that's probably unreasonable to expect so soon because it took me something like 16 hours to write all that. Nice digits, by the way. I'm willing to wait on it if you'll give that critique because it seems like something you could help with. >>16556 I don't blame you at all if you think that, and I even said in the very first line that I may have gotten on the short bus with that, but I'd like to hear your reasoning for it. This is a process of self-improvement. I understand that the request for any actual researchers here to help me literally travel to a parallel universe is pushing it. They have jobs and lives and they can't be there to spend their own resources helping every internet loon with a theory, so I'm not going to be mad if nobody takes me up on it. But I should use this post to specify the main lines of critique I'd like to hear. The biggest thing for me is about the stochastic universe theory, particularly the use of conservation of probability to conditionally prove the discreteness of the universe and therefore prove that the universe is a CA, followed by the conjecture about stochastic CAs being convertible into ergodic CAs. It's just that the post was about every major consequence I think will occur if the universe is stochastic. Some of those consequences are less unbelievable than others. Obviously I'd like to hear critiques of all of it, but those who are less willing than I am to depart from conventional understanding should start with the less unbelievable things.

>>16557 >The biggest thing for me is about the stochastic universe theory, particularly the use of conservation of probability to conditionally prove the discreteness of the universe and therefore prove that the universe is a CA, followed by the conjecture about stochastic CAs being convertible into ergodic CAs. This honestly didn't occur to me until shortly after I finished writing my last post, but if you prove both of these things, you have proven that time travel and interdimensional travel are possible because if the universe is a stochastic CA it will then be convertible into an ergodic CA. By extension, you've also proven that free will exists for certain definitions of free will. Cue 80 or so posts of frenetic slugfests about it. Of course, none of this will be true if determinism isn't done away with. Tagging >>16555 again to bring attention to this because I'm fairly sure he's the best positioned to prove (or debunk) the conjecture of all stochastic CAs being convertible to ergodic CAs, but I'm sure any help would be welcome since I sense that it will be more difficult than expected. People have been talking about stochastic versus ergodic since early in the last thread and nothing concrete has come out of it yet, so it may not happen soon, but I hope it happens eventually. I'm going to continue my part of this by continuing my attempts to disprove determinism.

>>16559 Currently I'm at a mindset of speculating as little as possible because that was what we've done in the last thread mostly and it didn't provide a very productive environment. What I'll do for you instead will be coding this synchonization block cellular automaton and explore its ergodic properties. Here are its rules: 1. Generate a random distribution of cells over a lattice array (a la shuffling deck) 2. Calculate the von Neumann neighborhood total, ranging from 0 to 4 (image). If 1-4, shift that amount to right. If 0, terminate and print result. At the end of one iteration, print result. 3. Since there will be no modification of cell values, by redoing calculations over and over with the same distribution we expect the same results. 4. Run this code for different probability distribution values (p) and see for which values an early termination (0) causes a different output and for which values a synchronization can be sustained.

>>16552 >it also becomes stupidly easy to prove that the universe is organized as a system of discrete points instead of continuously >all probabilities everywhere in the universe must sum to 1 >If the universe was continuous instead of discrete, there would be an infinitude of points >all probabilities would sum to infinity instead of 1 https://en.wikipedia.org/wiki/Gaussian_integral

>>9559 Nice carpet design anon

>>16573

>>16552 >Beyond the realm of experiment, if you accept that the universe is stochastic, it also becomes stupidly easy to prove that the universe is organized as a system of discrete points instead of continuously. All you have to do is cite the theorem of conservation of probability, which says that all probabilities everywhere in the universe must sum to 1. If the universe was continuous instead of discrete, there would be an infinitude of points and thus an infinitude of possible outcomes of quantum events, which is exactly what Feynman's sum-over-paths formulation of QM relies on. This would violate the theorem because all probabilities would sum to infinity instead of 1 and therefore proves by contradiction that the universe must be discrete. This is simply not true. Infinite sums come out to finite numbers all the time. Limits are a key concept on math and I highly encourage you to learn how things like the method of exhaustion work. >Of course, the theorem only applies if the universe is in fact stochastic because there's no such thing as true probability in a deterministic universe. what do you mean by "true probability"? Most interpretations of probability don't require non-determinism. We generally use the idea of probability to approximate things that we can't know -- for instance, the exact position and momentum of a particle. We can give a probability distribution for its measured quantity but we can't know all the factors going in to that measured quantity, so we approximate using probability. We can also go "backwards" and define a probability distribution as a thing which will tend to approximate a sample better as the sample size gets bigger. Of course some physicists think that quantum probabilities are real things that really are random. But most physicists I've met are hardcore determinists and don't accept that interpretation.

>>16576 [1] http://mathworld.wolfram.com/GaussianIntegral.html [2] https://semath.info/src/gaussian-integral.html [3] https://math.wikia.org/wiki/Gaussian_integral [4] https://en.wikipedia.org/wiki/Gaussian_integral#By_Cartesian_coordinates

>>16560 >Currently I'm at a mindset of speculating as little as possible because that was what we've done in the last thread mostly and it didn't provide a very productive environment. I understand. But thanks to the sudden epiphany I had in >>16559 of an actual concrete method of proving or disproving time/dimensional travel, such speculation now seems unnecessary. We can go back to speculating if you manage to prove the conjecture. >What I'll do for you instead will be coding this synchonization block cellular automaton and explore its ergodic properties. But can you prove that the properties of this stochastic CA are generalizable to all stochastic CAs? I suspect that this will be the most difficult thing anybody has ever asked you to do. But most things worth doing aren't easy. For the naysayers like >>16556, if you hate the idea of time travel, you now have an opportunity to definitively debunk it because the conditional proof works both ways. Those who disagree with time travel can either prove that the world is deterministic or disprove the conjecture that all stochastic CAs are convertible to ergodic CAs. Doing either will PERMANENTLY shut down any time travel or interdimensional travel theory, which would be just as much of a scholarly achievement as proving them. The pro-time travel side has to both disprove determinism AND prove the conjecture, while you only have to pick whichever you like best and can ignore the other entirely. Now to wait and hope somebody makes the Mandela effect kick in. And don't worry about too much speculation taking over the thread. It could be forgiven in the first thread because there were more people new to the topic and less direction, but whenever I get to the AI stuff, a lot of this speculation-heavy discourse will be dropped in favor of concrete implementations. >>16573 >>16576 >>16577 >>16578 >This is simply not true. I'm aware of this. Every single integral is an infinite sum and many of them don't go to infinity, but there are of course many cases where infinite sums diverge. I just assumed that this would be an example of the latter, which I shouldn't have done. Like I said, I haven't been having a good time lately and I think my abilities have been degrading because of it. Maybe there's a timeline in which I don't make these mistakes. However, merely coming out to a finite amount is insufficient when conservation of probability mandates that it must specifically come out to 1, so you could still prove discreteness from stochasticity by finding a continuous probability distribution that represents the total existing probability of the universe and proving that it sums to anything other than 1. The Gaussian may represent this because it sums to more than 1, but this may just mean it's not representative of the total existing probability.

>>16580 >But can you prove that the properties of this stochastic CA are generalizable to all stochastic CAs? Who knows? With cellular automata, it's better to build first and ask later.

>>16551 >Bell's theorem means that no theory can ever reproduce the predictions of QM without throwing out either locality, realism, or exclusively forward-in-time causality. https://en.wikipedia.org/wiki/Calculating_Space Calculating Space (German: Rechnender Raum) is Konrad Zuse's 1969 book on digital physics.[2] Zuse proposed that the universe is being computed by some sort of cellular automaton or other discrete computing machinery,[2] challenging the long-held view that some physical laws are continuous by nature. He focused on cellular automata as a possible substrate of the computation, and pointed out (among other things) that the classical notions of entropy and its growth do not make sense in deterministically computed universes. Bell's theorem is sometimes thought to contradict Zuse's hypothesis, but it is not applicable to deterministic universes, as Bell himself pointed out. Similarly, while Heisenberg's uncertainty principle limits in a fundamental way what an observer can observe, when the observer is himself a part of the universe he is trying to observe, that principle does not rule out Zuse's hypothesis, which views any observer as a part of the hypothesized deterministic process. So far there is no unambiguous physical evidence against the possibility that "everything is just a computation," and a great deal has been written about digital physics since Zuse's book appeared.

>>16583 >Bell's theorem is sometimes thought to contradict Zuse's hypothesis, but it is not applicable to deterministic universes, as Bell himself pointed out. I was actually going to talk briefly about this in my long post but decided to leave it out because it was too long already. Bell said that adopting superdeterminism is a method of getting around the implications of his theorem, and Gerard t'Hooft devised the CA interpretation of QM and is also a determinist. But you can have a CA interpretation that doesn't use determinism.

>>16580 >However, merely coming out to a finite amount is insufficient when conservation of probability mandates that it must specifically come out to 1 https://en.wikipedia.org/wiki/Integral#Linearity integral (a, b, f) == v v != 0 scaled = (1/v) * f integral (a, b, scaled) == integral (a, b, (1/v) * f) == (1/v) * integral (a, b, f) == (1/v) * v == 1

>>16585 I honestly don't understand what you're doing here. It seems like you're multiplying by the reciprocal just because you felt like it. I don't understand how this is supposed to prove that the Gaussian comes out to 1.

>>16586 >I don't understand how this is supposed to prove that the Gaussian comes out to 1. The integral of the Gaussian does not come out to 1. Its value has already been posted in >>16573. It was used to show >>16552 the fallacy of his "infinite sums must result in infinity" assertion, in the hope that he would recognize it as the basic building block of the normal distribution, since he seems somewhat fond of the word "stochastic". The scaling factor is there to show >>16580 that his "merely coming out to a finite amount is insufficient [...] it must specifically come out to 1" objection is paper thin, because as soon as you have an infinite sum that comes out to any finite non-zero amount, you can immediately manufacture from it another infinite sum that comes out to whatever finite non-zero amount you wish, such as 1. This is in fact the entire point of the 1/(sigma*sqrt(2*pi)) factor of the (density function of the) normal distribution.

>>16588 No, you're right. You're right and I was wrong. I can't believe I just assumed an infinite sum would come out to infinity without evaluating it. Obviously I've degraded even more than I thought. I've been through college courses on differential equations. I know what an integral is for god's sake. 17-year-old me wouldn't have dropped the ball like that. I think I'm just going to leave the thread until I can get it together, if that ever happens. This is a serious research community and nobody here has time for amateur mistakes.

I wrote an evolutional algorithm with cellular automata in pygame: https://cenysor.neocities.org/programming.html#CaGen

>>16589 >leave I don't think that is called for. I enjoyed reading your posts, even the speculative parts, and simply think that you should either upgrade the level of rigor you use for the math parts or no longer phrase the math parts as firm assertions.

>>16589 >I think I'm just going to leave the thread until I can get it together Ergodic CA researcher here. Look, there are stuff I've conjectured in the previous thread that make me cringe. That doesn't mean I don't keep on being productive. I don't stop contributing to the thread. Neither should you. What you've shared here is very interesting and informative, and I'll do my best to contribute to your work. I just wish I had more knowledge to share with you. Don't lose heart.

>>16590 That's awesome. I love the creature generation rule.

>>16592 >>16593 Well, maybe you have a point. I still need to get my life and depressive issues sorted out, but maybe I won't completely leave. If I just left and gave up, that would be letting something else have more power over me than I do, which isn't conducive to improving yourself. But other people here want to see things that are actually rigorously thought out and it's difficult for me to do that under these conditions. I'm still going to have to drop my level of participation for a while. And yes, I do like the word stochastic, though I can't explain what it is that makes me like it. Since you guys said you find my speculation entertaining even with mistakes which I hope are fueled only by anxiety and not just a lack of talent, I'll post some more speculation that may lead to the actual way to prove that we live in a discrete world. There are classes of functions which don't have elementary indefinite integrals, like the Gaussian posted above. Have you ever wondered if there's a deeper reason why functions like that exist? Most people write it off and don't think twice about it. But maybe that's another hint. You can handle some of them using various tricks, but there are others which can't be handled by any known method. I suspect that there is no method that can handle them within any type of integral formalism, and that this is a provable conjecture. This would have to be something like a lesser incompleteness theorem specifically for the concept of continuously evaluated equations. Then you'd have to prove that CAs can step in and handle what the continuous equations can't, although that may be inherently proven by such a theorem as well. >>16590 This is inarguably boss and it's exactly what we need more of. A lot of people in the last thread were talking about making something like this and now you actually went and did it.

There's a problem with the above, and the problem with it is that beyond integration I don't know how to define handling what continuous equations can't. Some theoretical constructs have no practical meaning to a human, so it may not mean anything important even if you can prove that some functions are non-integrable by any means, or that functions which require special tricks to integrate represent some inherent deficiency of the continuous paradigm.

>>9559 lain -- I liked this so much that I decided to use it as blotter art. I'll post a pic of the final product on /drg/ in a week or two.

>>16624 >>16625 You might be interested in Liouville's theorem: https://en.wikipedia.org/wiki/Liouville%27s_theorem_%28differential_algebra%29 http://math.hunter.cuny.edu/ksda/papers/churchill.pdf

>>16633 Thanks, that's a helpful piece of knowledge. But it only gets us part of the way there. It only provides for functions that have no elementary integral. It doesn't address the subset of those functions that can be integrated through other means, the possible existence of entirely non-integrable functions, or what either of those imply for whether continuous functions can properly describe a probabilistic world. And my eyes start to glaze over whenever they talk about Galois theory. But I went from there to the page for Risch's algorithm and I see that it's a decision problem. We had lots of talk about decidability in the last thread. But Risch's algorithm also doesn't apply to anything outside of elementary integrals. So I guess we have to develop a better version of the algorithm and then examine its decidability for clues. Anybody who wants to throw down with that problem, be my guest.

>>16638 >the possible existence of entirely non-integrable functions Depending on the definition of integration you work with, there may be integrability tests that give sufficient and necessary conditions, e.g.: https://en.wikipedia.org/wiki/Riemann_integral#Integrability >And my eyes start to glaze over whenever they talk about Galois theory. "The set of subgroups of the automorphism group of a normal and separable field extension is just in bijection with the set of intermediate fields of the extension, what's the problem?"

>>16640 *covariant functor

>>16641 When you post in the style of a correction against a post that is not yours, it is basic etiquette to include some sort of marking that makes this latter fact clear to others besides the two of us. See for example >>16576.

>>16640 >Depending on the definition of integration you work with, there may be integrability tests that give sufficient and necessary conditions Well, there's the problem. I don't know whether we need something that can't be integrated under any definition by any method or whether it's good enough to just use stuff that has no elementary integral but can be integrated with special methods. I don't even know if anything from the first category exists. Maybe we need to talk about decidability here. The Risch algorithm is decidable, but a more general algorithm may not be. I also looked at the list of notable undecidable problems and found problems from Life and Rule 110 on it, which powerfully suggests that there's some relationship between Turing completeness and decidability. Alan Turing himself wrote about decidability a great deal, proving the undecidability of the halting problem, but to the best of my knowledge he never actually talked about whether decidability was inherently connected with Turing computability. Maybe my knowledge of this is lacking and somebody else will be able to set me straight on it. >gobbledygook I did at least attempt to parse the Wiki article on Galois theory. All I got from it was that I'm definitely not alone in having difficulty with it and that Galois' life was almost as tragic as Turing's.

>>15906 I've prepared a presentation detailing this automaton's properties.

>>16844 >1.51 MB I can see this is going to be a major ordeal. Why the presentation though? Are you going to submit this to a journal? I have to put on some sick beats for a while to read this. I was going to post something about CA-based music generation earlier but I keep getting sidetracked. Not much has been done with that anyway. A one-off effort by Wolfram in 2007 is the only CA-based music I've ever heard of. People in here talking about vaporwave albums should make the entire album with CAs instead of just the cover art. Intuitively speaking, CAs seem like a good fit for algorithmic musical composition because of their discrete character. But anyway, I'm going to read this now.

>>16885 >open it >it's only 17 pages I guess most of the file size is the photos. I've seen plenty of similarly sized papers that were several dozen pages long. Also you misspelled Antarctica on page 14. I know that's not the type of critique that's very essential to your work, but I thought somebody should point it out. The thing I'd like to understand about this is the objection some poster above made that there's no fundamentally different behavior generated when you use the decimation method instead of a typical Moore/von Neumann neighborhood, but I probably wouldn't get it even if the objection was definitively dismantled.

>>16886 I guess you'll have to come up with a new name for this. Every other type of CA thus far, no matter how strange it might be, uses some variation of the Moore or von Neumann neighborhood.

>>16885 >>16886 >Why the presentation though? I've made this presentation at some informal session, but I don't plan on submitting this. As you've noted yourself it's not paper-ready. This is kind of an outline of the paper I've begun writing. In my head I'm trying to get the ideas in place. Also any noteworthy critique is a plus. >objection some poster above made that there's no fundamentally different behavior generated when you use the decimation method instead of a typical Moore/von Neumann neighborhood I'm not 100% sure if there's a novelty in my work, but I haven't seen anything like it. Maybe there is and I'm just oblivious to it. In terms of behavior though, the bifurcation at upper and lower bounds and the trigonometric roots (along with the tangent function) are properties I've never seen in a cellular automaton. That alone should be worth investigating imho. >Also you misspelled Antarctica on page 14 Thanks, noted. >>16887 Naming these has been a hassle. Despite my best attempts, I'm not even sure I've captured the essence of the properties enough.

>>16890 I think lots of things could stand to be renamed. Basically any principle or other important piece of technical knowledge that's named after a person, like Moore or von Neumann neighborhoods, should be given a name that actually conveys its meaning. What are you going to do with the paper though?

>>16896 I don't know. I plan on submitting it to a conference.

>>16896 These are the technical names >>16904 It'd be nice if you posted your speech notes along with these to give additional context/explanation.

>>16905 >It'd be nice if you posted your speech notes along with these to give additional context/explanation. I really wish I had more written content to share at the moment. That would make the paper writing process much easier.

>>16907 Bummer. Good luck nonetheless. If you get something you feel like sharing, I'd be interesting in reading.

>>16907 I'm going at the things from the reverse order. Usually the paper is written first and presentations are made to explain that paper.

>>16908 Sure thing. Whenever I get some tangible content I share it on here for feedback, so keep hanging around. Thanks

>>16905 >These are the technical names Well, yeah, and they ought to be different because they don't convey meaning as well as other names could. But this is getting too far outside of the topic. >>16910 >the name generator gave you von Neumann for this post Gotta love that. Get enough stuff for a full paper and make the man proud.

>>16911 >Well, yeah, and they ought to be different because they don't convey meaning as well as other names could. But this is getting too far outside of the topic. Could you offer a few suggestions?

>>16844 This is cool, good work. I can't claim to be knowledgeable of everything you mention therein, but I learned a little from it. I can't offer advice for improving it, beyond some opinions of grammar, but the grammar style seems consistent and isn't out of place for what seems presentation-oriented.

>>16930 Appreciated. I know it's a little too technical for general audience. I had to steer the presentation into its applications to make it more appealing, but that was in the Q&A session.

>>16928 Hi not posted here before, very interested in all this. I think the naming is fine, mathematics always requires learning new notations that aren’t necessarily conveying the most information, but if you want another view similar structures are used in membrane/plate modelling, the Moore neighbourhood is pretty much the representation of the 9 point Laplacian, so something like that could work. Also you misspelled avalanche on the sand pile section.

OP here, I am very pleased to see discussion is still as lively as I hoped it would be!! I will be spending the night catching up on everything.

>>16943 >the Moore neighbourhood is pretty much the representation of the 9 point Laplacian That was pretty neat to find out. >Also you misspelled avalanche on the sand pile section. Noted, thanks. I'm also interested to hear if there are any fundamental issues in the ideas I'm conveying through, like the objection this Lain >>16886 brought up.

>>16844 The random matrices you've explored in the last slide are relevant because your system appears to be a Tracy-Widom distribution. https://www.quantamagazine.org/beyond-the-bell-curve-a-new-universal-law-20141015/ https://www.quantamagazine.org/in-mysterious-pattern-math-and-nature-converge-20130205/

Open question: Why would the following equation yield a cotangent function for bifurcation values of the automaton? ((number of cells with value 0)/(number of cells with value 1)) - (number of cells with value 1))/((number of cells with value 0) = 1 The transformation that leads to the solution is: (number of cells with value 0) = cosu (number of cells with value 1) = sinu Its roots are the golden ratio and its negative pair.

>>17095 To be more specific, to generate the graph I added constants where: a = rate of change of (8 * (number of cells with value 1)) b = rate of change of ((number of cells with value 0)/(number of cells with value 1)) (a * sinu/cosu) - (b * cosu/sinu) = 1 simplified to: -b cot^2 u - cotu + a = 0 Which is the exact equation I've grabbed from WolframAlpha in the image >>17095 What is going on? How does a cellular automaton output cosine values for number of zero-cells and sine values for number of one-cells? This has been troubling me for over two years. I believe we are looking at a fundamental equation of this automaton but I can't read through it.

I wrote a simulator for a kind of cellular automaton in which the state of each cell is computed from the states of the surrounding cells. This way, each cell functions like an independent information processor. https://cenysor.neocities.org/programming.html#CellGates

>>17103 >This way, each cell functions like an independent information processor. That's just neat. How is the runtime?

>>17103 Isn't this the default way that CAs work?

>>17113 I am not sure, if I understand your question correctly, but each cell calculates its state (output) for every discrete time step of the simulation. >>17115 In the usual definition of CAs there is one transition rule for all cells. In my approach, every cell can use a different rule. This way, my model can simulate all conventional CAs

>>17117 >but each cell calculates its state (output) for every discrete time step of the simulation Right but you've written >there are 256^256 possible ways to map 265 bytes of input to 265 bytes of output Surely that must be taking a while to simulate, or not?

>>17097 >>17095 The upper bifurcation value of 1/2 + 1/(2sqrt(2)) is equal to cos^2 (pi/8). It's conjugate, the lower bifurcation value is 1/2 - 1/(2sqrt(2)) and by naturally sin^2 (pi/8), both adding up to 1. cos^2 (pi/8) - sin^2 (pi/8) = 1/sqrt(2) When broken into factors: cos (pi/8) - sin (pi/8) is a very good approximation to cos(1 rad) cos (pi/8) + sin (pi/8) ~ cos(1) + sin(1) sin(1) ~ cos^2 (pi/8) So the model is approximating its live values to cosine and sine of 1 radians as far as I can tell.

>>17130 A rule can be a lookup table for 256 values. But it can also be a function with a few lines of code that serves the same purpose. 256^256 is the number of possible distinct rules

>>17140 >256^256 is the number of possible distinct rules. That's impressive.

>>17140 >>17147 Isn't it 8*2^256 though?

>>17148 Unless I am missing something, when every 8-bit value (0-256) can map to any 8-bit value, that makes 256^256 possibilities. E.g. 0x01 can map to 256 different values, 0x02 can map to 256 different values, and so on...

>>17153 Do you think your code can be modified into any automata, to achieve universal computation, say this one? >>15906

>>17194 Theoretically, yes. My model uses a completely different approach than >>15906 though

>>17202 Right. But how would you establish the independent processing cells in >>15906 ?

>>17209 If I understand correctly, the rule for an individual cell can change during the simulation in >>15906 This was not a design goal for my automaton. It can nevertheless be used to compute the same mathematical model, although the implementation would not be as straightforward.

>>17213 >If I understand correctly, the rule for an individual cell can change Not quite. There is one overall rule but it's actually rules nested in rules nested in rules so it appears to be changing. You see the simulation is based on a conditional on the cell's initial value. If the initial value is 0, the cell does a neighborhood search and upon meeting a criteria either updates to 1 or not. If the cell's value is 1 -> run rules for upper, lower, right, left neighbors, run the reverse neighborhood check of the step above, and add coupling function in the end. What I had in mind is whether your processor design could be implemented into each subrule above.

>>17214 Now I understand it better. I will try to implement the algorithm. I won't have time for it in the next days, but I will definitely come back to it.

>>17232 Nice. Let me know if you need anything.

>>17131 >the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ solve for x

>>17131 cos(1)(cos(1)+sin(1)) ~ cos^2(1) + cos(1)sin(1) = 1/2 (cos(2) + sin(2) +1) ~ 1/sqrt(2) = cos(2) + sin(2) ~ sqrt(2) - 1 = 1 + sin(4) ~ 3 - 2sqrt(2) = sin(4) ~ 2 - 2sqrt(2)

>>17338 = sin(4) ~ 2 - 2sqrt(2) = sin^2(pi/8) = 1/2 - 1/(2sqrt(2)) =(-sin(4)/2)^2 ~ sin^2(pi/8) =1/(4-2sqrt(2)) = 1/2 + 1/(2sqrt2) = cos^2(pi/8) =1/(2+sin(4)) ~ cos^2(pi/8)

>>17339 = 1 - sin^2(4)/4 ~ cos^2(pi/8) = 3/4 + cos^2(4)/4 ~ cos^2(pi/8) = 1/2 + 1/2sqrt2 = cos^2(4) ~ sqrt(2) - 1 This is an even better result. Solve for x (closest approximation, in rad): sin^2(x) ~ 2 - sqrt(2) cos^2(x) ~ sqrt(2) - 1

>>17340 abs(sin(x)) = sqrt(2-sqrt(2)) abs(cos(x)) = sqrt(sqrt(2) - 1) arcsin(sqrt(2-sqrt(2))) = 0.871611162 rad arccos(sqrt(sqrt(2) - 1)) = 0,871611162 rad 0,871611162 rad = 49,939641 degree is the root angle of my cellular automaton. where sin(x) + cos (x) ~ sqrt(2) >>15906

>>17340 ROUND 2 FIGHT! sin^2(x) - cos^2(x) = - cos(2x) = 3 - 2sqrt(2) - cos(2x) = 0.17157287525 -sec(2x) = 5.82842712475 = cot^2(pi/8) This concludes my proof. >>17131

>>17356 where 5.82842712475 = square of silver ratio = (1 + sqrt(2))^2

>>17356 >-sec(2x) = 5.82842712475 = cot^2(pi/8) It's crazy because once hyperbolic transformations are applied for: e^u = cosx / sinx sec(2x) = cothu = cot^2(pi/8) and from there I can extend my hyperbolic definitions.

>>17408 >e^u = cosx / sinx Now apply complex transformations.

>>17357 BOOM

>>17493 and I've derived automaton's critical point (highest live cell count) as the Golden Ratio from here.

Could someone explain to me how the Strata rule runs in the link below? https://www.dhushara.com/CA/ It operates on a Mac based CA2D app with the 2D von Neumann setting as explained in the link. I've attached the image where it's referenced (second from left) and also a text file that contains the rules for the automata. The thing I don't understand about the rule is how to interpret the numerical input as a cellular automaton's rule. Strata [/b] [b]135000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000000000000000000000010000011000000011000000000000000000001000000000000000000001020002000000000000000000002021002000000

>>17508 """General von Neumann and Moore Neighbourhood Rules Binary (n-Ary) Rules You can input a binary rule for a von Neumann or Moore neighborhood using the rule tscnnn and select Neumann to portray a variety of well-known von Neumann neighborhood rules of up to s = 8 states or Moore up to 4 states and centered Moore up to 3. The first number t = 1-2 gives the rule type, binary =1 or totalistic 2), s = 2-8 is the number of cell states, c gives the neighbourhood type 4 is vN 5 is centered vN, 6 for hexagonal (use scale 6), 8 for Moore and 9 for centered Moore. The remainder nnn... give the value 0 ... s-1 for each of the sc neighbours in n-ary format with the bit order C,N,E,S,W,Nw,Ne,Se,Sw, or the (s-1)*c+1 summative states, adding the values of neighbours, for totalistic, since the largest possible sum is (s-1)*c and the smallest is 0."""

>>17510 Okay, so how does one come up with the number of s neighbors, especially when the count is over C,N,E,S,W,Nw,Ne,Se,Sw?

>>17514 Either your state transition function is expressible by taking at most the 3x3 block centered on your cell as its group of arguments, or it is not. If it is not, such as because you also need cells further away to perform your computation, then your state transition rule is not expressible in the vanilla version of the "General von Neumann and Moore Neighbourhood Rules" of "CA2D 4 Mac". The strata rule from your original question >>17508 does not need such extra cells, so CA2D:Neumann can express it and handle it.

>>17515 Thank you. Forgive me I'm a little slow these days. All the zeroes in 13500000000000000000000000000000000000000000000000000000000000000000000002 is the omitted cells but what about the 2 at the end? And how does one determine how to come up with these? Trial and error is a very daunting task.

>>17516 >All the zeroes in [...] is the omitted cells No, they are not. The explanation for rule encoding is in >>17510. >And how does one determine how to come up with these? Trial and error is a very daunting task. If you mean how to pick a state transition function to get an interesting CA, that's your job. If you mean how to encode a particular state transition function for vN+M for CA2D, the instructions are in >>17510 and there is no trial and error involved. >but what about the 2 at the end? Here is >>17510 expanded in detail. You have t, s, c = 1, 3, 5 therefore your cells are ternary and your neighborhood is centered von Neumann. This is why the encoding has 3+s^c = 3+3^5 = 246 characters. The '2' you quote is at index >>> rule.find ("2", 3) 73 in the encoding, so the nnn-array index is 70. Decimal 70 is ternary 2121. >>> int ("2121", 3) 70 The digit positions for centered von Neumann are CNESW with W in the unit position. Therefore the "2" at nnn-array index 70 encodes the fact that when your neighborhood state is C=0 N=2 E=1 S=2 W=1 the state of your cell for the next frame will be 2.

>>17528 Thank you so much. That was a very good explanation.

sand pile video: https://www.youtube.com/watch?v=hBdJB-BzudU

I've made a stackoverflow post to better illustrate the problem I'm facing. Are the bifurcation points inherently related to the critical point of the 2D square-lattice Ising Model? If so, have I portrayed it accurately algebraically? https://physics.stackexchange.com/questions/538656/particular-ising-model-of-cellular-automata-shows-mathematical-properties

>>17547 That's cool stuff. Had trippy moments with the animation in the YT description.

>>17340 Your >sin^2(x) ~ 2 - sqrt(2) value is the critical Ising parameter of Fortuin-Kasteleyn random cluster, where the model's duality is calculated from a p (1-p) * q, not unlike what you have mentioned here. >>16320 Look into Conformal Field Theory.

Do you see practical applications of cellular automata in cryptography, now that we are entering post-crypto quantum computer phase?

John Conway just passed away. RIP https://www.youtube.com/watch?v=C2vgICfQawE

>>17818 source?

>>17776 >now that we are entering post-crypto quantum computer phase elaborate?

>>17818 Great mathematician, RIP. I highly recommend the symmetries of things if you want to see the world differently

>>17818 RIP

>>17829 Quantum computers having the capability of breaking cryptography >>>/sec/7687

>>16255 >p = float(0.5 + (1 / (2 * sqrt(2)))) It's a qubit. Your automaton is calculating quantum superposition. 1/sqrt(2) (|0> ± |1>) = 1/sqrt(2) (1/sqrt(2) ± 1/2)

There's a weird borderline schizo dude named Garry who posts on 4chan about a cellular automaton based Unified Theory of Everything he calles "Optimum Theory". His automata run on Excel, but it's pretty interesting to watch. https://www.youtube.com/watch?v=6ClC50BsK5Y&feature=youtu.be

>>17915 His stuff looks cool but it also sets of my bs meter. Has he made any testable equations? Otherwise it is a pretty cool autonemon regardless and it seems to at least simulate real physical realities well.

>>17915 I watched this https://www.youtube.com/watch?v=CGqsqqLPi1I Seems like gobbledeegook to me. Muh math isnt good enough. I don't think he's actually done any serious mathematics in his life to say stupid things like this. Maybe I'm wrong though.

>>17917 >>17918 He is pretty lost. Here's his automata code. import numpy as np import matplotlib.pyplot as plt import matplotlib.animation as animation from numba import jit #code is very unoptimized pls no bulli @jit def optimum(input_arr,G): """The original Optimum Thoery CA. Takes a 2D array of values and returns the next iteration. G is the divisor.""" bottom = np.roll(input_arr,-1,axis=0) top = np.roll(input_arr,1,axis=0) right = np.roll(input_arr,-1,axis=1) left = np.roll(input_arr,1,axis=1) upper_left = np.roll(left,1,axis=0) upper_right = np.roll(right,1,axis=0) bottom_left = np.roll(left,-1,axis=0) bottom_right = np.roll(right,-1,axis=0) neumann_sum = bottom + top + right + left symm_diff_neumann_moore_sum = (upper_left+upper_right+bottom_left+bottom_right)/2 rand_offset = np.random.randint(-1000,1000,size=input_arr.shape) delta = input_arr+rand_offset return((neumann_sum + symm_diff_neumann_moore_sum + input_arr + delta)/G) @jit def optimum_quiet(input_arr,G): """A quiet version without the 'quantum foam'.""" bottom = np.roll(input_arr,-1,axis=0) top = np.roll(input_arr,1,axis=0) right = np.roll(input_arr,-1,axis=1) left = np.roll(input_arr,1,axis=1) upper_left = np.roll(left,1,axis=0) upper_right = np.roll(right,1,axis=0) bottom_left = np.roll(left,-1,axis=0) bottom_right = np.roll(right,-1,axis=0) neumann_sum = bottom + top + right + left symm_diff_neumann_moore_sum = (upper_left+upper_right+bottom_left+bottom_right)/2 return((neumann_sum + symm_diff_neumann_moore_sum + input_arr)/G) @jit def optimum_wave(input_arr,change,G): """The 'wave equation' version. Note that this one also takes the 'change map' as an argument.""" bottom = np.roll(input_arr,-1,axis=0) top = np.roll(input_arr,1,axis=0) right = np.roll(input_arr,-1,axis=1) left = np.roll(input_arr,1,axis=1) neumann_sum = bottom + top + right + left return(change + (neumann_sum-4*input_arr)/G) @jit def optimum_3D(input_arr,G): """3D version. Takes a 3D input array and iterates it (memory intensive, can easily be improved).""" forward = np.roll(input_arr,-1,axis=0) backward = np.roll(input_arr,1,axis=0) bottom = np.roll(input_arr,-1,axis=1) top = np.roll(input_arr,1,axis=1) right = np.roll(input_arr,-1,axis=2) left = np.roll(input_arr,1,axis=2) c1 = np.roll(forward,1,axis=1) c2 = np.roll(forward,-1,axis=1) c3 = np.roll(forward,1,axis=2) c4 = np.roll(forward,-1,axis=2) c5 = np.roll(backward,1,axis=1) c6 = np.roll(backward,-1,axis=1) c7 = np.roll(backward,1,axis=2) c8 = np.roll(backward,-1,axis=2) c9 = np.roll(top,1,axis=2) c10 = np.roll(top,-1,axis=2) c11 = np.roll(bottom,1,axis=2) c12 = np.roll(bottom,-1,axis=2) e1 = np.roll(c3,-1,axis=1) e2 = np.roll(c3,1,axis=1) e3 = np.roll(c4,-1,axis=1) e4 = np.roll(c4,1,axis=1) e5 = np.roll(c7,-1,axis=1) e6 = np.roll(c7,1,axis=1) e7 = np.roll(c8,-1,axis=1) e8 = np.roll(c8,1,axis=1) msum = forward+backward+bottom+top+right+left csum = (c1+c2+c3+c4+c5+c6+c7+c8+c9+c10+c11+c12)/2 esum = (e1+e2+e3+e4+e5+e6+e7+e8)/3 rand_offset = np.random.randint(-1000,1000,size=input_arr.shape) delta = input_arr+rand_offset return((msum + csum + esum + delta)/G) @jit def vec_map(arr): """Takes a 2D array and returns the (absolute) 'vector map'.""" y_s,x_s = arr.shape h = np.zeros_like(arr) for y in range(y_s): for x in range(x_s): vals = np.zeros(8) vals[0] = np.abs(arr[(y+1)%y_s][(x-1)%x_s]) vals[1] = np.abs(arr[(y+1)%y_s][(x)%x_s]) vals[2] = np.abs(arr[(y+1)%y_s][(x+1)%x_s]) vals[3] = np.abs(arr[(y)%y_s][(x+1)%x_s]) vals[4] = np.abs(arr[(y)%y_s][(x-1)%x_s]) vals[5] = np.abs(arr[(y-1)%y_s][(x-1)%x_s]) vals[6] = np.abs(arr[(y-1)%y_s][(x)%x_s]) vals[7] = np.abs(arr[(y-1)%y_s][(x+1)%x_s]) mi = np.argmax(np.abs(vals)) h[y][x] = mi return h #EXAMPLE #define main grid grid = np.zeros((128,128)) prev_grid = np.copy(grid) change = np.zeros_like(grid) #matplotlib setup fig = plt.figure(figsize=(8,8)) ax = fig.add_subplot(1,1,1) #iterates the main grid when called. Needs to be its own function for mpl animation. def animate(i): global grid global prev_grid global change ax.clear() prev_grid = np.copy(grid) grid = optimum(grid,7.125) change = grid-prev_grid ax.imshow(vec_map(change)) #animate anim = animation.FuncAnimation(fig,animate,interval=10) plt.show() # #save the first 512 frames as images (use this INSTEAD of FuncAnimation & plt.show()) # for i in range(512): # animate(0) # plt.savefig('frame_%05d' % i)

>>17919 Dope.

Inspired by this thread and Wolfram's book, I decided to spend my week trying an experiment with CA and crypto markets. My hope was that there was some sort of pattern matching with a CA rule in the minute to minute movement. I assumed that as movement over larger periods of time can be attributed to randomness, movement on a minute by minute basis could've matched some CA rule. So I wrote some code to take various increments of the minute to minute movement and generate all CA rules with that size and find matches. As an example, taking a 30 minutes of data meant generating all CAs with a width of 30 and depth 10000, changing the market data to 1s and 0s (1 if the price went up, 0 if the price went down), and then trying to identify matches. And while I've been able to find matches, especially in rule 45, they're completely random and follow no set pattern. I've tried plugging the results into https://oeis.org/ but still didn't get anything. Kinda bummed out about this. Did anyone else think of applying CAs to finance in some novel ways? Any suggestions for how I could improve on this?

>>18026 >I assumed that as movement over larger periods of time can be attributed to randomness >movement on a minute by minute basis could've matched some CA rule. Why wouldn't a minute by minute movement match randomness by your own account? I would say longer periods of time would be more or less predictable.

>>18041 Shit, I was tired when I wrote that. I mean that movement over longer periods of time can be attributed to information and rational actions, while on the minute by minute basis it would look random but maybe match some CA rule in a predictable fashion. I guess I'll stick with trying to predict movement over longer periods of time with more traditional algo trading methods.

>>18050 I think you've got it backwards. You need to define relations between different markets and then try to formulate it as a CA. For example: Dollar is propped up against petrol, aka petrodollar. In a fair market the rising petrol prices would mean the rising dollar because you would need to purchase dollars to make payments for a fixed amount of oil a country would consume on average. Thus this would formulate: If a cell is crude oil prices, if it's 1, its neighbor USD is also 1. Then you'd go on with the stock market. If a stock market is the display of the health of a capitalistic economy, the rising prices of commodities would withdraw cash invested from stocks. Thus the rising dollar and the petrodollar would be inverse to the stockmarket. Then the stock neighbor would be 0 or -1. Then you'd move on to precious metals. They rise and fall depending on the trust to dollar and the stockmarket. If the dollar and the stockmarket are both -1, the precious metals are 1. On and on you can formulate the entire economy as a CA.

>>18053 >>18053 That's really clever and closer to what I initially had in mind, I think. Although to me it looks like all those variables are tied to a value, in this case the petrodollar. How exactly would that look as a CA? Lastly, do you think there's any insight to be found in doing this? I assume you'd have to use a totalistic CA and get some shades in it. I can see it being interesting as an art/coding practice challenge, but I wonder if it has any practical applications in forecasting. Thanks lainanon.

>>18066 >How exactly would that look as a CA? #FEeD money into circulation def init(): global c, nc c = zeros([L, L]) for x in xrange(L): for y in xrange(L): c[x, y] += 1 nc = zeros([L, L]) # NOOOO NOT MY MONERINOOOSS NNOOOO def haha_money_printer_go_brrrrr(x, y): brrrrr_count = 0 for dx in range(-1, 2): for dy in range(-1, 2): brrrrr_count += c[(x + dx) % L, (y + dy) % L] # print brrrrr_count return brrrrr_count - c[x, y] #smart money escapes bobo def trade(): global c, nc, array array = [] for x in xrange(L): for y in xrange(L): array.append(c[x, y]) for dx in range(-1, 2): for dy in range(-1, 2): if c[x, y] < brrrrr_count / 8: nc[x, y] -= 1 nc[x+dx, y+dy] += 1 c, nc = nc, c >Lastly, do you think there's any insight to be found in doing this? I assume you'd have to use a totalistic CA and get some shades in it. I would use a sandpile model to describe the boom and bust cycles.

>>18068 As for forecasting, I think it's best to proceed with a try-and-see approach. What would the outcome be like?

>>18066 Further reading: https://www.sciencedaily.com/releases/2008/03/080331110052.htm https://arxiv.org/pdf/cond-mat/0405257.pdf

>>18068 >>18069 >>18070 Thank you based lainanon, I'm gonna get to work now. This should be interesting. Slightly off-topic since it's not strictly CA but follows the same principles, but what do you think about Wolfram's Fundamental Theory of Physics ideas? He's been posting and streaming his progress. https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/

>>18082 Wolfram is a self-aggrandizing hack. Unreliable claims.

>>18082 i tend to agree with >>18084 but i think some of what he's saying is worth listening to. for one thing, focusing on developing a deeper theory of emergent behavior seems like a good way forward in areas of physics that have been difficult to progress in using classical theories. fluid dynamics in particular comes to mind. ultimately, i think it will be essential if we ever want to have a rigorous way to reason about consciousness. however wolfram tends to give himself more credit in developing this idea than he actually deserves. don't get me wrong, his cellular automata stuff is cool, but people like mandelbrot, lorenz, cantor, have contributed more to so-called "chaos theory" than wolfram has. another problem i have with his approach is he seems to think "solving physics" is all about finding a single predictive rule rather than developing tools to analyze and describe the behavior of chaotic dynamical system more abstractly. even if such a rule could exist (which is a huge assumption, it would essentially imply that non-computable numbers don't "exist" in any physical sense, since by definition you can't converge to a non-computable number using a finite algorithm, and in mathematics non-computable numbers are infinitely more abundant than computable numbers), there's a question of whether it would even be useful for doing physics. and of course as with any rigorous "theory of everything" there's the incompleteness gremlin lurking in the background, but i don't want to overstate the importance of this as the implications of godel's theorem in theoretical physics aren't well established at this point.

>>18082 I agree with both >>18086 >>18084 Wolfram's work is cool but it's not as revolutionary as he'd like to believe.

>>18068 Slight correction: if c[x, y] < brrrrr_count / 8: nc[x, y] -= 2 #forgot the double count nc[(x + dx) % L, (y + dy) % L] += 1 #forgot the bounds

>>18082 it's not very great. The stuff with emergent geometry from CA rules is really cool but the physics stuff fails to hold water. Their work on using their framework to model general relativity is shakey at best. They use a hypergraph embedding into Minkowski space to model spacetime, but it's unclear how that embedding always exists or whether it can be reformulated to be intrinsic to the hypergraph. All the opinions I've read from physicists say similar things about the quantum stuff, which I am unqualified to judge. It really is just another "theory of everything" with Wolfram's customary flare and dishonesty. I would not even hold it on par with nonstandard gravity theories or string theory.

>>17341 >0,871611162 rad = 49,939641 degree when I round this up to 50 degrees and run the simulator with cos(pi/3.6) and sin(pi/3.6) as initial probability densities, cosine gives the largest live cells of the simulation, and sine gives the closest approximation to 2D square lattice Ising model's critical inverse temperature. Shit's wild

Here's an interesting Symbolics paper I've come across, discussing an approach to the Game of Life which optimizes it by splitting computation into cacheable chunks. It's in the form of six pages, so here are the first three. It's brief, so I don't see any shame in reading it a few times to fully understand its precise approach.

Here's the latter half. This was hosted from the following page in this same form: http://www.loper-os.org/pub/hl/hl.html

>>17931 Source, please.

>>18441 It's behind a paywall unfortunately: https://www.optimuminstitute.org/ He got derezzed on r/physics and lurks on 4chan /sci/. https://warosu.org/sci/?task=search&ghost=&search_text=gary https://warosu.org/sci/?task=search2&ghost=yes&search_text=&search_subject=optimum+theory+continuation

>>18239 >>18238 Here's those images in a single PDF

hello i wrote program in C MPI, that generates several text-based GamesOfLife and runs them in parallel. It somehow works, but text is mangled, some lines are longer than another, even tho i use print_method only once per cycle https://pastebin.com/w8Z5tyH0

>>18561 >text is mangled, some lines are longer than another I'm no MPI expert, but looking at your code, it looks like there's just one synchronization barrier before the print, and then all the processes will try to print at the same time (with a rank argument hardcoded to 7 for some reason?), possibly leading to messy output. Also, calling malloc in an infinite loop without ever freeing the memory is a bad idea if you want your program to run for even a reasonably long time.

My Ising model cellular automata has a second critical point around p=0.64-0.65, when the cells' evolution for a LxL = 100x100 must be above 4700. I repeated with various combinations but eventually averages down to 4600 level. 1. Is there a known fixed point around that range? 2. The initial above 4700 ride and the eventual drop down reminds me of hysteresis, yet I reset the automata at every rerun. Is this possible?

>>18082 >I’ve always assumed that any entity that exists in our universe must at least “experience the same physics as us”. But now I realize that this isn’t true. There’s actually an almost infinite diversity of different ways to describe and experience our universe, or in effect an almost infinite diversity of different “planes of existence” for entities in the universe—corresponding to different possible reference frames in rulial space, all ultimately connected by universal computation and rule-space relativity. >>>/drug/

>>18960 Sickest burn in physics history.

>>18513 spared me from having to google imagemagick commands. thanks

This website contains a wide array of cellular automata rules running on the background, along with a development kit called WebCA. Maintained by cofounder of Autodesk. https://www.fourmilab.ch/cellab/webca/?show=demo

I've recently come across this Accretor 3D cellular automaton used to create sculptures: https://notnot.home.xs4all.nl/accretor/Accretor.html I learned about it from this article; one distinguishing factor is the generation rule takes position into account, instead of merely the neighbor count: https://softologyblog.wordpress.com/2018/01/12/accretor-cellular-automata I don't have anything to contribute beyond making others aware of this work.

>>18698 >2. The initial above 4700 ride and the eventual drop down reminds me of hysteresis, yet I reset the automata at every rerun. Is this possible? most likely variables not being cleared after each run

>>18962 Hahaha oh man I could go on about the kurzweil and the Singularity doomsday cult, but that's a topic for another thread

>>12956 >- how does hyperbolic geometry fit into this, although I'm guessing it's related to the fact that you can embed trees in hyperbolic space The Ising model's critical fixed point and the self-dual clustering point are distances (1+sqrt(2)) and ((1+sqrt(2))*(1+sqrt(2)) units apart from the origin respectively. When transformed into unit circle, the reciprocal of (1+sqrt(2)) becomes (sqrt(2)-1), and the hyperbolic distance is: arctanh(sqrt(2)-1) = 1/2 * ln(1+sqrt(2)), or the critical temperature of the 2D Ising square lattice.

Explanatory video! Watch before it gets pulled down, as is the norm. https://www.youtube.com/watch?v=WuXCS_K_8qM

>>15906 What you're describing is a type III spin-ice vertex, which is a monopole (net magnetic charge), i.e. impossible according to Gauss's Law.

>>15906 Hello.

>>18698 >2. The initial above 4700 ride and the eventual drop down reminds me of hysteresis, yet I reset the automata at every rerun. Is this possible? Yes. >>18698 >2. The initial above 4700 ride and the eventual drop down reminds me of hysteresis, yet I reset the automata at every rerun. Is this possible?

I'll take the liberty of anonymity to skip all pretensive modesty in preamble; I have privately solved what many of you are intuitively heading towards. You will not reach the Singularity, though I am impressed in some of the dialogue here. I will answer a single question, FCFS.

>>19496 What is the first order ferromagnetic transition point below the Curie temperature, T<Tc under changing H magnetic field strength? Pic related, also the link below: https://physics.stackexchange.com/questions/179144/paramagnetic-ferromagnetic-transition-under-a-magnetic-field If you can't answer that, then: Which of the above discussions impress you the most? Why?

>>19499 >What is the first order ferromagnetic transition point below the Curie temperature, T<Tc under changing H magnetic field strength? Pic related, also the link below: This of course entails the exact solution of the 2D Ising square lattice for when H ≠ 0

>>19496 As a matter of fact, I think I've figured out the answer to the question I've posed in >>19499. It's a ferromagnetic-spin glass transition. Thus I'd appreciate if you could priorily expand on the notable dialogues and promising ideas you've seen in this thread. Of course I wouldn't mind if you'd answer the other question, but that is not a priority to me at this moment.

>>19499 >Which of the above discussions impress you the most? All that feature a very particular aggregate link between NN, CA and self replication. Why am I impressed? It highlights a number of necessary social parameters discordant with traditional research structures; unusual in my experience. The diffuse answer is the price of your open question. A squandered favour...

>>19508 Yeah, I don't think so. You failed to impress me. Thanks for playing.

>>19509 Who are you, and why should I care? You squandered a very rare favour; used wisely- evident in actuality. I've done this many times before, and intend on 14 more before my death. This is now over.

>>19496 >>19510 SEETA num?

>>19512 d937327f535d88f3 - "groundling".

>>19510 >neurosuggesting

>>19510 Do you enjoy playing computer games with cheat codes on?

>>19508 >>19510 You see, I didn't actually need your help with the Ising model. I will solve it on my own eventually. I needed to know the trajectory my research needs to take and you gave me the answer to that. For that I'm grateful. I would be more pleased if you could have provided insights, but that is up to you and irrelevant to me. P.S. I'm the one who brought the NN Quine into this thread in the first place.

One more step towards getting published.

This presentation >>16844 evolved into the one attached. One of our very own automata is about to get published.

>>19751 say my name Heisenberg model.

>>19751 Why do you have the symbol of Chaos on the first page?

>>20106 My Photoshop license ran out, I am not familiar with GIMP and I had to improvise.

any new stuff being done in the cellular automata world/intelligent agent field?

Does anyone know some resources to learn programming CA using compute shaders?

I read this paper lately, and it documents a non-deterministic, one-dimensional cellular automaton which is used for mirrored maze generation. I've also implemented it, and at first I wasn't going to make it in-place, but there's only a single cell which needs to be saved for such; otherwise, only four current cells need to be considered, the prior two, the current, and the next. My implementation doesn't use the mirroring as input for generating the final cell, which I believe the original does, but it still comes out looking similar. It's neat.

There are articles about Swarm Intelligence working with Cellular Automata. Is it any good?

>>22107 Run a Particle Swarm Optimization (PSO) and get the probability of nodes, plug them into a CA.

What is a good graphics library for C that I could use for procedural generation and cellular automata? I saw the OpenFrameworks book in the books thread but iirc it was just C++ examples although openframeworks also has a C interface. I want to start with 2D stuff but eventually try some 3D animations as well later on. I have read through most of thebookofshaders.com so I know some stuff about GLSL programming as well.

>>23999 I find SDL the best one. There is also raylib, which is good since you can get something done faster than using SDL, but I have never used for anything that is not some atari game clone or board game.

>>9558 I'm working on simulations of things called 'Skyrmions'. Imagine every atom inside a material was a little arrow, that could point in any 3D direction. This arrow represents the magnetic orientation of the atom, the direction its 'north pole' is pointing, also known as the atom's spin vector. It turns out that in certain materials, if you get them to the right temperature and set up a magnetic field pointing through the material, you get patterns in these arrows- these spin vectors- that act like larger-scale structures. We call some of these magnetic Skyrmions (pic related). What's interesting, in the context of this thread, is that the whole thing is sort-of similar to cellular automata. Every pixel/cell is an atom, the color represents the direction they point in (black means in or out of the plane, red is East, green is North, purple is West, teal is South) and there are a set of rules determining how the state changes from one frame to another. There are also tweakable parameters- the strength of the external magnetic field you're subjecting the atoms to, the temperature of the simulation, the size of the simulation, and the relative strengths of the different parts of the equation describing the overall energy level of the simulation, which is what decides how the simulation evolves.

>>23999 OpenCV is decent for this kind of thing because it's oriented towards more scientific rendering (unlike SDL which, while still good and certainly capable of what you want to do, is designed with video game engines in mind). However, it's not a GPU compute library. Your actual simulation would be handled on the CPU with the results displayed using opencv. If you want to actually process your model on the GPU then that's kind of a different story. You'd probably just want to write your own renderer in that case, perhaps with some intermediate library to make interacting with OpenCL/Vulkan/cuda more pleasant.

>>24070 >>10556 >>12865 >Ising model Huh, should have checked whether that had come up before I posted. To be clear, what I've posted is a variant of the classic Ising model- where the Ising model as most people understand it is 2D with 1D unit vectors (a 2D lattice with vectors pointing up or down) the spin model I'm using here is D=3, d=3- a 3D lattice with 3D unit vectors at every lattice point. The similarity to CA is that the time evolution of a spin depends, probabilistically, on its neighbours (and neighbours-of-neighbours). For a general overview, in case anyone's interested, we have an expression for the energy associated with a particular spin vector in the lattice (one of the unit vectors). This expression depends on the spin's neighbours. There's one term to the energy function (the Heisenberg term) that grants assigns lower energy to neighbouring spins co-aligning, but another (the DM interaction) that grants lower energy to them being at right angles. These terms are in direct competition, and the interplay between them leads to neighbouring spins ending up at slight angles to each other. The last term is the influence of the outside magnetic field. It just encourages spins to point in the same direction as the field; if you make the field strong enough, the field interaction swamps out the other two, and all the spins just point along the field axis. Very boring. How the simulation iterates is interesting. For a given spin in the lattice, we generate a potential new spin vector at random. Then we compare its energy, according the the energy function and neighbouring spins, to that of the old spin. If the new spin would decrease the overall energy- good! We just accept it. If it would increase the energy, we accept it with a probability that decreases with the difference between the old and new energies, and increases with the temperature of the system. Skyrmions are the round structures- patterns in the lattice of spins. You can see the curving caused by the interplay of the H and DM interactions. The direction of the field, in this simulation, is along the viewing axis. What's cool about all this is that these things exist in real life. This is all a model of real-world physics, like the spin glasses mentioned earlier. People are able to get these patterns to form, and they're able to move them around like particles- even though they're just quasi-stable patterns in the texture of spins. (In pic related, I'm not rendering any arrows significantly in line with the viewing axis. There's a spin-vector being simulated at every lattice/grid point.)