https://community.wolfram.com RSS Feed for Wolfram Community showing questions tagged with Wolfram Science sorted by new. https://community.wolfram.com/groups/-/m/t/3567195 Hi I'm an independent researcher exploring a potential link between the shortcut Collatz function (`T(n)`) and ECA Rule 54 (single cell evolution). I need help verifying a formula from the MathWorld page for Rule 54. The page presents this piecewise formula: $$ a(n) = \begin{cases} \frac{1}{15}(4^{n+2} - 1) & \text{if } n \text{ is even} \newline \frac{7}{15}(4^{n+1} - 1) & \text{if } n \text{ is odd} \end{cases} \quad (n \ge 1) $$ [https://mathworld.wolfram.com/Rule54.html][1] My work proves that `T(n) = wH(a(n)) - 1` (where `wH` is binary Hamming weight), assuming `a(n)` represents the normalized decimal value (odd-part) of the n'th row pattern. Could someone please help clarify: 1. Is the interpretation of `a(n)` (as normalized value) correct for Rule 54 (single cell) ? 2. What is the source or derivation for this specific `a(n)` formula? 3. Is the formula original to MathWorld? 4. If a primary source isn't readily available, is citing the MathWorld page directly considered acceptable practice for this formula in a formal paper? Verifying this is crucial for my paper connecting Collatz to Rule 54. While Dr. Weisstein might be the most direct source, perhaps he isn't very active here. Any insights, references, or verification from the community would be greatly appreciated. Thanks so much! Rudi B. Stranden [1]: https://mathworld.wolfram.com/Rule54.html Rudi Bjørnsen Stranden 2025-10-29T17:38:56Z https://community.wolfram.com/groups/-/m/t/3549651 Hello! I am trying to develop a set of canonical examples of what one might call "a piece of Ruliology", and I am asking Community members to provide examples of what they think represents good Ruliology. Even if you don't know of an example that you consider to be canonical, you can still contribute to the discussion. What do you think makes a good "piece of Ruliology"? What section headers would cover the components you would expect to see? What specific questions would a Ruliologist seek to answer in an investigation? What other questions should I be asking? As a point of interest, the Atlas of Simple Programs provides us a useful basis for describing the units of study in the discipline of Ruliology. You might want to check it out, even if you haven't seen it recently. Any suggestions for how to best use/improve this asset in this context would also be relevant to the conversation here. https://atlas.wolfram.com/ This is a great chance to contribute to a wide ranging conversation about Ruliology, and to impact the future of this computational discipline. I appreciate any engagement you can allocate to this topic! -Elly Ellynne DEC 2025-09-23T13:31:53Z https://community.wolfram.com/groups/-/m/t/3054160 Is the reduction of machine language into cellular automaton thorough? Or are there elements outside of this reduction? Are certain "Mathematica Rules" usually here but incomplete in any reverse engineering into cellular automaton? Digital Linguistics 2023-10-22T21:44:57Z https://community.wolfram.com/groups/-/m/t/2986471 I'm interested to learn more about what ideas and work has been done with respect to Lagrangian / principle of least action etc. with respect to the Wolfram Physics Project; it is mentioned in Chapter 8 of the Technical Introduction, but I'm wondering if anyone knows of any "further reading" of sorts on this topic. Thanks! Quote from Chapter 8.3: "quantum action density (Lagrangian): total flux (divergence) of multiway causal graph edges". I would be very curious to understand the derivation or connection. Robert Mendelsohn 2023-08-08T16:28:13Z https://community.wolfram.com/groups/-/m/t/2919022 Was wondering if there was a Discord for things related to NKS. And if there isn’t, if there are people out there who would be interested in one? Currently studying it and think it would be useful if there was a channel there to exchange notes (on the notes), assisstance with concepts. Vince Nguyen 2023-05-13T05:17:25Z https://community.wolfram.com/groups/-/m/t/2706895 Does anyone have experience with: "Mining The Computational Universe"? Is Mining primarily for Algorithms/ meta-algorithms? What kind of equivalences would you search for? *Beginner to: NKS. Evyn Tyndzik 2022-11-20T22:57:17Z https://community.wolfram.com/groups/-/m/t/2677088 I have a few questions about Wolfram's theory for fundamental physics that I would like to clarify... I. First, as I understand it, and please correct me if I'm wrong, is that similar observers to us will most likely perceive at least the same "general laws" (or most fundamental laws) of physics and mathematics. But for vastly different observers compared to us, couldn't they be able to perceive vastly different general laws? II. Recently, Stephen Wolfram wrote an interesting article about his proposed relationship between maths and physics (https://writings.stephenwolfram.com/2022/03/the-physicalization-of-metamathematics-and-its-implications-for-the-foundations-of-mathematics/#some-historical-and-philosophical-background). There, Wolfram talks about the physicalization of mathematics and adopts some sort of platonic position saying that mathematics does really exist in some sense or another because mathematics and all the relations between abstract concepts would exist in the ruliad (more information in the article). This reminded me of Tegmark's thesis of the "Mathematical Universe Hypothesis" (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) where all mathematical structures would exist as separated universes. (There's even a comment in that article asking what is the relation between Wolfram's and Tegmark's ideas, but nobody replied). Therefore, basically my question is: Since Wolfram says that mathematical concepts and structures would exist in the ruliad, and the rulial space is what makes reality (and every possibility is realized by it), couldn't we say that all the universes proposed by Tegmark would exist in some way according to Wolfram's ideas? Couldn't all logically possible worlds exist in some way according to this? Nodu Agga 2022-10-29T01:52:30Z https://community.wolfram.com/groups/-/m/t/2668494 Anyone have a clue about what the code at the back of New Kind of Science is supposed to be? ![back of an NKS book][1] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=CleanShot2022-10-24at00.27.20.png&userId=2600645 Vince Nguyen 2022-10-24T04:28:02Z https://community.wolfram.com/groups/-/m/t/2607031 Has any of the reversible extensions of the elementary one-dimensional cellular automata in NKS (e.g. Rule 37R) been shown to be computationally universal (like Rule 110)? If so, please give me links. Otherwise, could this be the case? Or is there a proof that no nR can be universal? Giulio Prisco 2022-08-28T07:20:12Z https://community.wolfram.com/groups/-/m/t/2422777 I've been really interested in the field of small universal Turing machines, having read some of "A new kind of Science" that I took from the library. I saw that Stephen Wolfram wrote a few words about how 4-state, 2-colour machines exhibit complex behaviour given the right rules, but I've only found proofs of universality (and indeed p-completeness due to emulation of rule 110 cellular automata) in 2-state, 4-symbol machines. I've been wondering whether the 4-state, 2-colour machines have been proven universal or have been proven non - universal or whether or not this is still an open problem? Likewise, Wolfram claimed that it is unlikely, although not impossible for a 3-state, 2-colours machine to have a universal rule, and I wish to know if this is also still open or has been resolved, and if it is possible, then it will share the property of being the smallest UTM together with the 2-state, 3-symbol machine that has already been proven to be universal. Thanks in advance! Yann Tal 2021-12-11T14:09:00Z https://community.wolfram.com/groups/-/m/t/2397482 I am trying to understand the causal invariance property from known (physical) phenomena. Are natural ocurring cycles in astronomy, biology, geology, climate, weather but also in physics and mathematics a kind of manifestation of the causal invariance property? What about the mathematical phenomena of attractors when a system tends to evolve to a certain specific state given a wide range of initial conditions? Constantin Cozma 2021-11-01T08:15:26Z https://community.wolfram.com/groups/-/m/t/2397051 Does the Wolfram Model describe and confirm Ilya Prigogine`s theory on irreversibility and dissipative structures by which there is a cyclic mechanism (biorhythm) from unstable to stable connections? If the answer is yes, which I believe it is, then where can this process be found in the model? Constantin Cozma 2021-10-31T11:16:55Z https://community.wolfram.com/groups/-/m/t/2395893 Hello, I am trying to understand the relation between time dilation and gravity through the hypergraph model. Is gravity a function of the hypergraph density of nodes in a way that affects the local computational speed and generates a time dilation? Thanks ! Constantin Cozma 2021-10-29T06:46:36Z https://community.wolfram.com/groups/-/m/t/2380863 Are the four paradigms of science described in this Stephen Wolfram's writing (https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/) more general as we reach the 4th paradigm of the 2020's? In other words, does the 4th paradigm (the multi computational one) englobe or include the rest of them? Is it a "generalization" of all of them? (I mean, mathematical models are not included in the first "structural" paradigm, since they usually consider time, while in the structural paradigm time is not considered. But the mathematical paradigm includes all the models from the structural paradigm, since time is or is not considered in mathematical models. So, would this continue up until the last paradigm, which would include all the models from the rest)? If it is, then, according to the framework of the multicomputational paradigm, every possible model from the rest paradigms (all possible structural, mathematical and computational models) would be included in it. But if that is really the case, then, how could there be an incompatible model with Wolfram's framework (which is basically the multicomputational one) as he says here ( https://www.wolframphysics.org/questions/scientific-general-interest/ ): > Any particular rule could be proved wrong by disagreeing with > observations, for example predicting particles that do not exist. But > the overall framework of our models is something more general, and not > as directly amenable to experimental falsification. Asking how to > falsify our framework is similar to asking how one would prove that > calculus could not be a model for physics. An obvious answer would be > another model successfully providing a fundamental theory of physics, > and **being proved incompatible**. If his multicomputational framework englobes every possible conceivable model, how can there be any model incompatible with it? Nodu Agga 2021-10-06T23:21:02Z https://community.wolfram.com/groups/-/m/t/2379277 At the Wolfram Physics Project Q&A site, there is a question asking whether the model/framework is falsifiable. The answer is: * > Any particular rule could be proved wrong by disagreeing with > observations, for example predicting particles that do not exist. But > the overall framework of our models is something more general, and not > as directly amenable to experimental falsification. Asking how to > falsify our framework is similar to asking how one would prove that > calculus could not be a model for physics. An obvious answer would be > another model successfully providing a fundamental theory of physics, > and being proved incompatible. * My question is: How could a model be proven to be incompatible with Wolfram's model? Is it because there could be a model involving hypercomputational processes? But if that is the case, couldn't the assumptions of the model be relaxed to allow hypercomputation? Nodu Agga 2021-10-04T12:37:43Z https://community.wolfram.com/groups/-/m/t/2346160 I've watched this live minting (Stephen Wolfram's Picks of Cellular Automata from the Computational Universe): https://youtu.be/pMfrRFNCKhE and looks like it uses the Cardano blockchain, but I can't find it in the Cardano blockchain explorer. I asked for it on the Youtube video, I sent a question about it to [SUPPORT][1] and I asked a question on the Tweet at https://twitter.com/manofstring/status/1425921930834714629 but so far all messages from me are ignored. Does someone here have the information where I can find the NFT, and maybe even buy it? [1]: https://liveminting.com/contact Frank Buss 2021-08-19T18:04:58Z https://community.wolfram.com/groups/-/m/t/2290617 I know this is an open question, but I wanted to run down a few arguments on it. [Rule 30][1] is a well-known [ECA][2]. Wikipedia has a fine introduction to that topic if you're unfamiliar. Here's a 60-second version of the mechanism: 1. Let $s_0 \leftarrow 1$. 2. Let $s_{i+1} \leftarrow s_{i}00$, but altered by stepping through the bits left to right and flipping all of them except for those followed by more than one $0$. (The last two bits can wrap around to the beginning for this purpose.) 3. Repeat step $2$ ad infinitum. Where not otherwise specified, I'll be referring to the Wolfram standard configuration of a single "seed" cell with a value of $1$ set against an infinite $0$ background as the initial row. For the purposes of this question, I'm also asserting that a few as-yet-unproven properties of Rule 30, which are widely believed to be almost certainly true, are true, in particular that every finite binary sequence on a row or column appears somewhere within the light cone generated by the initial seed cell. [![rule 30, first 50 rows, eye candy][3]][3] There are several defining and/or typical properties of TC systems that seem to be in a gray area w.r.t Rule 30. (Skip down to **Bi-simulation** for the part I find the most vexing.) ### Limits on computation as pure function mapping? First, I strongly believe that every [total computable function](https://en.wikipedia.org/wiki/Recursive_set) will be computed as the structure unfurls. In particular, there will be infinitely many row segments which could have $k$ bits set as the input to a function, where each of the $2^{2^k}$ distinct mappings from $k$ input bits to a single output bit will eventually appear in infinitely many cells down the road, in every column. It's important to note that the computing taking place to arrive at these answers is legitimate, and would give the correct result if we went in manually and altered the inputs, or found another copy of the algorithm with different "hardcoded" inputs resulting from the single $1$ seed cell. The computing will almost certainly be extremely slow and scattered, but each individual logical step required will actually take place, so there's no trickery in that sense. That said, my understanding is that what's going on is the enumeration of increasingly complex boolean circuits, or whatever analogous interpretation you prefer. This means that every complete computation it can do was guaranteed to terminate from the outset, which would seem disqualifying. You could maybe make an argument that in one sense, given a sufficiently hard problem, it will continue to try more and more complex algorithms and if one does solve it, it will eventually compute it; if none exists, it never will. In that narrow sense, the behavior seems the same as a TC system to me. ### Lack of organization Also problematic is the apparent utter unpredictability of the system. While we *may* be able to prove that all of these functions must be represented and computed *somewhere*, it seems unlikely that there's ever going to be any kind of map or organized order which would make these computations effectively accessible. For now, the only way (that I know of) to find a set of input cells and output cell that exhibit your goal behavior is to mostly brute force your way through all the computations in the whole structure, keeping an eye out for what you want. And again, it's unclear to me how disqualifying this is or not. A well-behaved TC system should have translator programs so it can bi-simulate other TC systems, but such a thing almost certainly can't exist for Rule 30; or to be more precise, it *could* exist, but it would be doing an amount of work (largely bookkeeping, redundant, unrelated) that most likely dwarfs whatever would be involved for the target computation itself, and I'm not sure that counts. But the structure *is* lousy with computational tidbits, and as proof of concept, it was trivial to find an early spot that functions as a half-adder. #### An emergent virtual machine? Okay, this one is probably more of a stretch, but since digital computers are essentially big finite state machines (or bounded linear automatons? whatever it is), and they fundamentally run on basic logic, such structures should also exist somewhere in the Rule 30 structure, a virtual computer perfectly set up to tackle whatever problem we're interested in. Conceivably there could even be a mechanism wherein memory is not limited and the process continues farther down as needed. Note that the Rule 30 structure is constantly expanding, increasing capacity for computation and with it some form of memory, and this unbounded expansion is one sense in which it seems more Turing-Machine-like after all. ### Bi-simulation This is maybe the same point as the last, but it deserves emphasis. The gold standard for proving something Turing complete is the ability for a known TC system and your system to be able to emulate each other perfectly. Any operation that one system can carry out, the other must be able to, even if it seems like a very different computational architecture. Clearly any TC system can emulate Rule 30 trivially. The question is whether Rule 30 can emulate, say, a Universal Turing Machine. If we were handed the specs for a UTM, we could not (for now) sit down and run it through a simple translator and run it in Rule 30. *However*, if every computation really is carried out, then if you run Rule 30 long enough, it is guaranteed to precisely mimic every last one of the UTM's states and instructions. Even if it involves an open-ended loop, there will be further computation downstream in Rule 30 that behaves as though it used such a loop. So if we know the simulation is there, perhaps scattered invisibly in the noise, but functional and real nonetheless, does that count or not? Particularly because we really _could_ build a translator program, in theory, even if it did use brute force and would be slower than sin. That seems like a mark in favor of TC-ness for Rule 30. ### Summing up I am interested in opinions on which of these points are persuasive or not, and overall, whether or not a case could be made for Rule 30 being Turing complete. If someone can point out where one or more required criteria for TC-ness absolutely are or are not met, or if someone can point me towards resources substantially addressing this question, that'd be great. [1]: https://en.wikipedia.org/wiki/Rule_30 [2]: https://en.wikipedia.org/wiki/Elementary_cellular_automaton [3]: https://i.stack.imgur.com/Ek17H.png Trevor Cappallo 2021-06-15T00:39:23Z https://community.wolfram.com/groups/-/m/t/2286938 According to the physical interpretation of the Wolfram model given by Wolfram and Gorard, the evolution of a multiway graph represents the evolution of a quantum system. The different paths that a system may take are to be summed together in some way similar to a Feynman path integral. From Gorard's [Some Quantum Mechanical Properties of the Wolfram Model][1]: > Such an interpretation brings forth strong connotations of the path integral formulation of quantum mechanics, in which the overall trajectory of a quantum system is taken to be described by a sum (or, more properly, a functional integral) over all possible trajectories, weighted by their respective amplitudes or > [A branchlike hypersurface] may be considered to be a linear superposition of the basis eigenstates of the multiway system My question is: **how do I calculate the complex amplitude/path weight of a particular state in a multiway system?** The simplest answer that comes to mind is: just sum the number of ways you can get to a particular state. This is what Gorard has apparently been doing in [his bulletin about the double-slit experiment][2]. The problem with this approach is that the number of ways you can get to a particular state is a natural number. The amplitudes will be positive integer (or rational if you normalize the state) numbers, while in quantum mechanics 1. amplitudes are in general complex numbers 1. amplitudes contain often irrational or transcendental numbers 1. amplitudes can also be negative. Destructive interference is not possible if the amplitudes will only add In fact, the interference pattern that Gorard obtains in the bulletin is only due to the peculiar choice of a sorting algorithm for the strings, as explained by Matt Kelly in a comment. The question remains. I couldn't find a satisfactory answer in Gorard's [paper][3] either: > these amplitudes are concretely specified by a sum of the incoming path weights for the associated vertex in themultiway graph, where these path weights are computed using the discrete multiway norm, as defined below Therefore he seems to be using the discrete multiway norm, that he defines later at p.27. Basically he defines a multidimensional lattice (with a time axis) and embeds the multiway graph in the lattice. Then he calculates the Minkowski norm between the points in the lattice. Unfortunately this norm is totally dependent on the particular choice of embedding (as discussed [in this post][4]), and therefore cannot be used to calculate amplitudes (which of course should not depend on the embedding) Some months ago I was discussing the issue of path weights with Pavlo Bulanchuk and José Manuel Rodríguez Caballero in [Quantum random walk][5], but we couldn't converge to a definite answer. It has been 5 months now. If anyone has found some insight on the issue it would be much appreciated. I think that understanding how to calculate path weights would be a critical step in the comprehension of quantum mechanics in the context of the Wolfram model. I would also be satisfied is someone could tell me: "No-one has figured this out yet", but this would mean that our present understanding of the model is way behind Wolfram's claims in the [One-Year Update][6]. [1]: https://www.wolframcloud.com/obj/wolframphysics/Documents/some-quantum-mechanical-properties-of-the-wolfram-model.pdf [2]: https://wolframphysics.org/bulletins/2020/08/a-short-note-on-the-double-slit-experiment-and-other-quantum-interference-effects-in-the-wolfram-model/ [3]: https://www.wolframcloud.com/obj/wolframphysics/Documents/some-quantum-mechanical-properties-of-the-wolfram-model.pdf [4]: https://community.wolfram.com/groups/-/m/t/2132822 [5]: https://community.wolfram.com/groups/-/m/t/2147321 [6]: https://writings.stephenwolfram.com/2021/04/the-wolfram-physics-project-a-one-year-update/ Ruggero Valli 2021-06-09T15:33:06Z https://community.wolfram.com/groups/-/m/t/2285459 Does anyone know of or is able to track down a 2-color, 1-D cellular automaton admitting an aperiodic initial configuration and a ruleset that is asymmetrical, and which has at least one (but not infinitely many) columns that are periodic or eventually periodic? I realize that sounds like a lot, so a word on motivation. It's still unproven whether or not the center column of Rule 30 ever becomes periodic. (It doesn't, but a proof is elusive.) This would be a generalized attack on the problem, showing that whole class doesn't exist. There are a couple of examples out there of rules that do have a central column which eventually or immediately cycles while other columns remain aperiodic (e.g. Rule 150), but every single one that I've seen has a symmetrical ruleset, meaning any symmetrical initial configuration can only reach states with the same axis of symmetry. To keep things reasonable, I limit us to 2 colors (read: cell states) since if we add a third, there are no doubt uninteresting solutions which paint a line of the third color down the center, and then ignore it while carrying on processing actively with the other two colors. That said, I *think* we can allow range to be arbitrarily large without a similar pitfall, so rulesets could draw from e.g. the 5 cells above them instead of the typical 3, although going too high might turn out similarly unhelpful. As additional clarification, such a CA would need to be on an open grid, not a torus. Periodic tiling in the initial row is fine so long as there's some finite, aperiodicity-seeding deviation from it at some point, e.g. the $0^\infty10^\infty$ classic start for Rule 30. In short, I'm wondering if anyone knows of *any* example where an asymmetrical ruleset could maintain a periodic column, or has any thoughts on this issue as to why such an arrangement is or is not plausible. My guess is that such examples do not exist or are very rare, partially based on having searched some and not found any, but I'd love to have someone else try their hand at finding one. And again, another way to state the main constraint is that there must be one or more periodic columns along with aperiodic columns occurring both somewhere left and somewhere right of the periodic column—one whole side devolving into periodicity is not what we're after. Trevor Cappallo 2021-06-08T14:23:27Z https://community.wolfram.com/groups/-/m/t/2284879 &[Wolfram Notebook][1] In the first step there are two arrows, because the rule (which is the identity) can be applied in two ways to the initial state. I don't understand why there are four arrows for every subsequent step. [1]: https://www.wolframcloud.com/obj/36166089-f617-4236-8466-6c88706c3dce Ruggero Valli 2021-06-08T09:46:41Z