https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Wolfram Fundamental Physics Project sorted by most viewed. https://community.wolfram.com/groups/-/m/t/587818 I would like to measure how complex various networks are. There are certainly many approaches, but I am interested in what we can do with tools available in the Wolfram Language. There are many discussions in the [New Kind of Science][1] book about networks. But since its publication Wolfram Language evolved significantly. Imagine we have something more general, like a mixed multi-edge graph: network[v_, e_] := Graph[RandomInteger[{1, v}, {e, 2}] /. {x_Integer, y_Integer} :> RandomChoice[{DirectedEdge[x, y], UndirectedEdge[x, y]}]] SeedRandom[131]; g = network[20, 60] ![enter image description here][2] Symmetries should affect complexity, and some general measure should detect that in, say, pure trees or symmetric and/or planar graphs: TreeGraph[RandomInteger[#] <-> # + 1 & /@ Range[0, 100], GraphLayout -> "RadialDrawing"] ![enter image description here][3] GraphData["IcosahedralGraph"] ![enter image description here][4] **Does anyone know or have any ideas for some complexity measure that we can quickly compute with Wolfram Language?** Perhaps [@Hector Zenil][at0], [@Todd Rowland][at1] or [@Marco Thiel][at2] could point to the right direction. Maybe there were some [Wolfram Science Summer School][5] projects about this. [at0]: http://community.wolfram.com/web/hectorz [at1]: http://community.wolfram.com/web/rowland [at2]: http://community.wolfram.com/web/mthiel [1]: https://www.wolframscience.com/nksonline/toc.html [2]: http://community.wolfram.com//c/portal/getImageAttachment?filename=q34t5wrhgsfdas.png&userId=11733 [3]: http://community.wolfram.com//c/portal/getImageAttachment?filename=ertq354thrwgdsf.png&userId=11733 [4]: http://community.wolfram.com//c/portal/getImageAttachment?filename=3q4t5htrwgdfavsdc.svg&userId=11733 [5]: https://www.wolframscience.com/summerschool Vitaliy Kaurov 2015-10-21T22:10:32Z https://community.wolfram.com/groups/-/m/t/1956061 If the universe is a deterministically evolving hypergraph - is the state of previous cycles preserved? This is another way of asking (in this model) if the past is real? Is the future real? or is this model an example of Presentism rather than a Block Universe? Barry Silverman 2020-04-26T23:37:06Z https://community.wolfram.com/groups/-/m/t/1945186 Improved illustrations of the interweaving grid-graphs: An example of how to achieve a graph from which a 2D space with Euclidean distance emerges at a large scale, and therefore rotational also symmetry: ![Euclidean Plane Graph][1] Analogous example for a 2D SpaceTime with Lorentz symmetry: ![Minkowski Plane Graph][2] I noticed that Wolfram said in a live-stream, that you need to have random-looking connections as opposed to a grid-like structure in order to achieve the Euclidean non-Manhattan distances. But I found that you could instead just use a very large number of interweaving grids that are connected to each other in such a way that they each represent different angles of rotation while maintaining the same scale, which can actually be achieved by simple local rules on graphs. The long range graph distances would approximate Euclidean distance as the number of grids A,B,C,D... is growing. Analogous constructions can be made for a Minkowski space time, where the different grids represent frames of reference of different boosts. A very large number of such interweaving grids (called A,B,C...) then analogously leads to approximate Lorentz distance on the large scale and hence also to Lorentz symmetry. They can be arranged as a tree of girds or as a grid of grids. Note that this leads to a density of nodes that could be an astronomical number per plank cube, while the length of the edges could just be the plank length itself. Such 2D spaces that already contain rotational and Lorentz symmetries could then be used as building blocks to generate a 4D space time, but this would be more complicated and I would not be surprised if quantum phenomena emerge from these complications. [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=IMG_2262.jpg&userId=1941950 [2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=IMG_2264.jpg&userId=1941950 Gabriel Leuenberger 2020-04-19T00:31:07Z https://community.wolfram.com/groups/-/m/t/982238 This is the Community Post presenting the 2016 Wolfram Summer School Project of József Konczer, a Hungarian PhD student of theoretical physics, assisted by Todd Rowland. The project notebook with the code is attached to this post. ## Fundamental theories in Physics ## The Holy Grail of theoretical physics would be a theory, which could describe all known phenomena in every situation. This would be called the Theory of Everything ([ToE][1]). In present times, all approaches to a ToE candidate—like String/M theory, Loop quantum gravity ([and many others][2])—incorporate quantum mechanics from the beginning. This approach served surely very useful effective theories like the Standard Model itself, however this approach does not help to find an underlying deterministic theory from which quantum effects would emerge, like Einstein dreamed. The standard argument for the unavoidability of quantum mechanics and uncertainty is the result of the [Bohr–Einstein debates][3] which was "won" by Bohr because the [EPR paradox][4] was tested by [measurements][5] (there is an ongoing test as well called the [Big Bell test][6]) and the result excludes local hidden variables. (The arguments can be found in details [here][7] and [here][8]). It has to be emphasized that in these arguments locality is a key assumption. ---------- One can ask, what kind of nonlocal theory can be constructed, which still have predictive power, and is not based of conspiracy of Nature? There are only a few researchers who post this question openly, one is them is Gerard ’t Hooft who published recently a [book][9] based on collected [papers][10]. His approach is conservative (from main stream point of view), and mainly suggests, that if one quantize time, then in same basis the unitary time evolution considered in quantum mechanics become a permutation operator between special basis elements or "beable states". However not every time evolution has this property and typically the interacting theories fail to fulfill the requirements. A more bold, however much less understood theory (or framework) is what Stephen Wolfram described in [NKS][11]. The brief summary of his ideas can be found in this [blog post][12]. The main idea here, is to find a simple data structure, for instance a sparse graph, a simple discrete dynamics governed by a replacement rule, an interpretation for this cellular automaton (CA) and then investigate if we can observe similar phenomena what we see in our Universe. ## Hints pointing toward the CA description ## This is a speculative and highly subjective argumentation, however I think this blog post is an appropriate place to articulate my motives and do not stick to the objective style of research papers. So first of all without going too deep into metaphysics I don't want to state things about Nature it self, I only talk about our description of it. The first successful and highly useful description of Nature was Newtons description, which heavily used the idea of continuity of space and time. This idea proved to be useful in description of solids, liquids and gases as well. However some ideas became so useful and popular, that we forgot that all of them is only our description and not Nature its self. Quantum effects and effects related to relativity reminded us, that under non standard circumstances old descriptions can fail. As I see, quantum effects have two message for us. The first is that quantities could be and should be described by discrete variables, and secondly that under a certain level systems can not be observed without disturbance. If we take into account that space and time even as we observe them are influenced by these quantized quantities, it is straightforward to deduce, that space and time should be quantized as well. Before these findings in physics probability theory was developed. First it was used to analyze gambling situations where one don't know every information about the system. From this point of view it is clearly a strategy to manage our ignorance toward some details in a deterministic situation. However after some point physicists started to use probabilities as they were part of the phenomena, and not only our clever way to make inference from systems where we do not know every detail. Many physicists—including myself—where educated in the spirit of frequentist [interpretation of probability theory][13], which is useful in some cases but as I think, prevents some questions to ask. I think this promotion of probability to an objective property contributed to the interpretation of quantum mechanics as well. As Jayns wrote in his [book][14]: > In current quantum theory, probabilities express our own ignorance due to our failure to search for the real causes of physical phenomena; and, worse, our failure even to think seriously about the problem. This ignorance may be unavoidable in practice, but in our present state of knowledge we do not know whether it is unavoidable in principle; the ‘central dogma’ simply asserts this, and draws the conclusion that belief in causes, and searching for them, is philosophically naive. If everybody accepted this and abided by it, no further advances in understanding of physical law would ever be made; indeed, no such advance has been made since the 1927 Solvay Congress in which this mentality became solidified into physics. But it seems to us that this attitude places a premium on stupidity; to lack the ingenuity to think of a rational physical explanation is to support the supernatural view. However even if one thinks, that theories incorporating quantum mechanics are "only" effective theories, probably we can get intuitions from them. There is a [recent result][15] from [AdS/CFT][16] correspondence as an example for the [EPR=ER][17] conjecture. And a connecting [paper][18] of Leonard Susskind, concluding that: > What all of this suggests to me, and what I want to suggest to you, is that quantum mechanics and gravity are far more tightly related than we (or at least I) had ever imagined. The essential nonlocalities of quantum mechanics (the need for instantaneous communication in order to classically simulate entanglement) parallels the nonlocal potentialities of general relativity: ER=EPR. The cited papers state, that spacetime structure can be understood as a net of entanglements, however maybe the statement can be reversed, and say that the phenomena of entanglement can be described by a nonlocal spacetime structure. Among the mentioned hints, the existing theoretical constructions can help to find an appropriate interpretations as well. For example it can happen, that to describe our seemingly 3 dimensional space one has to describe space with higher effective dimensionality and interpret the entangled parts not just as connected regions, but as global structures in the extra dimensions. After taking hints from existing theoretical constructions one can investigate, what kind of phenomena can appear in simple CA-s which mimic some parts of Nature. Perhaps the most well known CA is Conway's [Game of Life][19] this is a 2D Cellular automaton where localized objects (called [spaceships][20] or gliders) can propagate, and can interact with each other. This behavior can remind us to particles, however the built in rectangular structure is reflected on the properties of spaceships, and there are no nonlocal connections between these object because of the locality of the rule. Both problems can be solved, if one tries to construct a CA without built in topology. (This construction will be described in detail.) Another nice feature of special CA-s called substitution systems, is that for a structure living in the automaton, can not observe the absolute number of steps, or other structures beside him, only the causal net of implemented changes can be recognized from inside. This feature unites the relative space and time for observers or structures inside the system. It can remind us to causal network description of General Relativity. A third hint from CA point of view is the typical appearance of complex behavior, which can lead to an effective probabilistic description of the system with a higher symmetry what the framework originally allowed. (For example CA description of flows) From disorder new effective order can emerge possible with higher symmetry. The conjectured computational irreducibility of CA would replace the promised "free will" possibility of quantum mechanics with a different but in some sense similar concept. In this framework the faith of the Universe would be determined, but even an observer outside the system—God if one wishes—could not know the consequences only by letting the simulation run up to the desired point. Furthermore a multiway CA dynamics is compatible with the many world interpretation of quantum mechanics, with the advantage, that the splitting points of histories are not observer dependent. In this framework the overall dynamics is deterministic, however structures living always on one branch of the evolution will witness an unavoidable true random behavior from the inside point of view. ## Nature and our understanding of it ## Of course it would be an arrogant attitude to force Nature to fulfill our philosophical expectations, however one can imagine how our description of it can change during time. There are several situations what one can imagine: - There is a deterministic description which is valid in any situation (which can appear in our Universe) - This can be totally discreet - Or it can be continuous partially or in whole - It can be, that after some point a truly random (or at least appearing to us) mechanism will appear, which can not be unfolded - Or it can happen that construction of laws to describe Nature will never come to an end, and our understanding of reality will based on infinite set of possibly deterministic rules. - And of course it can happen that something unexpected will turn out. Without favoring any of the listed cases above, my main point is that the very first situation, namely that our universe can be described as a deterministic discrete system is not totally excluded. And the most natural way to understand it can be a CA description. ## CA description candidate for our Universe ## To have a CA description, one has to choose a data structure, a dynamics and an interpretation. (It has to be pointed out, that any CA can be simulated on another Turing complete CA with different interpretation of the states. Because of that any CA description is highly non unique. However one can try to choose a description which has the "simplest" interpretation.) For a fundamental CA description one can choose simple graphs as data structure. This seems as a natural choice, because of its simplicity and because of its non fixed topology. To have a chance to describe deterministic dynamics on this data structure, we even restrict the degree of nodes on the graph. One can try to find the threshold of complexity of the CA, and it seams that cubic graphs can already produce complicated enough structures. So one can set the data structure to a simple cubic graph. The next step is to define an appropriate dynamics on this data structure. A natural approach is to introduce subgraph replacement rules, which means the following: If one finds a given subgraph pattern $H_1$ in the present graph $G$, then replace it with a compatible new graph $H_2$. It sounds simple, however there are many details which have to be fulfilled to get a dynamics with desired properties. I mention here two properties of the patterns, which seems to be essential to get a substitution systems which generates a complex behavior and appears completely deterministic from the inside without the specification of the order of replacements in the system. The first one is a **non overlapping property** of the pattern graph(s) $ H_1 $. This means, that $H_1$ has a special structure such as there is no cubic graph $G$, where two subgraphs can be found which are isomorphic to $H_1$ and have nonzero intersection. The following rule does not fulfill this requirement because there is a cubic graph, where two intersecting copies of $H_1$ can be found ![overlapping rule][21] ![Interesting patterns][22] The second requirement is **non triviality**, which gives a constraint for $H_2$. In this case we wish to have $H_2$, that there exists a cubic graph $G$, which contains a subgraph $H_1$, where after the replacement $H_1 \rightarrow H_2$ there can be found a new pattern $H_1$, which intersects with $H_2$ but has parts outside $H_2$ as well. (Without this property only self similar or frozen (where there is no more pattern which can be changed) graphs can be generated from finite initial graphs.) The pictures show visually the requirement: ![enter image description here][23] ![enter image description here][24] After setting some rule, which fulfill these requirements, we have to find an initial graph, apply the rule many times and find an interpretation for the result. It has to be pointed out, that the actual graph structure at a given step can not be observed from an inside point of view. What an inner observer, or a structure can explore is the causal structure which is generated by the replacements. (For details see [NKS chapter 9, section 13][25] ) This is similar to the [causal set program][26]. So a natural way of interpretation of the emerging causal net is that it is a discretization of some kind of spacetime. And local propagating disturbances relative to the overall average structure are particle like excitations, which can have nonlocal connections relative to the average large scale structure. However from AdS/CFT insights it can happen, that we have to interpret particles for example as global structures in a higher dimensional bulk spacetime, which have ends on a boundary-like smaller dimensional surface. ## My contribution to the project ## During the 3 weeks of 2016 Wolfram summer school I set a framework where the steps of a substitution are precisely defined, and in which the substitutions can be effectively performed even for relatively big graphs. Furthermore I tested a numerical approach to measure the effective dimensionality of the emergent graph structure after sufficiently many steps. Unfortunately I could not test this framework with rules which could give complex, deterministic behavior, so I could benchmark this machinery on a simple, point to triangle rule, which gives a fractal-like structure. If we interpret this graph as space, then this simple dynamics results a $D=\log(3)/\log(2)=1.58$ dimensional fractal space. Here is a graph of the generated fractal Universe after 100 steps, started from a tetrahedron: ![Generated fractal Universe after 100 steps, started from a tetrahedron][27] And the neighborhood structure of this space: ![Local structure in the fractal Universe][28] ## Further directions ## This project to find deterministic CA description for our Universe is in its infant stage. The framework is more or less set, but it needs tremendous work to investigate possible dynamics and analyze the results of simulations. An outline of a huge project would be the following: - List the possible rules, which fulfill the non overlapping and non trivial conditions - Investigate their long term behavior starting from simple initial graphs - Find quantities and a method of their measurement which can be determined from generated causal graphs - Find fixed points of the dynamics which preserve long scale dimensionality and possibly other quantities - List and investigate local disturbances near these fixed points - After setting an interpretation analyze the particle-like structures (gliders of this dynamics) - Develop an effective field theory which can describe an effective behavior of the system near to the fixed points - Match these field theories with the Standard Model of particle physics - Find out new predictions of the derived effective field theories, which can be tested by measurements ## Conclusion ## In my project I could set a framework and show a trivial example for a deterministic graph evolution model. During the summer school I was not fortunate enough to find dynamics which produce complex behavior, however to find an appropriate rule seems reachable in the near future. Hopefully a dynamics producing complex topology would be interesting enough to inspire much more people and after some point a serious investigation of the field could be started. I think personally, that proving or even disproving that this framework to describe Nature can be worked out is an extremely interesting challenge and deserves further theoretical research. In the end I would like to thank my mentor Todd Rowland, and the whole Wolfram summer school team for the organization and I really hope that there will be a continuation of this project. Last but not least I thank for all the summer school participants for great discussions and a lifelong experience! ![enter image description here][29] ---------- ## Further comments ## I try to collect here some useful comments of my friends and collegues, who kindly read my post, and responded in person: There is a concept named [Digital physics][30] which has a much longer history what I suggested, and probably the earliest pioneer of the field was Konrad Zuse. Fortunatelly his thesis—[Calculating Space][31] or “Rechnender Raum”—is now translated into English and has a modern, LaTeX typesetting. Beside NKS there is another relevant book, which can serve as an extended list of references and valuable material in its own, written by Andrew Ilachinski with the title [Cellular Automata A Discrete Universe][32]. There is an ongoing "mini revolution" in the description of AdS/CFT based on [Tensor Networks][33]. The original paper on the topic can be found [here][34]. [1]: https://en.wikipedia.org/wiki/Theory_of_everything [2]: https://www.quantamagazine.org/20150803-physics-theories-map/ [3]: https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates [4]: https://en.wikipedia.org/wiki/EPR_paradox [5]: https://arxiv.org/abs/1508.05949 [6]: http://thebigbelltest.org/#/science?l=EN [7]: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521818629 [8]: http://www.springer.com/in/book/9783662137352 [9]: http://www.springer.com/us/book/9783319412849 [10]: https://arxiv.org/abs/1405.1548 [11]: http://www.wolframscience.com/ [12]: http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really/ [13]: https://plato.stanford.edu/entries/probability-interpret/ [14]: http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521592712 [15]: http://www.nature.com/news/the-quantum-source-of-space-time-1.18797 [16]: https://en.wikipedia.org/wiki/AdS/CFT_correspondence [17]: https://en.wikipedia.org/wiki/ER=EPR [18]: https://arxiv.org/abs/1604.02589 [19]: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life [20]: http://conwaylife.com/wiki/Category:Spaceships [21]: http://community.wolfram.com//c/portal/getImageAttachment?filename=H1H2.png&userId=981213 [22]: http://community.wolfram.com//c/portal/getImageAttachment?filename=GH1H1.png&userId=981213 [23]: http://community.wolfram.com//c/portal/getImageAttachment?filename=H1H2_2.png&userId=981213 [24]: http://community.wolfram.com//c/portal/getImageAttachment?filename=GG.png&userId=981213 [25]: http://www.wolframscience.com/nksonline/section-9.13 [26]: https://en.wikipedia.org/wiki/Causal_sets [27]: http://community.wolfram.com//c/portal/getImageAttachment?filename=PresentationTemplate_KJ_2.png&userId=981213 [28]: http://community.wolfram.com//c/portal/getImageAttachment?filename=PresentationTemplate_KJ_3.png&userId=981213 [29]: http://community.wolfram.com//c/portal/getImageAttachment?filename=vladstudio_higgs_boson_fluo_800x600_signed.jpg&userId=981213 [30]: https://en.wikipedia.org/wiki/Digital_physics [31]: http://www.mathrix.org/zenil/ZuseCalculatingSpace-GermanZenil.pdf [32]: http://www.worldscientific.com/worldscibooks/10.1142/4702 [33]: https://arxiv.org/abs/1306.2164 [34]: https://arxiv.org/abs/0905.1317 Jozsef Konczer 2016-12-16T07:22:29Z https://community.wolfram.com/groups/-/m/t/1950834 In collaboration with Johan-Tobias Schäg, we were able to create a rule producing an ordinary 2D-grid (https://community.wolfram.com/groups/-/m/t/1946413). Wolfram himself also mentions grids several times (see for example: https://www.wolframphysics.org/technical-introduction/limiting-behavior-and-emergent-geometry/recognizable-geometry/). It is often assumed in several of the texts I read on this subject (the last link is an example) that in the limit of making grids finer and finer we obtain the well known 2-dimensional continuous space. This is not correct, however. In fact, it seems quite complicated to come up with a graph structure that resembles classical 2D-space even remotely (the same is true for higher dimensions as well). It is important to solve this problem since any model resembling physics should look like ordinary continuous space on large scales. In order to illustrate this, I will consider the simplest case known to us all; the regular 2D-grid (see the figure below). I have drawn a piece of a 2D-grid, as well as four points on the grid labeled A, B, C, and D. ![enter image description here][1] In the texts, I have read so far, there is no mention of how we think about *distances* in graphs. However, there are indirect hints. For example, when the subject of dimension and curvature is discussed, the distance $r$ from a point plays a crucial role. In these cases, the distance between two vertices is indirectly defined as the *least* number of (hyper)edges you would have to cross in order to get from one point to the other. This is a very logical definition, and I will keep using it here. Now keep in mind, that the graphs we draw do *not* illustrate these distances correctly. They are only representations of the graph. For example, take another look at the picture of the grid above. The distance between A and B is 1 (you could call one edge for the fundamental length unit, or you could simply omit units for simplicity) and the distance between A and D is 5. The distance between B and C is also 1. However, as the graph is drawn on this picture, the distance between A and C should be $\sqrt{2}$, while the true answer is that the distance is 2! In fact, you could never create graphs with non-integer distances. Okay, this is weird, but not problematic yet. After all, the graph should only look like ordinary space on *large* scales, and my example here is on the absolutely smallest scale possible. However, even on large scales, the answer is still the same. Imagine a huge grid with point A. Choose a direction and go $n$ steps in only that direction (left, right, up, and down are the possibilities on a grid). Now denote the point you end up at by B. There is no shorter path between the two points, which means that the distance between A and B is $n$. Now choose a new direction away from B, which is not the same (or opposite) as before. If you chose to go to the right away from A, you could choose to go upward now, for example. Go another $n$ steps up and label this point C. We are now in the same case as before, but by varying $n$ we can vary the scale of the setup as much as we want. What is the distance between A and C? One path to take is to go from A to B and then from B to C as above. In fact, there is no shorter path than this either! So the distance is $2n$. In the usual 2D-space, we expect the *ratio* between the distance from A til B (denoted $d(A,B)$) and from A til C to fulfill the following identity: $$\frac{d(A,C)}{d(A,B)}=\sqrt{2}$$ A non-integer value like this is no problem anymore (irrational numbers are, however, but never mind that for now). What is the case for our graph? Well, that is easily calculated: $$\frac{d(A,C)}{d(A,B)}=\frac{2n}{n}=2$$ The result is 2 no matter how huge $n$ is! You can make the grid as fine as you ever want it, but you will never get to the usual 2D-space. You could also say that the Pythagorean Theorem doesn't apply, which we know it has to in any physical space. The same problem arises for any regular grid. There are two main points up until now: 1) do *not* believe in the distances drawn on a graph (even simple, regular ones), and 2) just because you get an infinitely fine grid, you do *not* automatically get continuous space. All is not lost, however. Up until now, I have only mentioned regular grids. It is, however, not surprising if nature is not regular on the smallest of scales. In fact, I would be very surprised if that was really the case. Instead, a graph that appears 2-dimensional on large scales could very well look quite chaotic on the small scales. The vertices and edges could have very complicated and seemingly random connections (I have heard rumors of Wolfram mentioning this, but I am not sure). This could solve the problem, but the natural question then is this: **How do we create a grid which obeys the Pythagorean Theorem on large scales?** Other natural questions to ask are: - **How do we deal with angles on large scales? How and when are they defined?** - **What happens to the irrational numbers that arise from the Pythagorean Theorem? Are they approximations?** - **Can we create a large scale isotropic graph?** [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=GridProblem.png&userId=1945692 Malthe Andersen 2020-04-22T15:01:20Z https://community.wolfram.com/groups/-/m/t/2574806 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=SierpAttatched.PNG&userId=1982950 [2]: https://www.wolframcloud.com/obj/ef3c1941-c0bf-43df-9083-66d423e09ce3 Simon Fischer 2022-07-19T20:20:04Z https://community.wolfram.com/groups/-/m/t/2027996 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=9678wormholefpost.png&userId=1919403 [2]: https://www.wolframcloud.com/obj/9d6c5a25-c87c-4901-b65e-40af5764f610 Tobias Canavesi 2020-07-14T14:16:12Z https://community.wolfram.com/groups/-/m/t/2034502 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/53def694-c172-47b6-9da2-b027f62b53b8 Macy Maurer Levin 2020-07-15T20:17:53Z https://community.wolfram.com/groups/-/m/t/2029731 ![enter image description here][1] &[Project Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=BH_Cover.png&userId=2028772 [2]: https://www.wolframcloud.com/obj/2301da72-c139-440d-a3ba-56b4c0a6acff Carlos Rafael Garduno Acolt 2020-07-14T16:59:38Z https://community.wolfram.com/groups/-/m/t/2029759 ![A branchial graph][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=LeadingImage.jpg&userId=1981340 [2]: https://www.wolframcloud.com/obj/d8332be9-4fb5-4966-8c19-8baf255f11ec Marc Sperzel 2020-07-14T17:07:34Z https://community.wolfram.com/groups/-/m/t/1961439 I'm trying to grasp the essence of what the Wolfram Physics Project is saying about the universe. In Jonathan's paper, "Some Relativistic and Gravitational Properties of the Wolfram Model" he states in paragraph 2.1, "The essential idea here is to model space as a large collection of discrete points..." To help me grasp this I need to compare it to my understanding of what other theories say are fundamental about the universe: - Quantum Field theory says that "fields" are fundamental to the universe and particles are "excited states" of those fields. These fields permeate all of "space" (for whatever that is and for how ever many "dimensions" it has). - String theory says that what we originally thought of as "point-like particles" are actually one-dimensional strings that propagate through "space" (for whatever that is and for how ever many "dimensions" it has). How these strings vibrate is what determines their properties and how they interact with other strings. - M-Theory extends the one-dimensional strings to include larger dimensional objects. So, what I take away from Jonathan's statement about how to model space is: "Space" "is" a collection of points. These points are fundamental. I'll call them "Space Points" (that's a terrible term). Space Points aren't in space. They are space. If Space Points are in anything, it's not space; it would be something else entirely different. There is some underlying rule which governs how Space Points interact with each other. On scales larger than the scale of the interactions between space points emerge behaviors that look like dimensions, times, distances, speeds, fields, forces, particles, probabilities... everything. Is this what the Wolfram Physics project is trying to say? Spenser Spam 2020-05-01T18:25:01Z https://community.wolfram.com/groups/-/m/t/2965206 ![Multiway graph][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Untitled.png&userId=2964459 [2]: https://www.wolframcloud.com/obj/b6ad5a15-e8d2-4359-ae89-b72d3e3d06d4 Eric Archerman 2023-07-13T23:11:59Z https://community.wolfram.com/groups/-/m/t/1729148 ![enter image description here][1] Mentor: Jonathan Gorard [GitHub][2] We examined confluent set substitution systems using [SetReplace][3] package. We specifically obtained three results. First, we looked into applying critical pair completion idea from automated theorem proving to get quantum behavior. We did not explicitly reproduce any quantum effects, however, we found several possible directions to explore. Second, we emulated elementary cellular automata with non-overlapping set substitution systems, which demonstrates their universality. # Critical Pair Completion ## Confluence and Relativity In rewrite systems, the reference frame is determined by the order of rewrite events [\[NKS, 516\]][4]. In general, different order of rewrite events will yield different states of the system (i.e., rewrite events do not commute), which, at least if one only considers a single branch of the multiway system, does not reproduce special relativity. To get around this issue, one idea is to only consider confluent rewrite systems, i.e., rewrite systems in which every divergent pair of rewrites (critical pair) can be converged to a single branch. Furthermore, due to Newman's lemma [\[Newman, 1942\]][5] local confluence is equivalent to global confluence, i.e., the possibility of converging a critical pair immediately is both sufficient and necessary for the global confluence (and therefore relativistic-ness) of the system. One approach that is explored more in the next section is to begin with a rule that is locally confluent. However, in this section we explore a different approach based on real-time critical pair completion. ## Critical Pair Completion for Strings Critical pair completion is an idea from automated theorem proving, where one starts evolving an arbitrary rewrite system as usual, but as soon as a critical pair is encountered, one adds a new bi-directional rule between the critical pair outputs. For the sake of demonstration, consider the following non-confluent string rewrite system: In[] := StringReplaceList["ABA", {"AB" -> "X", "BA" -> "Y"}] Out[] = {"XA", "AY"} Note the critical pair arises because there are two ways to make a substitution, and once made, the system terminates. One can however add new rules to deal with this: {"XA" -> "AY", "AY" -> "XA"} Now, the multiway network of this system is confluent, as there is a way to get from any final state to any other state In[] := NestGraph[ StringReplaceList[#, {... (*original rules*), ... (*new rules*)}] &, "ABA", 2, ...] ![enter image description here][6] ## Critical Pair Completion for Networks A similar approach can be used with networks. Consider for instance a particle represented as a single-vertex edge moving on a path graph. In[] := {{v0}, {v0, v1}} -> {{v1}, {v0, v1}} // ... (* HypergraphPlot *) ![enter image description here][7] In[] := SetSubstitutionSystem[... (* rule *), {{v0}, {v0, v1}, {v1, v2}, {v2, v3}, {v3, v4}}, 4] // ... (* HypergraphPlot *) ![enter image description here][8] This system is confluent already, in fact its multiway system is In[] := multiwayGraph[... (* rule *), ... (* path graph *), 4, ...] ![enter image description here][9] But what if we have a branching in the graph? What if we start with the initial condition In[] := {{v0}, {v0, v1}, {v1, v2}, {v2, v3}, {v3, v4}, {v2, v5}, {v5, v6}} // ... (* HypergraphPlot *) ![enter image description here][10] We then get a multiway system with two branches In[] := multiwayGraph[... (* rule *), ... (* branching graph *), 4, ...] ![enter image description here][11] If we now perform a critical pair completion, we get an extra rule to move the particle back and force between branches In[] := branchCollapseRules[ multiwayGraph[ ...], { ... (* rule *)}] // ... (* HypergraphPlot *) ![enter image description here][12] and the multiway system becomes In[] := multiwayGraph[ branchCollapseRules[ ...], ... (* branching graph *), 4, ...] ![enter image description here][13] Note, a single collapse is insufficient to produce a confluent system, we need to perform a collapse again. After the next iteration we get In[] := Nest[branchCollapseRules[ multiwayGraph[#, ... (* branching graph *), 4], #] &, ... (* original rule *), 2] // ... (* HypergraphPlot *) ![enter image description here][14] with the multiway system In[] := multiwayGraph[ Nest[ ... (* branch collapse rules *)], ... (* branching graph *), 5, ...] ![enter image description here][15] Note an interesting point that the rules created at this step allow one to go "back in time". The loop created by this does not constitute a problem as it is not an actual time loop (in a sense of the causal network), but an evaluation loop, which the observer existing in any of the states of the system cannot see. If we keep collapsing critical pairs until the fixed point, we will get the following rules In[] := FixedPoint[ branchCollapseRules[ multiwayGraph[#, ... (* branching graph *), 4], #] &, ... (* original rule *)] // ... (* HypergraphPlot *) ![enter image description here][16] and the multiway system that is a complete graph In[] := multiwayGraph[ FixedPoint[ ... (* branch collapse rules *)], ... (* branching graph *), 5, ...] ![enter image description here][17] Note the multiway system we get after the first branching point is a complete graph. That is in fact a general feature. At first sight that seems like an issue, because the multiway graph loses all structure, therefore we don't gain anything by performing critical pair completion. ## "Quantum" States However, one has to realize that the rules used for completion are slightly more general than necessary to collapse the branches, and as such they might produce new states that did not exist in the system with original rules. To see this happen, consider a graph with a branch that is not accessible in the original (classical) system In[] := {{v0}, {v0, v1}, {v1, v2}, {v2, v3}, {v3, v4}, {v2, v5}, {v5, v6}, {v7, v2}, {v7, v8}, {v8, v9}} // ... (* HypergraphPlot *) ![enter image description here][18] In this system, the particle can never be at v7, v8 or v9 because they are not accessible by following directed edges from v0. However, after a single critical pair completion we get the multiway system that "leaks" into the forbidden branch In[] := With[{multiway = multiwayGraph[ Nest[ ... (* branch collapse rules *)], ... (* graph with inaccessible path *), 5, ...]}, HighlightGraph[multiway, Complement[VertexList[multiway], VertexList[multiwayGraph[ ... (* original rules *)]]]]] ![enter image description here][19] This effect exhibits some similarity to quantum tunneling, in a sense that we can get classical-looking states which are nevertheless not accessible from the original system. However, more investigation of more realistic systems is necessary to, for example, compare probabilities (which are not even clear how to compute in this model) of tunneling to what we expect from quantum mechanics. ## Other Ideas A notable feature of the systems considered above is that multiway systems with original rules might significantly diverge. If one thinks of the original systems is classical, it would be plausible, for example, for the Earth to exist and not exist on different branches. Which essentially implies many-worlds interpretation of Quantum Physics. In other words, observer's memory is different on different branches. Another possibility is that all branching disappears by the observer's scale, and the observer essentially sees the course-grained version of the multiway system. In the description of "tunneling" above, we considered the first possibility. In what follows, we will only examine non-overlapping systems, which is the simplest case of the second possibility. # Simulating Cellular Automata ## Problem Definition To begin our study of non-overlapping systems, we will demonstrate their universality by simulating rule 110 elementary cellular automaton (CA) with it. Rule 110 if started with a single black cell produces a collection of interacting structures In[] := ArrayPlot[CellularAutomaton[110, {{1}, 0}, 500], ...] ![enter image description here][20] One can in fact use these interacting structures to demonstrate its universality [\[NKS, 675\]][21]. We will use this fact to in turn demonstrate the universality of non-overlapping set substitution systems. ## Finite Grid Cells Representation We begin by simulating a CA with a finite grid. To achieve that we begin with a schematic representation of a CA cell caBlock[id_String, neighborIDs_ : {_String, _String}, color_Integer] := { {cellCenter[id], "nextStepLeftNeighborInput", nextStepLeftNeighborInput[id]}, {cellCenter[id], "nextStepRightNeighborInput", nextStepRightNeighborInput[id]}, {cellCenter[id], "nextStep", nextStepCenter[id]}, {cellCenter[id], "inputFromLeftNeighbor", rightNeighborInput[neighborIDs[[1]]]}, {cellCenter[id], "inputFromRightNeighbor", leftNeighborInput[neighborIDs[[2]]]}, {cellCenter[id], color}, {leftNeighborInput[id], "nextStep", nextStepLeftNeighborInput[id]}, {leftNeighborInput[id], color}, {rightNeighborInput[id], "nextStep", nextStepRightNeighborInput[id]}, {rightNeighborInput[id], color} } // Map[ToString, #, {2}] & For a finite grid with 4 cells ![enter image description here][22] we get the following structure, where a subgraph corresponding to the first cell is highlighted in red, and its connections to the neighboring cells are in yellow. In[] := Join[caBlock["0", {"3", "1"}, 0], caBlock["1", {"0", "2"}, 1], caBlock["2", {"1", "3"}, 0], caBlock["3", {"2", "0"}, 1]] // ... (* HypergraphPlot *) ![enter image description here][23] Note there are 6 vertices used to represent each cell: there are three colored ones for the current step. Out of these three, one is used to determine the color at the next step for the current cell, and two others are used by neighbors. Three other cells correspond to the cell at the next time step, and are not assigned any color. ## Color Updating Rules We can then make a rule that would take as an input ![enter image description here][24] Note there is no overlap between different rule applications (some vertices are not highlighted because we are only concerned with the overlap of edges here, and as vertices at the next step do not carry any information, we do not need to associate any additional edges to them apart from highlighted references). ![enter image description here][25] These rule inputs have 5 tentacles going from the main (middle) cell vertex: three point to the next time-step representation of the same cell, and at the rule application, they need to be assigned a particular color. The other two point to the next step of the neighbors. The vertices these tentacles point to would not be updated, however, the references for the next step would be moved to the next steps of the neighbors (also blocking subsequent updates from happening until neighbors are updated as there is no color initially assigned to the next steps of the neighbor cells). We can schematically write the rules as caRules[rule_Integer] := With[{oldLeft = #[[1]], oldMiddle = #[[2]], oldRight = #[[3]], newColor = #[[4]]}, { {cellCenter_, "inputFromLeftNeighbor", leftColorReference_}, {leftColorReference_, "nextStep", nextStepLeftColorReference_}, {cellCenter_, "inputFromRightNeighbor", rightColorReference_}, {rightColorReference_, "nextStep", nextStepRightColorReference_}, {cellCenter_, "nextStepLeftNeighborInput", nextStepLeftNeighborInput_}, {cellCenter_, "nextStepRightNeighborInput", nextStepRightNeighborInput_}, {cellCenter_, "nextStep", nextStepCenter_}, {cellCenter_, #[[2]]}, {leftColorReference_, #[[ 1]]}, {rightColorReference_, #[[3]]} } :> Module[{nextNextStepLeftNeighborInput, nextNextStepRightNeighborInput, nextNextStepCellCenter}, { {nextStepCenter, "inputFromLeftNeighbor", nextStepLeftColorReference}, {nextStepCenter, "inputFromRightNeighbor", nextStepRightColorReference}, {nextStepCenter, "nextStep", nextNextStepCellCenter}, {nextStepCenter, "nextStepLeftNeighborInput", nextNextStepLeftNeighborInput}, {nextStepCenter, "nextStepRightNeighborInput", nextNextStepRightNeighborInput}, {nextStepCenter, newColor}, {nextStepLeftNeighborInput, newColor}, {nextStepRightNeighborInput, newColor}, {nextStepLeftNeighborInput, "nextStep", nextNextStepLeftNeighborInput}, {nextStepRightNeighborInput, "nextStep", nextNextStepRightNeighborInput} }]] & /@ Map[ToString, 1 - Flatten /@ Thread[{IntegerDigits[Range[0, 7], 2, 3], 1 - IntegerDigits[rule, 2, 8]}], {2}]; For an example where both neighbors are white, and the center is black, we get this step where old edges are in gray, and the new edges are in red In[] := With[{states = SetReplace[Join[ ... (* CA cell blocks *)], caRules[110], #] /. { ...} & /@ {0, 1}}, HighlightGraph[ Graph[ ...] &[ Union[Complement[states[[1]], states[[2]]], Complement[states[[2]], states[[1]]]]], {Style[ Complement[states[[1]], states[[2]]], LightGray], Style[Complement[states[[2]], states[[1]]], Red]}]] ![enter image description here][26] The entire graph after this step looks like this (with new edges highlighted in red), where one can see that the evolution of the same cell is blocked as there is no input from the neighbors In[] := With[{states = SetReplace[Join[ ... (* CA cell blocks *)], caRules[110], #] /. { ...} & /@ {0, 1}}, HighlightGraph[Graph[ ...] &[states[[2]]], Complement[states[[2]], states[[1]]]]] ![enter image description here][27] We can confirm that this system does not overlap after running it for 10 steps In[] := SetSubstitutionSystem[caRules[110], Join[ ... (* CA cell blocks *)], 10, "CheckConfluence" -> True]["ConfluentQ"] Out[] = Missing["Unknown"] *Missing is returned because the code only checks for overlaps between edges at the current step, but it is possible for overlaps to occur between space-like pair of events even if some of the vertices for one of them are already deleted. For this particular system, it is easy to see that this is not occurring, hence it is in fact confluent.* ## Localization In the above, we simulated labeled edges and vertices by creating global vertices with names such as "inputFromLeftNeighbor", "nextStep", "1", and representing labeled edges by hyperedges passing through these global vertices, i.e., Labeled[x -> y, "nextStep"] <> {x, "nextStep", y} However, it would be interesting to see if we can make the rules local, i.e., not involving any global vertices, and only depending on the vertices nearby the application site. This is indeed straightforward to do by making use of hyperedges with varying numbers of vertices, i.e., caLocalization = { {x_, "inputFromLeftNeighbor", y_} :> {x, x, y}, {x_, "inputFromRightNeighbor", y_} :> {x, y, x}, {x_, "nextStepLeftNeighborInput", y_} :> {x, x, x, y}, {x_, "nextStepRightNeighborInput", y_} :> {x, x, y, x}, {x_, "0"} :> {x}, {x_, "1"} :> {x, x}, {x_, "nextStep", y_} :> {x, y, y} }; Now, we can localize both the rules and the initial condition to get the local evolution In[] := SetSubstitutionSystem[caRules[110] /. caLocalization, Join[ ... (* CA cell blocks *)] /. caLocalization, 3] /@ Range[0, 3] // GraphicsGrid[ Partition[HypergraphPlot[#, ImageSize -> Scaled[1]] & /@ #, 2], ImageSize -> Scaled[1]] & ![enter image description here][28] To decode it, one can look at the number of loops in the cell centers: 4 loops means 0, 5 loops means 1, so decoding evolution from above, we get `{1,0,1,0}, {1,1,1,1}, {0,0,0,0}, {0,0,0,0}`, which matches the evolution of the actual CA: In[] := ArrayPlot[CellularAutomaton[110, {0, 1, 0, 1}, 3], ...] ![enter image description here][29] We can demonstrate that this system is confluent as well In[] := SetSubstitutionSystem[caRules[110] /. caLocalization, Join[ ... (* CA cell blocks *)] /. caLocalization, 10, "CheckConfluence" -> True]["ConfluentQ"] Out[] = Missing["Unknown"] We can trivially extend this to any number of cells. For 10 cells, for instance, we get In[] := SetSubstitutionSystem[caRules[110] /. caLocalization, Join @@ Table[ caBlock[ToString[Mod[index, 10]], ToString /@ {Mod[index - 1, 10], Mod[index + 1, 10]}, If[index == 1, 1, 0]], {index, 1, 10}] /. caLocalization, 8] // HypergraphPlot[#1[-1], ImageSize -> Scaled[1]] & ![enter image description here][30] that maps to `{1,1,0,1,0,1,1,1,1,1}` after 8 steps, same as rule 110 CA In[] := ArrayPlot[CellularAutomaton[110, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 8], ...] ![enter image description here][31] It is interesting to see how the causal network looks like in this case In[] := SetSubstitutionSystem[ ...]["CausalNetwork"] // ... (* Graph3D *) ![enter image description here][32] ## Infinite Grid Proving universality however requires an infinite grid. We can achieve that by initially attaching the ends of the grid by specially labeled "left/right end" cells, and then adding new rules to expand these ends. Specifically, we can construct the following four-cell structure where one of the end cells is colored in red, and the grid cell it is attached to is colored in yellow In[] := Join[Catenate[{caBlock["0", {"leftEnd", "1"}, 0], caBlock["1", {"0", "2"}, 1], caBlock["2", {"1", "3"}, 0], caBlock["3", {"2", "rightEnd"}, 1]} /. {"rightNeighborInput[leftEnd]" -> "leftEnd", "leftNeighborInput[rightEnd]" -> "rightEnd"}], {{"leftEnd", "inputFromRightNeighbor", "leftNeighborInput[0]"}, {"leftEnd", "leftEnd"}, {"rightEnd", "inputFromLeftNeighbor", "rightNeighborInput[3]"}, {"rightEnd", "rightEnd"}}] // ... (* HighlightGraph *) ![enter image description here][33] For the rule, we essentially need to replace the end cells with new white blocks connected to new ends, i.e., endExtensionRules = {{{xCenter_, "inputFromLeftNeighbor", leftEnd_}, {leftEnd_, "inputFromRightNeighbor", xLeftNeighborInput_}, {leftEnd_, "leftEnd"}} :> Module[{cellCenter, nextStepLeftNeighborInput, nextStepRightNeighborInput, nextStepCenter, newLeftEnd, leftNeighborInput, rightNeighborInput}, { {cellCenter, "nextStepLeftNeighborInput", nextStepLeftNeighborInput}, {cellCenter, "nextStepRightNeighborInput", nextStepRightNeighborInput}, {cellCenter, "nextStep", nextStepCenter}, {cellCenter, "inputFromLeftNeighbor", newLeftEnd}, {newLeftEnd, "leftEnd"}, {newLeftEnd, "inputFromRightNeighbor", leftNeighborInput}, {cellCenter, "inputFromRightNeighbor", xLeftNeighborInput}, {cellCenter, "0"}, {leftNeighborInput, "nextStep", nextStepLeftNeighborInput}, {leftNeighborInput, "0"}, {rightNeighborInput, "nextStep", nextStepRightNeighborInput}, {rightNeighborInput, "0"}, {xCenter, "inputFromLeftNeighbor", rightNeighborInput} }], ... (* left <-> right *)}; Running these rules from a single black cell will extend the edges indefinitely In[] := SetSubstitutionSystem[endExtensionRules, Join[caBlock["0", {"leftEnd", "rightEnd"}, 1] /. {"rightNeighborInput[leftEnd]" -> "leftEnd", "leftNeighborInput[rightEnd]" -> "rightEnd"}, {{"leftEnd", "inputFromRightNeighbor", "leftNeighborInput[0]"}, {"leftEnd", "leftEnd"}, {"rightEnd", "inputFromLeftNeighbor", "rightNeighborInput[0]"}, {"rightEnd", "rightEnd"}}], 3] /@ Range[0, 3] // ... (* Graph *) ![enter image description here][34] Combining the two sets of rules together, we can get evolution of the CA on an infinite grid In[] := SetSubstitutionSystem[Join[endExtensionRules, caRules[110]], Join[caBlock["0", {"leftEnd", "rightEnd"}, 1] /. {"rightNeighborInput[leftEnd]" -> "leftEnd", "leftNeighborInput[rightEnd]" -> "rightEnd"}, {{"leftEnd", "inputFromRightNeighbor", "leftNeighborInput[0]"}, {"leftEnd", "leftEnd"}, {"rightEnd", "inputFromLeftNeighbor", "rightNeighborInput[0]"}, {"rightEnd", "rightEnd"}}], 3] /@ Range[0, 3] // ... (* Graph *) ![enter image description here][35] Note, the evolution does not happen row-by-row, but instead we get past cones, for instance after 3 steps, we get the cells highlighted in blue In[] := CellularAutomaton[110, {{1}, 0}, {1, {-3, 3}}] // ... (* ArrayPlot *) ![enter image description here][36] We can localize these rules similarly as before just adding new types of edges for left and right ends caLocalizationInfinite = { {x_, "inputFromLeftNeighbor", y_} :> {x, x, y}, {x_, "inputFromRightNeighbor", y_} :> {x, y, x}, {x_, "nextStepLeftNeighborInput", y_} :> {x, x, x, y}, {x_, "nextStepRightNeighborInput", y_} :> {x, x, y, x}, {x_, "0"} :> {x}, {x_, "1"} :> {x, x}, {x_, "nextStep", y_} :> {x, y, y}, {x_, "leftEnd"} :> {x, x, x, x, x}, {x_, "rightEnd"} :> {x, x, x, x, x, x} }; Running that for 3 steps, we then get In[] := SetSubstitutionSystem[ Join[endExtensionRules, caRules[110]] /. caLocalizationInfinite, ... (* initial condition *)/. caLocalizationInfinite, 3][-1] // HypergraphPlot[#, ImageSize -> Scaled[1]] & ![enter image description here][37] After 10 steps we get In[] := SetSubstitutionSystem[ Join[endExtensionRules, caRules[110]] /. caLocalizationInfinite, ... (* initial condition *)/. caLocalizationInfinite, 12][-1] // HypergraphPlot[#, ImageSize -> Scaled[1], GraphLayout -> "SpringEmbedding"] & ![enter image description here][38] which corresponds to a slice of the CA In[] := CellularAutomaton[110, {{1}, 0}, {6, {-12, 12}}] // ... (* ArrayPlot *) ![enter image description here][39] Of course, by changing the order at which the rules are applied, any space- (including light-)like slice of the CA can be obtained. Note, it is straightforward to extend this approach to produce periodic initial conditions as well, which is what's required for the proof of universality of rule 110. We can finally show that this system is also confluent In[] := SetSubstitutionSystem[ Join[endExtensionRules, caRules[110]] /. caLocalizationInfinite, ... (* initial condition *)/. caLocalizationInfinite, 12, "CheckConfluence" -> True]["ConfluentQ"] Out[] = Missing["Unknown"] Thus we demonstrated that it is possible to construct a universal non-overlapping set substitution system. # Future Work In the previous section we found a universal confluent set substitution system. However, that system is rather large. In fact, the total number of vertices references in the combination of the rules and the initial condition is In[] := Length@Flatten[{ ... (* rules & initial condition *)} /. RuleDelayed -> List] Out[] = 587 It would be interesting to see what is the smallest system possible that is both confluent and exhibits complex behavior. Critical pair completion also requires more investigation, in particular, it would be interesting if known quantum effects such as entanglements, Bell's experiment, and double-slit experiment can be reproduced with this set up. 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https://community.wolfram.com//c/portal/getImageAttachment?filename=caInfiniteLocalizedResultSmall.png&userId=286762 [38]: https://community.wolfram.com//c/portal/getImageAttachment?filename=caInfiniteLocalizedResultLarge.png&userId=286762 [39]: https://community.wolfram.com//c/portal/getImageAttachment?filename=caInfiniteComparisonLarge.png&userId=286762 Maksim Piskunov 2019-07-10T18:00:05Z https://community.wolfram.com/groups/-/m/t/2312929 ![Comparison of discrete Kretschmann scalar (left) and the real Kretschmann scalar (right)][1] &[Wolfram Notebook][2] [Extra files and data][3] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=test500.png&userId=2066780 [2]: https://www.wolframcloud.com/obj/76b89950-1409-4573-90cd-bb23906ffd22 [3]: https://drive.google.com/file/d/1hqBLsApv7IYT70lMiRi5QI_-rUEuSTuz/view?usp=sharing Ariadna Uxue Palomino Ylla 2021-07-13T19:23:07Z https://community.wolfram.com/groups/-/m/t/2029319 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/4f443850-1789-41e0-a638-f117c66ab440 Patrick Geraghty 2020-07-14T16:23:19Z https://community.wolfram.com/groups/-/m/t/1949326 [The Wolfram Summer School Fundamental Physics Track][1] Are you an advanced research-oriented student in mathematics, theoretical physics or theoretical computer science, who wants to make potentially cutting-edge contributions to our understanding of general relativity, quantum field theory and the foundations of mathematical physics? Or perhaps your physics is a little rusty, but you’re a strong algorithmic programmer who’s enthusiastic about writing robust, scalable code that may help increase our understanding of the universe? The [Wolfram Physics Project][2] comprises a small team of mathematicians, physicists and computer scientists, led by [Stephen Wolfram][3], dedicating to investigating [Wolfram models][4]: a new class of discrete models for the fundamental structure of spacetime. And through the [Wolfram Summer School][5]’s newly-inaugurated [Fundamental Physics track][6], we’re offering you the unprecedented opportunity to join our team for a few weeks, and help contribute to our investigation. [Our formalism][7] appears to have [deep connections][8] to many branches of pure mathematics (such as higher-order category theory, topos theory and geometric group theory), theoretical physics (such as loop quantum gravity, twistor theory, string theory and conformal field theory) and theoretical computer science (such as term rewriting theory, geometric complexity theory and quantum information theory). However, many of these connections are currently only barely explored or understood, so we’re particularly excited to hear from graduate students and other researchers with expertise in these fields. We also especially welcome applicants from professional algorithmic programming backgrounds with experience in relevant areas, such as graph theory, distributed computing and virtual reality. A project in the [Fundamental Physics track][9] has the potential to be turned into an academic publication, or even to form the basis of a bachelor's/master's/doctoral thesis; particularly successful projects may also be featured as part of the [Wolfram Physics Project][10] itself. It is our hope that, in addition to your expertise being valuable to our project, some of the recent insights from our project may also help to inform and energize your own future research work. This year, our summer school will be run as a fully immersive digital experience from June 28 until July 17. You can [apply via our website][11] until May 24; we hope to hear from you very soon. Looking forward to meeting you all in a few weeks' time! Jonathan Gorard<br/> Director of the WSS Physics Track<br/> Principal Researcher, Wolfram Physics Project [1]: https://education.wolfram.com/summer/school/programs/physics/ [2]: https://www.wolframphysics.org [3]: https://www.stephenwolfram.com [4]: https://www.wolframphysics.org/technical-introduction/ [5]: https://education.wolfram.com/summer/school/ [6]: https://education.wolfram.com/summer/school/programs/physics/ [7]: https://www.wolframphysics.org/technical-documents/ [8]: https://www.wolframphysics.org/questions/relations-to-other-approaches/ [9]: https://education.wolfram.com/summer/school/programs/physics/ [10]: https://www.wolframphysics.org [11]: https://education.wolfram.com/summer/school/ Jonathan Gorard 2020-04-21T16:26:37Z https://community.wolfram.com/groups/-/m/t/2579591 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=2-dimensionalSM.png&userId=2318743 [2]: https://www.wolframcloud.com/obj/d065ae8b-dcb1-4fe2-80bf-9201bae29c01 Damian Musk 2022-07-21T19:15:51Z https://community.wolfram.com/groups/-/m/t/2034307 ![wm44586][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2020-07-15at2.04.53PM.png&userId=2033438 [2]: https://www.wolframcloud.com/obj/7c12a402-8697-418f-b5f9-7ef63f79dcc3 Quinn McIntyre 2020-07-15T20:05:31Z https://community.wolfram.com/groups/-/m/t/2034637 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=mssMap1rotate.jpg&userId=1938138 [2]: https://www.wolframcloud.com/obj/acb9bde6-ed0e-4c85-a897-63669d2a3606 Yorick Zeschke 2020-07-15T20:22:29Z https://community.wolfram.com/groups/-/m/t/2028486 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig4.jpg&userId=2027374 [2]: https://www.wolframcloud.com/obj/1879c0c6-5cf9-462d-9364-a5da1da74de8 Xiyuan Gao 2020-07-14T15:15:12Z