Wolfram Physics Project: Working Session Tuesday, Oct. 26, 2021 [Generalized Multiway Systems]

okay hello everyone welcome to a working session for our physics project so we have a bunch of topics here um because i don't think jonathan and i have done these live streams for a little while and so we we have a lot of pent up supply of topics um the one that we are really going to dig into is glocal multi-way systems but of course i have a distractive initial topic that i wanted to bring up which is so i've been thinking about quantum quantum mechanics and the analog of lorentz transformations maybe we shouldn't do this maybe we should just go to a vocal multiplayer system i think that might be a better thing let's not distract we'll we'll get we'll get way off track if we do that um no no no jonathan is ready for a distraction okay but that's okay so the issue is this if you have a a the question is what's the analog of a lawrence transformation in branchial space and is there a way to set up something which is kind of like an inertial frame and so what i was thinking is if you have a quantum circuit there's just a bunch of wires and they are connected like you would do for a transposition sort where the wires are just in this kind of uh grid of connections and then you ask the question if you i mean i don't quite know how to formulate this but but if you're looking at you have input on the left and output on the right is there a how would you normally given that quantum circuit how would you normally think about it computing something is that is it a rhetorical question or no that was a question for you because i'm i'm i mean if you you have a tensor product initial state of qubits or qudits right and each element of the tensor product goes into one of the wires and comes out the other side i mean that's yes but but but but what's the what's the timing story i mean you're basically multiplying by a bunch of matrices right right right which is why circuits come in layers so for instance quantum circuit operator has layering built in but but so how should i think about i mean in terms of time there is you know the those unitary operators can be constructed presumably from hamiltonians and represent the elapsing of a certain amount of time every every layer is a certain amount of time is that is that correct or is that not correct well you can think of it that way but you know the whole point of a circuit representation rather than say you know an annealing representation or something is so that you only have to deal with discrete time you don't have to care about time evolution operators or something because the point is each layer is a discrete time step fair enough but so so in other words if i'm going to say let's do a lorentz transformation and let's change the relative timing of the of the different parts of each layer that you're claiming that sort of a nonsense thing to do in the circuit representation well it requires reformulating what the circuit representation is right i mean okay well let me to make this concrete right so like if we have look at represented in language it comes in layers right and the layers are telling you that the timing information right so we go from left to right and we're saying the the successive vertical column the successive horizontal columns there are correspond to successive quotes moments in time right right right so in the naive version of the or in the in the yeah the naive version of a transformation that you're discussing you would have to deal with situations where for instance this s gate and its control qubits are like the the input for the s gate and the input for the control are ingested at different times and then it's kind of not clear how you deal with the outcome of that computation now as i say there is a self-consistent way to think about lorenz transformations on quantum circuits but it you know it requires what i was talking about earlier which is what well it requires essentially not modding out your representation by the conformal group but so so imagine we had a lot of wires here right then it would be the case if i simply were to look at i mean we could just just have i mean you could make one um just you know like 20 wires okay then the notion of a maximum entanglement speed does have a i mean imagine okay imagine we take the limit of the number of wires going to infinity okay so that then we we can think about lorentz transformations again as something which is initially discrete but then limits to a continuum so so i mean if if i'm not mistaken if we have enough wires it will be the case that that we we have some kind of entanglement cone that we can see within the uh you know even even with these discrete time steps we can see the entanglement cone build up across the wires is that a true statement well not not in the circuit picture because in the circuit picture you say you have effectively modded out by the conformal group there is no there is no space like structure here um of course physically there is a you know there are bounds from things like you know there's the margolis leviton bound and other kind of uh bounds on you know how quickly information can can propagate in a quantum circuit but those aren't dealt with in the circuit picture you need something but i'm confused by that i mean when you say it's not dealt with i mean if i just look if i have n wires where n is a large number i do know which wire has you know i have some basis and i know what the amplitude is for a particular qubit to exist on a particular wire don't i and therefore i would surely know if i started off with something where all qubits they're all zero qubits i have some some state in which everything is a zero qubit and then i start poking the circuit somewhere most wires will still have zero qubits after some number of rounds of some number of layers of the circuit just because they haven't yet had a chance to interact with the wire that didn't have the zero qubit am i confused that depends on the structure of the circuit no i know but assume that the circuit is related you know the only thing close to a geometry on the circuit is the gate set exactly exactly the geometry is defined by the the structure of the gates so imagine that what we did was something like a transposition sort where basically what you're doing is every every pair of of wires every adjacent pair of wires is connected um for example a trivial example would be do you have something where uh you know an even odd transposition sort type thing i mean not that we're doing sorting here particularly but we're just saying we you know we put a gate that does something i don't know is there an exchange type gate that um does something on pairs of wires right so so it's a kind of checkerboard of connections between wires yeah would that not define i mean that would implicitly define a geometry for this system wouldn't it well in the sense that you'd have an undirected graph you'd be looking at essentially the evolution of information on an undirected graph okay but i mean that would mean that that in if i stop there's nothing quantum about this i mean you could do the same thing with a boolean circuit well except that you could do that for a quantum circuit you could do that tracing quantum amplitudes couldn't you with that circuit or with a boolean circuit this whole thing can be done with boolean functions there's nothing nothing nothing about what you've just said involves quantum mechanics well it depends what the gates actually do if the gates are just swapping then presumably then presumably that can be all they're doing is that's swap gates are that i mean that's a classical operation there's no i i understand that i understand that but if we have some kind of thing which changes i don't know in the zx calculus setup for example there are aren't there sort of elementary gates that well i i forget that i mean that you know with there's a certain set of you can get i mean obviously we can have universal quantum gates that can get to any quantum amplitude by just having something which has some rotation or something like that in it but but i'm not sure that really matters because what i'm what i'm curious about is just whether sorry the fact that a quantum gate is a gate set is universal does not mean that you can get to any amplitude anymore knowing that a cellular automaton is universal tells you you can get every possible register configuration that's not what universality means fair enough the okay but but okay well anyway i the the thing that okay maybe maybe we can um i mean my view is that there's a very very concrete way of thinking about this um which doesn't involve circuits it just involves multi-way systems right go for it maybe it's actually a segue to our second topic about goku multi-wave system because maybe it's actually related to that but tell me what the point is that in the order in all notions of quantum computation that we generally think about like circuit models there's an idea that you have synchronization across all eigenstates that appear in your superpositions right so in particular if we take a multi-wave representation of a quantum gate or the evolution of some quantum circuit let me look okay here's an example then you know we are always assuming effectively a cosmological rest frame in the sense that uh we assume everything that's on those you know everything all the eigen states that appear on the zero time step appear in the same superposition all the eigen states on the one time step appear et cetera if you relax that condition so you now generalize the notion of superposition so you can have eigenstates with different that are not strictly synchronized um then that lets you that lets you capture the exact analog of conformal transformations okay fine this is consistent with what i was talking about but i'm i'm i'm confused about why it could not be actualized in a quantum circuit but i i get it but but you it's you would need to generalize the notion of a quantum circuit in the way that i described earlier you'd have to not mod out by the conformal group okay explain to me why when you talk about okay so here we can take different reference frames for looking at this i mean if we take the example a trivial sort of example of a grid multi-way system we could see how different you know we can pick different reference frames if they're just sort of parallel if they're just sort of straight slices we can just pick ones which are differently boosted and they will have a different set of of states considered to be the quote simultaneous states aka the states combined in two positions right right right so we're we're talking about two different levels of thing right so there are relativistic transformations that occur at the level of the superpositions and there are relativistic transformations that occur at the level of the tensor products so and it's not you can't just deal with one which is what you were discussing with the circuit example right with with relativistic transformation of circuits when you're just looking at effectively desynchronizing what happens on each wire you're doing relativistic transformations of tensor products of the qubits but you're not relativistically transforming the actual eigenstates that appear in the superpositions for those key bits um and that's not mathematically consistent you need to transform both okay let me walk through that again because that was too fast for me so what what are you saying here so you're saying that one thing you could do is we got this multi-way system imagine it's a grid multi-way system we've got all kinds of uh at each stage relative to the cosmological rest frame we have a certain set of of um of states which are being of quantum eigenstates which are going to be combined in a superposition is that the right setup yes i mean what i'm saying is really very simple so in this multi-way system here there's only one qubit right it's a it's a two level system you can see that each state vector is a two okay right so there's no tensor products here so in in the circuit representation this has only one wire but there's still the point is this is still relativistically transforming because you're you know you're constructing some arbitrary you know that there are there are multiple ways of constructing anti-chains in in this multi-way system okay so the point that you're making which by the way is is deeply connected to the local multi-way system business i think is that there is one as soon as you add more qubits you've got your tensor producting up those states and so you're in general going to have two to the n uh why are there i don't quite understand what why are there what why isn't a qubit what exactly is going on here what what is this what is this thing so th this is showing a superposition of eigenstates okay so there's two wait a second so explain what the we've got no i don't understand what's happening here we've got two eigenstates and each one has a quantum amplitude right so but in this particular case uh to simplify things with where we treat the imaginary and negative directions as being distinct which of course in reality you wouldn't but to treat that in full generality requires a coordination theorem on branchial space that we don't yet have okay so this is a discretization of the quantum amplitude represented as um a pair of of uh of numbers i don't quite understand that so yeah so what so what's going on here one step yeah okay all right that might be simple enough for me to understand okay so what's going on here oh sorry let me basis uh let's let's start with a oh no actually that no that's fine we can do one plus solve one minus sign as our initial state something like that okay so we've got one plus i one minus i that's our initial state so you can see we've got a one zero an i zero a zero one and a zero minus i and they reach weighted one so wait a minute so those are representations of quantum eigenstates each with magnitude one right and the phase is not yet being defined is that correct well i mean this is why we have so in reality of course you just have one zero and zero one if you if there were a general way of extracting phase from branchial coordinates because we don't yet have a general coordination on branchial space we can't do that so i'm baking in the phase by also having by also treating i zero and zero minus i etcetera as being eigenstates as well which of course is unphysical okay so by baking in the phase you mean what we actually think determines the phase is in a sense position in branchial space but here instead of having instead of having a representation of position you're putting in the actual name of the state you're putting something which describes its phase is that right in effect yes i mean so what's actually happening is i'm defining all possible uh you know all possible orthogonal directions in this hilbert space as being separate eigenstates and then so then all the multi-wave system is doing is just shifting around weights between those vectors okay but so the positions the the branchial positions the relative branchial positions of the states that you get after three or four steps what is the significance of those relative to their quantum phase in other words of course in the physical case that is their quantum phase there is a one-to-one correspondence between their branchial coordinate and their face i understand but for example this thing i did recently about um multi-wave systems with numbers okay so i looked there at this question of if you actually put numbers on the nodes basically and then you use those numbers as a coordinate system uh there's a reasonable question which i did look at of what is the correspondence between the branchial coordinate system the implied branchial coordinate system and the explicit numerical coordinate system did that make sense yes and and there was some vague correspondence in some cases but okay but so here what we're doing is we're approximating the implied coordinates of branchial space by burnt in coordinates of the actual numbers so to speak so that's right yes okay so then the point you're making is that there is one sense of making a lorentz transformation which will be relative to that either genuine branchial space or or implied i'm sorry implied variational space so to speak or you're burnt in numerically determined branches you know numerically determined coordinates is that right so you could imagine just doing lorentz transformation with respect to your numerically determined coordinates which might not be an accurate representation of branchial space but it might be some approximation is that is that right no what i'm what i'm saying is that i'm stating the consistency theorem that i was able to prove for the you know for this case which is that you cut it you have to have the tensor products lorentz transformations and the eigenstate lorenz transformations covari if you want to have a mathematically consistent theory so this is what you're when you say tens of so so i mean if i took this thing here and what you've got here and i added another qubit so instead of having vectors with two elements i have vectors with four elements is that correct um so then but then what what's the i mean the other so that that will then become much more complicated right the yes um okay okay okay so no wait what happened there so that that is how did you make i don't understand what's the operator here what this the thing called operator well i realized that but what is its significance oh this is uh i think this is a square root swap or something something like that but but when you took that chronicle product you're taking the product you're saying independently on each wire do a square root swap is that right that's what the chronicle product means and so then you've got two things that you've got a chronic product on the original description of the qubits and then you've also got a kronecker product in the operator is that right yep and the point that you're making is that in order to get a consistent thing you have to have uh you have to make transformations both on the state and on the operator is that right no no the point i'm making is so you were talking about quantum circuits and relativistic transformations on the wires of those circuits those are relative those are lorentz transformation or conformal transformations that are applied to the level of the tensor product but with this qubit that's being or q-dip that's being tensor producted there are eigenstates and those also relativistically transform and it's not good enough to only transform one and not the other you have to that they are covariant quantities um you can't just treat one okay no i i can see that i mean look the thing that i'm wondering is i mean the the thing that got me sort of piqued my interest here was the possibility there might actually be a physical experiment one could do to measure maximum entanglement speed that it might be a physical experiment that effectively involves something with a large number of quantum wires is that conceivable so i mean the maximum entanglement speed has has has the consequence does it not that it should have the consequence that there are degrees of freedom that get affected only slowly so to speak yep and it i mean i it's a little confusing to me that well i'm i i guess i'm confused in in um in a sense in in traditional quantum mechanics if i just do a bunch of wires or a spin chain for that matter there is a the hamiltonian determines how quickly information spreads across the spin chain but we're imagining that there is a kind of a a lowest level hamiltonian which describes how the information spreads yeah all right this is a complicated story well okay go ahead it's kind of it's kind of clear what the prediction is right it's the the you have a quantum circuit that's described by a tensor product of cuditz and the tensorboard whose initial state is described by a tensor product of qdix and whose circuit is described by a tensor product of wires and then you just say well i can compute an entanglement entropy for any pair of wires and the and the prediction would be the information propagation is higher it takes longer between wires that have higher entanglement entropy um and that well i mean which is to say when there's a higher density of stuff in branchial space it takes longer to hop because there's more there's more hops you have to make in branchial space which is not what you would necessarily have thought about if you didn't believe in a discrete underlying setup sure because without a discrete underlying setup there's no reason for it to take longer is there to hop between different parts of branch or space is that a true statement in other words in a traditional setup with with traditional continuum quantum mechanics you would not expect unlike with physical space where we readily believe that there's a maximum speed of light that there is no such similar thing is there in ordinary quantum mechanics am i mistaken well not in not in quantum mechanics no in quantum gravity there is or is conjecture to be we've discussed right but i mean okay but quantum gravity has space time you know stuck in it as well whereas what we've got is a is a thing like that but without any space time we've got a branch go but quantum gravity impli certain conjectures in quantum gravity in particular as we've discussed the oracle's epr conjecture imply that uh that there should exist this effect in quantum mechanics because the corresponding effect exists in space-time i get it i get it i understand that i understand that but but then at a concrete level the question is is there a way of actually making a quantum experiment that could measure this and um uh in which you know what what does one expect to have happen if one has a maybe this is is too um you know you have a bunch of of you know essentially quantum wires and they're connected by various quantum gates and you know what is the normal expectation for i mean i i suppose it's just the ordinary quantum hamiltonian determines how quickly the amplitude builds up in wires in distant wires i'm trying to understand the difference between what happens in our setup and what happens in traditional quantum mechanics where you're basically just looking at propagation of of a of a lump of amplitude so to speak through different connections according to some hamiltonian is it obvious what the difference is between that setup that quotes classical quantum setup so to speak and what we have yes okay and it and what is that what is the difference it's the difference i just described so so you in the in the conventional quantum mechanical setup the notion of discrete wires the notion that you have a discrete set of uh you know of operators or of qubits or qudits that you're tensor producting together is thoughts to be an approximation right and when we actually when one actually constructs quantum circuits experimentally you are trying to set up a continuous quantum system in such a way that it approximates this discrete one that you know right and so in particular in the in the in the true quantum mechanical picture that you know that's being dealt with in in an experimental quantum computing setup the notion that you could compute an entanglement entropy between a pair of you know tensor products cubic cubits doesn't really exist because those are thought to be a fiction whereas in a discrete setup they do i mean that's well right i mean the measurement of entropy in a setup where everything is continuous and there's no counting of information is a tough business i mean is that what you're saying or something different well yeah i mean well okay so so i'm just don't understand i mean in is there an easy way to say what is the the i mean it's just like what you see i don't know it's like the classic uh sorkin calculation of entanglement entropies for black holes right the first step in that is say let's assume that we can discretize space times we basically treat you know the causal structure of spacetime as being a collection of you know couple harmonic oscillators and then we do the entanglement entropy calculation for those harmonic oscillators you know if your picture is continuous then anything that involves discrete computations including computations of entanglement entropies is necessarily an approximation if your picture is discrete then of course it's not um i i don't see why but i guess what i'm asking is is it obvious to you if somebody actually had a great big quantum lab that allowed them to set up you know arbitrarily complicated quantum circuits is there a measurement that you could suggest that they do that would measure maximum entanglement speed what is the analog of the fit so toothed wheel or something for measuring the speed of light for this case i don't have any refinements on what i already proposed so say again what you proposed just construct a quantum circuit yeah measure the entanglement entropy between different pairs of wires how do you do that in an actual circuit uh well you would design you you would design the circuit in such a way that the different pairs of wires have calculable entanglement entropies even though as we say to calculate it requires a discretization of some kind yeah sure but okay and then the question would be as you increase the entanglement entropy does it increase the amount of time it takes for certain quantum effects to propagate is that the theory yep and then the notion would be as you increase the entanglement entropy that you should sort of see a linear increase in the amount of uh of time to propagate the quantum effect somehow i don't know about linear but certainly monotonic right well somebody should do it um okay maybe we should change topics here and and go to what we originally thought we were going to talk about which is local multi-way systems and um i think you have some lovely functions now for doing local multi-base systems so maybe you could bring those up and we could just take a look so a bunch of them have been submitted to the function repository although they only got submitted yesterday so they haven't obviously been approved yet so um like the ones that have been documented to things like string local multi-way systems uh which maybe are the easiest to understand i don't know all right let's let's start there uh okay so here's the basic idea we have stephen do you have a favorite string multi-way system um um let's see the one i've been using is is um the heck is it it's a goes to a b no i don't know i have to look it up yeah we can help you it's a fairly boring one but we can use that no no that that one is that one is way too boring there's some let's pick one hold on hold on i have a favorite one this is the problem whenever people ask me like like even for cellular automata that's why it's good to have numbers because i can remember the numbers i can't remember the bit pattern actually boring might be good to begin with just because it'll be easier to yes that would be that's a good thing okay okay so if i announce these string glock multiple systems with the same input same rules okay here's what i get can you make that window wider oh yeah sure sorry i was uh of course um okay so let's go back to the the let's go first to the the traditional multi-way system right so with a traditional multi-way system your states are global and your events are global right so your state contains the whole state of your abstract rewriting system and the event ingests the whole state of the abstract rewriting system and outputs the whole state of the abstracted writing system right that's what we're seeing here with a local multi-way system everything is kind of decomposed so you have local tokens and local events global multi-way systems were my attempt because my personal philosophical prejudice is that it makes sense to treat tokens as local but events as global because the reason for that is that the the events are the things that effectively assemble the tokens together into global states and so if you want to be able to do things like isomorphism testing and you know a non-trivial state deduplication it makes sense to have the the events be global uh but it also makes sense to have the the tokens be local so you don't have to store the whole state of the system at every possible you know every different time step so that's what a global multi-wave system is doing it okay so let's let's try to understand this so a what was the original the initial state is a a the initial state is a so so let me explain what's going on so here you can see we've got one token as the initial state a now that token is being ingested two copies of that token are being ingested into this event because the input for this event is a a then how does that token know that there are two of it or does it not know it doesn't know it's a deduplicated token where as far as it's concerned it's the only one uh we'll come back to that so so token duplication is a slightly subtle thing um but okay so so there are two copies the okay the point is that the there isn't any notion of like um the notion that that tokens are a bounded resource is a notion that is respected by the events rather than respected by the tokens if that makes sense this event knows to only ingest two copies of a it's not that a itself knows that there are only two copies of itself yep um okay so so it ingests two copies of a and it outputs two copies of a in one copy of b because the output is aba so then we got these two tokens here how does it know the order of the output it doesn't from the tokens it doesn't just from the tokens just like in a local multi-way system again it's the events that care about the order it's not the tokens so the default case is that everything is deduplicated everything is is not deduplicated but we can say deduplicate tokens goes to true and then we get something that becomes a bit harder to analyze it's a lot more compact but but it becomes much harder to analyze so here you can see now rather than having copies of a's and b's for every for every possible output and input we've now just got one copy of each token at each time step but you know the number of evolution edges becomes sort of unfeasibly large quite quickly okay so in a purely local multi-way system with a essentially a token event graph what would happen here is that that event i mean here i'm not actually seeing any multi-events right i mean in this particular system i don't think this is any different from a token event craft is it uh it i mean in the sense that the system is confluent yes no no i mean i mean those events if those events were oh you oh you mean there's no right because there is a difference there is one i see the difference the difference is the spectators are included in the events here whereas the spectators are not included in the events in a token event graph right exactly because the events are global so the point is that unlike with unlike with a token event graph it's so the nice okay the thing that there are several things i like about global multi-way systems one of them is that it's trivial from this to reconstruct both the purely local token event graph and the purely global you know evolution causal graph um like somehow both of you know information about both of them is encoded in this one object why do you need the tokens why do the tokens why do you need to have tokens here to be able to do that reconstruction is that obvious uh well you need to know how many how many copies of each thing were ingested in output but you can read that off from the event can't you yeah but that's like saying you could you know you don't need an evolution causal graph you can just reconstruct the whole thing from the causal graph i mean that's also true right but which is ultimately true although in both cases there's a question of reference frame right which also gets exciting what's that which also gets kind of exciting okay well okay so okay so i'm just trying to understand this so the main difference here is that the the events include their spectators whereas in a token event graph they do not in a token event graph they only include what actually what actually was ingested and changed here they include spectator basically spectator tokens that are just going to be that just have a null event happening to them that are just copied correct right exactly exactly okay and and what you do so in the next step you are de-duplicating tokens and that means that okay so let me understand this but you're not de-duplicating across time steps you're only de-duplicating within right given time if i de-duplicated across time steps it would become essentially structureless i mean i mean it would basically just become a causal graph is it or is it a petrinas of some kind uh it would be a petri net if the if the tokens had um you know some kind of some kind of uh you know resource limitation semantics it would become something like a patreon right but i mean in but am i not mistaken am i mistaken in thinking that one way to view the evolution of a petri net is that you have certain as they call them tokens but we have to call them something different because they're not the same as our tokens some kind of markers that oh this might be the this might be the incomplete function repository function yeah okay there we go here's a multi-way petri net it's very swank okay so let's understand so there um how should i understand this in terms of the thing on the underneath ah maybe it's not worth understanding that but i mean the the main point is that in a different foliation your choice of de-duplication will be different is that correct for the thing underneath yes right exactly and but your claim is that because the events are global there's a unique sort of bipartite layering in some sense of events and tokens is that correct yep indeed so how would i make it a different foliation of this whole thing what i would be doing is saying there are different set of events that i claim are simultaneous events yeah exactly so there there's it's the same as making foliations of just space-time and in fact that plays it okay let me show you something else uh if i take this thing run that okay so let's yeah let's take this and just show its bronchial space so this is what a branch of graph looks like a global branch or graph so this contains information both about okay so you'll notice there are essentially three different kinds of edges here there are token to token token to event and event to event the token to token and token to event edges are colored in purple and the event to event edges are colored in orange for reasons that will become clear in a moment so effectively this is encoding both evolution ancestry because okay an ordinary branchial graph like for something something like this just encodes evolutionary ancestry right so it's saying these two have a common ancestor one step back in the evolution grow on the statecraft um a glocal branchiograph encodes evolution ancestry of the same kind so when you have a pair of tokens that are connected like you know these two noting that these are non-deduplicated tokens non-deduplicated exactly if that was duplicated that that would crush the branch of graph to practically nothing yeah so in the deduplicated case it's a bit less interesting but anyway let me show this okay there's still causal structure but there's no there's no one state structure exactly so when you have a pair of tokens that are connected by a branchial edge that means that they have a common and a common evolution ancestor same when you have a a token an event it's it's saying that this um it's saying that this token was spawned by this event then when you have two events and they're connected by not by a branch alleged by a causal edge that's telling you that they have shared causal ancestry so these two events share a causal uh ancestor so let me come back to something for just one second here so i mean in the purple so the structure of this thing is sort of a bipartite layered thing so every token has its ancestor an event yes exactly right so to say that to connect two tokens by a purple edge is to say that those two tokens have a common ancestral event is that correct no that they have a common ancestral token okay but but in that picture there you've got over on the right for example you've got an event and that event clearly connects to the tokens a and b because those tokens are generated by that event a goes to a b generates an a token the b token right yes exactly two of those edges the two edges coming the two so the the events to token edges are telling you this token spawns this sorry this event spawns this token yep token to token edges are telling you these tokens have a shared evolution ancestor just like in a regular branchiograph having a shared state ancestor i understand but but then that means that so can you show that the the the actual evolution event graph for this thing for a second oh yeah sure that was a goes to a b and we can look at you in the global case or the or the multi-way case the glock case okay okay so let's look at the a that thing over on the right a goes to a b and then there's a bbb at the i think it is where is it oh yeah it's over it's the fourth from the right can you just highlight the fourth thing from the right that one there isn't it okay all right so now those a b tokens are precisely the ones that show up with many i see oh no this is very very confusing that those a b tokens there's just one of them in that evolution event garth but am i am i seeing multiple copies of those a's and b's in the branch of golf no you you're seeing their ancestors why are their ancestors there i thought we'd moved on to a new generation of of tokens here i mean why are the ancestors there i thought the branchial graph was just recording the branchial the ancestral relationship between the a and b why are there multiple a's which which a and b are you talking about you mean you mean the the two and the two b's that are connected to this thing here yes that's what i mean why are there two a's and b's connected to that event why is it not just one a and b ah okay so sorry so you know in that case it's because there are two a's and two b's that are relevant in the output here to um common evolution ancestry one level back there's one spectator b that's left off okay i'm sorry you have to try that again well in a normal branchial graph i would expect that though the a b that comes out of that event comes out of the ego's a b b b spectator event that's the a and b that come out of that event they oh i see what you're saying you're saying that their common token ancestry includes those the a b that are somewhat to their right coming from the other event exactly okay so there's actually two levels of this kind of branchial graph that you could draw you could draw a branchial graph that purely includes the event to token piece right without including the token to token piece and what's that that's true um okay so the reason for wanting to choose this specification was that unlike the unlike global branchial graphs this specification lets it gives you all the information needed to reconstruct the full evolution causal graph if you know just the the sequence of branch-like hypersurfaces which you wouldn't otherwise be able to do sorry it gives you enough impatient so normally you would have enough information to reconstruct the multi-way graph but not the causal graph right that's correct you could reconstruct the state's graph but not the evolution causal graph this glocal multi-wave branch of graph gives you enough information to reconstruct both okay and you were about to explain the event to event edges there right right and so the event to event edges are telling you that there's shared causal ancestry so in addition to so okay in maybe maybe it's easier to see on that one of these things so in addition to we normally think about branch pairs as being state branch pairs right so like aba goes to abba we considered the other case we considered causal branch pairs but we thought they were trivial right yeah it turns out that they're not trivial um and the way that you can see that they're not trivial is by looking at these um but yeah so these are telling you causal branch pairs if you like what happens if you just take the multi-way causal graph and look at its branchial graph and we did try that i remember maybe we had bad luck in what examples we tried no i mean it it when you have global states as well it becomes less interesting i mean like you can i think it's called just branching is it event branch i forget yeah it's kind of dull well no no i'm not convinced it's always dull explain why it's dull we had an argument back a year and a half ago for why it was dull but it wasn't totally convincing and now now you're telling me it actually isn't dull if you okay what is it that is additionally knitted by the presence of tokens that makes it definitively not dull because it separates out um what's being ingested versus what's actually a spectator in a way that isn't separated out in a global multi-way system okay but but is it obvious i mean if we pick a different i i'm sure i'm sure i looked at this i mean i certainly back in those days i was in a very enumerative mood and i'm sure i tried every possible rule of a certain type for for its causal um branchiograph but is there an obvious reason let's see is that is can you can you reason through why why the causal bond show graph is usually boring no i i mean i never claimed that i knew that it was it's just that in the examples in all the examples i've seen the structure is rather trivial um i think global multi-way systems make it a lot clearer that it's not trivial okay all right okay so let's understand that so you're saying so if i were to just lift out here but i don't understand why is the why is the brown connectivity or the orange connectivity any different from the connectivity that you just showed me of the causal branchiograph so it occurs because the the way we assess causal structure in the glocal multi-wave system only looks at the you know the sort of the live tokens rather than the spectator tokens okay let's go through that that's a tricky issue in all of our setups and in fact when i was looking at this multi-way systems based on numbers thing numbers as such encode nothing you know you have a you know a 14 it doesn't really have a notion of of the part of the 14 that gets acted on and the part that's a spectator it's just like 14. what's that you have prime factorization depending on what your operation is yes yes i mean actually the thing i was looking at this multiplication thingy actually that's a that's a fair observation that i could probably draw the causal graph um based on factors i mean i i had a version of that when i first produced the multi-wave function system function i had a version of it that did exactly that and i think you thought it was too esoteric so we took it out yes i thought it was very weird i mean because it because it depends on the dynamics whether factorization has any natural character to it i mean you could as well say consider the number in unary for example and do it with addition sure um okay in any case so in this case here explain so you're saying that let's see what are you saying you're saying that these things are considered causally connected you're saying that spectators can lead to causal connection so for example the identity event could be causally can could lead to causal connection a sequence of identity events could be causally connected in the traditional global causal graph story and you're saying that's not true here right so if we take something like this sorry really trivial example that makes the point so the a's that are ingested by this event those are spectator a's but the fact that they aren't the inner the a's on the two sides of the island sorry the two a's that are ingested yes but you can't really see that they're spectator arrays until you break everything apart into tokens and then you see oh actually these tokens you know these two tokens were unchanged yes and so the point is that the causal structure is a bit works the notion of causal precedence works slightly differently for a global uh multi-way system to so put it a different way if you think of the world in terms of tokens it's easy to tell what changed and what didn't if you think of the world in terms of complete states you say the state changed even though actually most of the state didn't change whereas here what you're doing is you're accounting for change in a more fine-grained way exactly um okay okay i get it so yeah i mean i've wondered about this thing about spectators and their involvement in causal connection so this is this is homing in on the question of is a spectator part of a causal connection and it's basically saying you have a set up where spectators are not where just hanging around isn't enough to have an effect so to speak exactly just being touched by the event doesn't isn't enough um you have to be changed by the event now what happens so in other words if it was the case that it was a x x a and it turned into a x y a only the y would be considered to be a a genuine effect yes yeah it's nice okay i mean i i agree that that's a that's one of many horrifying kind of you know did it change what counts as equivalent type things all right so so i get it so now what uh so if you lift that causal graph out of the full if if you lift the causal branchio graph out what do you get uh well yeah then you get the i mean i don't actually have code that does this although it's quite easy to write then you'd get the glocal version of this thing here exactly it would have a lot more structure than this thing here well can we see it i mean uh i mean yes we can probably those those edges must be tagged so you can just select edges from that uh the edges are not tagged i'm afraid well i mean they're tagged with rendering i guess that's disgusting what we could do is okay let me figure out how i'm going to do this so uh oh no wait this should be quite easy because i can just say you just select the edges that have events on the ends wait wait yes that's right doesn't matter either one of them no because you don't want the token to event things as well okay fine um [Music] i think that would work and then we just say nope that uh hang on let's see can you make that video just let's see what it got wait wait um okay there we go oh interesting oh okay let's see that for um okay can we see that after a few more time steps okay why don't you just write a little wrapper function that extracts that from it doesn't really matter um uh you want to you want to get rid of the the rules you want to put the explicit rules in there oh sure yeah that's right yep okay let's do that we do we really want the um event rendering or just really no why don't we just do structure okay we go to seven can we push it that far let's try a trivial thing that has some kind of grid what what did what did this oh lovely okay may do a graph 3d of that uh just go up graph.3d i know i just said graph3d and now just make it bigger and go around unhelpful but amusing okay it looks like it could be some part of a polymer or something i thought it was very very summery based on its colors but anyway yes um uh looks like the kind of thing one might be allergic to if it was in some kind of um anyway never mind um uh okay let's go back can we try a more trivial rule and look at the analogous look at the corresponding i mean this is unfortunately this is a pretty trivial rule um it goes to a b oh my gosh what happens if it's a goes to a a does that do nothing because of its um uh no but that'll do something pretty similar i think i mean look i mean we can make this a bit well did you do three steps of that what that was a complete graph basically right yeah that isn't a complete graph if we do tokens we can make it we can make it simpler but it'll lose a lot of interesting structure well let's let's look at it with the duplicated tokens okay well i mean if i show the he goes to a b case this will now be much higher connectivity let me understand that so the reason for that is that there are events that got connected because their tokens were were considered equivalent and so the events became related by virtue of the observer equivalencing event equivalencing tokens right exactly so this is the implied observer you know observer equivalence version of this thing huh okay so what what this is basically saying is if we thought about this in space time if you say all pieces of space are equivalent though nothing that happens anywhere is connected and that's why all these events get connected by virtue of the identification of different parts of space basically right whereas if the different parts of space if the it isn't quite the level of atoms of space but but if the different uh sort of fragments of space were completely separate then the things that happened to them are happening separately but why is there such an asymmetric okay so if a goes to a b could you could you show a goes to a b in a single a and don't deduplicate the tokens okay why is that null wait maybe for an even number okay let's let's have a look yeah that's pretty dark okay so what is that what is that what is that hold on because we were extracting the causal branchograph and there's only ever we're going to be one event at each step that's why it's still i see so it has to be thickened up because look that's what it looks like all right okay so we might need a rule that has a bit more what what what is the rule let's see a goes to a b normally in the global multi-way graph that produces a grid correct uh for an aaa initial condition yeah not not for an a initial condition ah for an initial condition to produce path yeah okay fine fine so this is really doing something perfectly consistent with that yep okay fine yeah this is a slightly cheap way to get a two-dimensional thing but yes um okay so what you're saying is okay but the main thing here okay so there's a a carrier of a i don't know what the heck this causal graph is what what do we call this causal graph this is a a a sort of a token connected causal graph what is the right term for it it's a it's a um it's a token-based causal graph as opposed to a state-based causal graph it's a token causality-based causal graph token-wise causality versus statewise causality as in it tracks only its way of deciding what's what affects what is what had tokens changed as opposed to what had states changed okay so what does this tell one about okay so if we're thinking about space-time versus branch time basically what this is doing is by by if this was a hypergraph based system then the tokens do you have one of those yep let's load her up oh well that's very exciting uh okay the problem is it's this code is super slow this is the reason i haven't submitted it to the function repository yet because i'm still working on optimizing it let's do one step oh yeah sorry it's gonna have to pull in the okay but i mean basically what's gonna happen here oh gosh okay it's more useful i think to see at a conceptual level so what's going to happen is the tokens are now hyper edges the deduplication of hyperedges is isomorphism of hyper edges right yep here's the d wait a minute wait a minute wait a minute wait a minute wait a minute wait a minute that's a very very weird thing to do because an actual space those hyper edges contain different atoms of space all the hyperedges in the universe they are knitted together by their common atoms of space but many hyperedges have you know hyperedges on two opposite sides of the universe typically have completely distinct atoms of space in them yep so when you de-duplicate all the hyperedges you're saying that even though it's you know the hyperedge is in some quasar at the other side of the universe i'm going to consider it because it has the same structural form as a hyper as a local hyperedge i'm going to consider it i'm going to ignore the labels i'm going to ignore the knitting of those hyperedges together i'm just going to conflate them all correct right i mean it's corresponding to the fact that branch-like locality is not the same as space-like locality so it's entirely consistent with the branchial causal structure for you to have hyper edges that lie on opposite sides of the universe being ingested into the same event as long as they match the pattern right but that's saying i mean that that's the question of whether you know whether entanglement whether the speed of entanglement in physical space is infinite or not and this is saying if you can um you know it's just as well that that um nobody has decided you know there are all these things you can't do on you know these days you know you can't smoke a cigar or something it's just as well that nobody's decided you can't eat whatever we're eating never mind i'm sorry i didn't no no no no no i i it's i i did that too and it's it's um um the uh um it's uh for me it's necessary because the alternative is either the glucose supply to the brain you know dies off i don't need a piece of chocolate or something and then no multi-way systems come out um anyway but but so the point that you're making is um okay so is that without deduplication no no that's with deduplication without would be kind of um i mean i oh sorry uh duplicates this is going to be a horrible mess it will be actually maybe i can make it a bit faster if we don't do the rendering um but yeah it'll be something like that okay all right the the the rendering is definitely the bottleneck with um both remodel systems okay so hold on let me just think about what this means what you're saying is on two branches of history okay in general the version of space that is generated on two different branches of history will be connected by the common ancestry of those branches in the sense that there may be two atoms of space that exist on those two branches of history which are the same atom of space because they both came from the same event in but they were on different sides of a branch pair right right and then what you're saying is that one thing you can do is to say i'm just going to conflate all i'm just going to ignore every label on every atom of space and i'm going to put them all together and that's what token deduplication is doing here is that right that is correct yeah what on earth is the physical interpretation of that it's basically saying all of space we're ignoring space we're ignoring the we're ignoring the fact that there is extent to space right well you only care about branchial locality you don't care about space like locality right which is what which is the story of quantum mechanics if not quantum field theory right right quite um but so okay but now for example in that picture there you seem to have two uh two hyper edges down at the bottom there the figure eight things which look maybe they're not the same maybe they have different arrow structure i think they must have different arrow structure okay fine um now i mean we could we could imagine a generalization of this where instead of de-duplicating i mean maybe this is two or nate instead of de-duplicating as a as we progressively try and approximate genuine space-time we could say let us not de-duplicate things just at the level of individual hyperedges but let me look at sub-graphs and de-duplicate only across a sub-graph well there's there's an intermediate i agree that's that's also useful but there's also an intermediate case for the warfare model system that doesn't exist with these other systems which is that you could have deduplication where you don't do isomorphism check checking you just merge you know hyperedges as lists that are equivalent if you get what i mean yeah i understand what you mean well okay if they're lists that are equivalent then that means that the name of the atoms of space the names of the atoms that's going to be the point so then you have a notion of deduplication that preserves spatial structure that's the reason indeed and what does that do how much deduplication happens in that case is that significant to duplication well i haven't looked at it i'm saying that that's that seems like an interesting so so what that would successfully deduplicate is that's a good idea because what it will successfully de-duplicate is things on different branches of of history which are just the same thing basically precisely right so in other words an event that picks up a given collection of atoms and there are multiple events so for example in the case where there are spectators that will be a dramatic savings if you included spectators that would be a dramatic savings because then you are conflating together well it depends i don't know no this is complicated so i mean obviously so i did a bunch of experiments with this in this whole multi-space idea did you do a turing machine version of this again this is not very efficient which is why i haven't put this in the function repository yet but um let's take an example here's a non-deterministic turing machine very dull okay could is that de-duplicated uh no okay so the spectation so to speak whatever the right noun from that is the reason this is quite weird is because the head you are you are simply extracting the head independent of where it is and calling that the token well okay so this is this is why i need to figure out a less dopey way of rendering it right because the the point is i'm these tokens are just um positions on the tape when you call rule plot turing machine with a one you know with a one uh place tape it will plot the head at that position but you say you're really storing there the full state of the head and position of the head is that correct yes yeah it's just that there aren't you know because of how it's being rendered there isn't there isn't a place to put the head apart from in the one square where it can go if you see what i'm saying yeah i understand what you could do rendering-wise is you could just ghost the rest of it you could just have the framework of the rest of the tape i need to say i need to write a less silly rendering function but okay but basically each one of those so at a given level in a deterministic turing machine okay i am totally confused here in a deterministic turing machine why isn't there just one head configuration at every step there is you can see the the two head states are the same i mean they're the same they're both pointing but why is it on a different color of tape what do you mean the tape is four squares zero one zero zero so there are two white squares and one orange square being ingested into this event oh but but wait a minute but i see but it is incorrect to put i see so the problem is in the rendering here the fact that there is a head on that white square right that's the most squares so it's showing the right head state but it's just it's not the head is not located necessarily on this square right but isn't it the case why are these those de-duplicated because aren't those white squares things which should be labeled by their position or are you not considering i mean in the isomorphism testing the question is do you account for position of the cell no i mean we we don't do that in strings for instance so i don't see why we would in turing machines that's really weird so i mean a turing machine unlike a string which is a finite length a turing machine is conceptually of infinite length so at every step there's an infinite number of blank tapes cells blank tape squares that should be being ingested right so you'd have infinitely many arrows coming into each event right so that gives a a serious weight to the spectators like they're an infinite number of spectators right so getting rid of spectators but why i see for the purpose of the event no no i'm just confused totally confused again in a local world multi-way system the events still contain their spectators yes yes so the events contain the infinite blank tapes right but the tokens do not boy this is slightly head spinning not to think of turing machine heads too much which um okay the the question really as far as i'm concerned is one of the things that we would like to do is to get a good simultaneous representation of physical space and branchial space does this help us with that well i mean that's what the that's why i was kind of excited about these local rancho graphs right because everything about this physical spatial structure is reconstructable from the event ancestry and everything about the branchial structure is reconstructable from the token ancestry and then the coupling between the two comes from the edges that connect both right that's that's that's reasonably exciting yes um okay let me think about that for a second because in effect these are telling you space-like separations right they're telling you anti-chains these orange edges yeah one way you can think about them and then these are telling you anti-chains in the bronchial graph the purple edges and then there's mixtures between the two which are also encoded all right okay okay okay okay now i'm more excited okay so you're saying that hanging around some of the spatial structure they're little pieces of bronchial froth basically but if we think of this thing having a backbone that's the spatial causal connections right this thing here yes but so what that's saying in the picture above is that there is no branchial connection between one side of space and the other is that right at least there's no one-step branch shell connection between one side of space and the other right there's an entanglement horizon here well but this is only recording one-step ancestry right right right so you'd actually you need to see a hypograph version of this which would be a bit complicated well what would that hypograph so the hypergraph version would go back and i mean the hyper graph here would read backwards through a series of token event a token event type a sequence would it not right so and what would you expect to see there so what you would expect to see well ultimately everything goes back to the big bang and everything is related to everything right so what you're seeing here is some things are recently entangled as in recently branchially related and what you're seeing here is the mapping but this is very nice okay so you're seeing the mapping that the correspondence between what is spatially recently connected and what is branchially recently connected right so the obvious question then is this is this question of maximum entanglement speed that which is to make a you know an epr type setup um you need what you would be the way you would make that is you would have something where there's a branchial edge here that spans a lot of spatial you know where instead of if you want to get from here to there you might go spatialized spatialized spatial edge to you know tooling across the universe so to speak or you might hop on one branchial edge that a fair assessment so so the question then is how local the spatial structure we believe to be local so imagine that it made a manifold for example that would be a nice spatial manifold that would represent space then the question is does the branchial structure that sits around that also inherit some kind of manifold type structure right because what we're seeing here is a pretty local branchial you know froth what happens if we go another couple of steps check so get rid of the get rid of all the rendering should we go to six all right so what we're seeing here is that too much there we go okay let's admire this thing for a moment okay so it has some bronchial backbones this is the causal bit right so it's roughly some kind of one dimensionally ish thing spatially and what you're seeing there let's go back up to the what the heck is that let's go back up to the thing stuff which is a bit less exciting um okay so what that's showing is in that picture it seems that the branchial edges are hanging off the spatial edges in effect right there are very few highways they're very few you know wormholes where you can go it is literally it's what the heck is it it's an er equals epr wormhole something like that yeah so for this particular system the the causal structure is kind of the the the spatial causal structure if you like is needed to kind of knit together the whole branchial graph otherwise it would sort of fall apart i get that right i get that but but i mean just to interpret this picture for a second a wormhole in physical space would be assumed that that space that causal structure forms a nice manifold a wormhole in physical space would be a a shortcut from one part of that manifold to another in the causal structure okay a an entanglement wormhole so to speak uh an epr as opposed to an er um would be a place where the branchial edges jump sort of deliver you to tokens that are near uh events that were otherwise difficult to get to in space right so in a sense okay so what would er equals epr mean in this context it would basically mean you can as well you know it's sort of equivalent to jump around in physical space well it is it is rare to have a a fast jump from one part of physical space to another but you can do it and it's rare to have a fast jump in bronchial space but you can do it i mean at some level one interpretation of the er equals epr conjecture is that it's saying that the coloring in the glocal branchial graph is irrelevant right yeah it's saying that it you like from the point of view of causality and locality it doesn't matter whether you're traversing a branchial edge or a causal edge right but what that's saying is that at a microscopic level well okay so that's surprising honestly it's a bit surprising well maybe it isn't surprising maybe it is the reason that quantum mechanics isn't in your face every day because basically most of the time spatial structure is enough to account for what's going on right right and it's only in very extreme conditions that you need to actually have a separation between the two we need to enforce the separation between the two let's understand that for a second so so what you're saying is a quantum effect in this picture would be something where there is a a very separated bronze shield connection that is not achievable through spatial connections i mean that's what it would mean to have a a um [Music] uh locality and entanglement for instance yes and so let's see what would that look like so that that means that in a sense you have to you know you have to beat computational irreducibility it can't look too random it's got to have some systematic prong that connects one part to another that's nice though i mean that's a nice picture of the relationship between causal edges and bronchiologist but let's just go through this one more time so so what we would expect if we did this for a hypograph system the causal edges are genuine representations of they account for the different names of atoms of space yes you know the problem with this is going to be okay wait a minute and this doesn't token deduplicate so this hasn't crushed this hasn't forcibly crushed out space when it comes to the branchial direction right right if we did that you get nothing you just get we're just crushed to nothing so the interesting case is not de-duplicating tokens right but the advantage is we have a little bit less information about the braunshill um [Music] what is the point here so the full we're basically making equivalence classes yeah i mean what we're doing is we're crushing every complete spatial state and saying just give me your tokens right and then we're saying how are your tokens connected i don't care about how the rest of the spatial state is connected i think right right interpretation i want to try this one more time explaining what this picture actually is so each one of those blue nodes is a token no i i don't understand this i'm confused can we look again at one of those um sort of space-time pictures before we get um i mean like one of these yeah one with labels okay i i that's right this is a this is a branchograph so this is at a certain moment in time this is showing ancestral connections between things okay actually here's the thing worth remembering to reconstruct space from a causal graph you can do that by a branchial construction right if you've given a causal graph you can the structure of space in reality is is reconstructed the thing that is real is the causal graph and you're reconstructing the structure of space by looking at essentially branchial a branchial construction on the causal graph that's what is essentially what you're doing here it's not essentially what we're doing here i mean that's just how you that's the definition of an anti-chain that's how defined relative wait a minute was was this defined relative to tokens or is this defined relative to other events this is defined relative to events okay so this is the true event to event spatial reconstruction is that right yep okay so that is a genuine special reconstruction so then the stuff that's around it is is showing the evolution ancestry of the tokens that appear with that are generated by those events that appear within those events right so i think what this is showing is essentially the one-step quantum fluffing of space-time well in this case of space yeah space right taking a specification of a space like hypersurface and it's effectively telling you what ambiguities exist in the tokens as a consequence of the fact that you you know that this system is quantum mechanical one step quantum mechanical right so in principle one could do the same thing with a two-step you know the two steps away tokens and that would then show how space basically disintegrates at a quantum level right which is very interesting okay so let's go through this again so basically the backbone is the instantaneous state of space which no wait a minute now i'm confused by something this is not the single instantaneous state of space because it allows many different histories to have been chosen right right so what actually is it it's not the state of space it's the state of of what of all possible space or what like i i don't i never know what you mean when you just say space right this is showing the the antichain structure of the causal gra of the branchial causal graph or the global causal graph well i understand but the anti-chain structure of the or of a causal graph for the single thread of history is a reconstruction of the instantaneous state of space surface on one branch of history this would be the analog here this is a reconstruction of the modulized space of space like hypersurfaces under explain that okay so you're saying it's a why is it the by modulized space you mean the set of all possible ones knitted together is that right well all possible at that for you know for that particular time step and that particular rule which is yeah so right but i mean but the fact is there's a possible space like hypersurfaces with a topology defined on them which forms a modular space okay but if we were to pick a particular evaluation history we would get a small part of this graph right yep i mean we can see that if we want um yeah let's see it don't mess up what you've already got it's beautiful okay don't destroy what you already have so it's your favorite event selection function or should we just do sequential it's maximally boring yeah we get something totally trivial i mean it's a string what do you expect for what for that's okay that that's showing okay that's a little bit confusing to me because that's showing that's saying that most of the spatial structure is multi-way spatial structure it's a string or almost all of it will be oh that's right this is a string oh boy okay so in fact we're a little bit confused here in fact that thing we were seeing was mostly not in fact ordinary spatial structure again i don't know what you mean when you say ordinary spatial structure it's a graph of causal anti-chains of clockwork i understand that but i mean okay we we have at least the concept that we are currently embedded in ordinary space where you are on one side of the atlantic i'm on the other side of the atlantic there's a definite notion of space we do not have this idea of essentially a quantum space or a you know a multi-way space we we imagine that we're living in a definite space well the single history space like hypersurfaces for a string substitution system are always going to be trivial i i understand that i understand that but i'm trying to understand that that that insofar as we are trying to map what we're seeing here onto our usual interpretation of space and quantum mechanics i'm trying to understand how that works and i agree that the model probably for hypergraph systems this is going to be more you know there might be structure here even in the single history case right if this were model rule there would be structure even in a single history case or at least there could be correct yep and and so then the plot that you made that was previously just that boring branchograph could be a much puffier more interesting thing full of causal edges right that would represent a single branch of history um and which would show for that single branch of history what the kind of neighboring branches were based on branch based on genuine branchial edges right correct i think that's correct you're showing the branchial sort of you're showing the the um you know delta i i wouldn't say delta x right because we're not it's not in it's delta b what you're showing here is the in where b is branchial direction that b is a branchial vector so to speak branch of branchial variable or something is that not correct that what you're seeing here is is sort of in a sense by showing the one-step branchial graph you're showing delta b you know plus minus about not really plus minus but something like plus minus delta b is that the right interpretation i mean in the sense that one is a possible value of a branch like distance which you could call delta b that's what i'm saying yes yes but i mean in general there can be much higher branch-like distances but these are not showing that this is just showing the local this is just showing the neighborhood the branchial neighborhood of this uh multi-spatial space-like structure is that a correct statement that is correct all brand geographs that we've investigated thus far have been one step for rancho graphs no i understand that but but but by by using in the spatial structure it is a delta t but not delta x in ordinary spatial structure in our interpretation of space we're seeing you know our vision of the universe as we see all of space at a particular time yes um well in both cases the in both the i don't know in both the branchial case and in the causal case of course the anti-chain structure is determined by considering what happens one time step back so you're perturbing in t in order to obtain b or x exactly that's how you assess the anti-chains the the that's the same in space time as it is in branchial space no i realize that but the fact that you're doing this the fact that you've got this glocal set up with local tokens individual tokens means that there isn't a symmetry here between space and between causal space so to speak and and branch and bronchial space right the very fact that you're you're you're you're having global events local tokens means that your treatment of the branchial direction is different from your treatment of the spatial direction well only in the sense that the branchial direction has higher granularity if you collapsed all tokens into single states they would be identical they would be compatible i mean all tokens into single states what do you mean you have you have crushed down the states into their individual tokens yeah so every pair of tokens here can be collapsed into a single state in some global multi-way system so there's always enough information present here to do that okay so what you're claiming is that this picture shown below still has the difference that the events don't account for spectator if i'm if i'm not mistaken the events ignore spectators yes yeah but that doesn't imply that um branchial directions and causal directions are non-commutative no i agree i agree but what you're saying is you could draw a version of that picture that has those kinds of events but where the states have been reconstructed and instead of individual tokens there you've just got a bunch of states floating around the the events correct right exactly that might be an interesting picture right so that picture then is it is a a causal graph backboned version of what states are near what causal what what events that make sense now i'm confused by something else if there are two identical events on two different branches of history uh how does this work in a wolf model rule those two events will be ingesting differently numbered atoms of space right so again it depends on the deduplication semantics so in this case where there's no deduplication semantics you can have equivalent events like this one and this one appearing on different branches of history and they don't get merged if you did have a deduplication semantics they might and the thing that's worth us remembering is in a sense deduplication semantics is the story of the observer ultimately the observer is what does the deduplication so in a sense the universe you know i don't think we have to say is deduplicating at all it's just that we conflate all those identical branches and that's how we build up our synthesize our version of reality well these are quite lovely i mean they're clearly you know it's amazing the number of different kinds of graphs that seem relevant it's just absurdly great and i wonder how we should think about that what is this oh wow i got to see this one sorry this is just me playing around with what would combinators look like in a global multi-way context i was like gosh okay let me see if my brain can maintain integrity here what's that so we have s k i x and y as our initial tokens why aren't you just using sk i mean i can do oh yeah okay okay fine but but it's going to matter because if you just use s and k at every well hang on we can get some find some examples if you go to my post you'll find um or you can go there let's try something like this yeah we have some comments on live stream evp is asking what's different from observer and spectators okay totally totally different things the observer is us trying to parse what's happening in these systems and doing things like saying uh those two hypographs are isomorphic so to me they just seem the same that's one thing the other thing is in the low level structure of these systems is there something which is in this hyper graph and elements atoms of space in the hypergraph which is simply not affected by a given event ah let's see um various comments here is the interpretation that branch of space is independent of the dimensions of causal space yes branchial space has no bronchial space is probably exponential dimensional as we expect from it being hilbert space like thing as opposed to something finite dimensional like ordinary physical space um duplication of bronchial collisions boy yeah eli is commenting observers preserving the identity of a real multi-way system yeah the observer in the royal case is yet another level of of of mind-bending stuff uh which i'm just trying to write about and untangle a bit so i may be able to untangle it um try only using s okay what do we got here we've got dominators doing that thing oh no sorry these were the combinations i was showing this is this is general like wolf remodel expression evaluation so this is what group theory looks like when it's glocalized okay let me see if i understand this so we're defining axioms of group theory as just replacement rules so every event is the action of an axiom yeah so g being a composition operator um and then we're starting from the initial expression g of x comma y and evolving it and this is what we get okay so what this is doing effectively is this is applying valid axioms to get expressions which are equivalent to g of x y correct the ordinary multi-way operator system if you look at like this this is you know at some other you can just think of this as being uh you know a proof of all theorems that says gxy yeah the the state and equivalence to gxy um and so i was just curious like what would this look like what you know effectively what would evaluation you know what would um pattern evaluation look like in the global multi-way case and i don't know it's kind of right so let's understand this picture i mean so so in the first approximation okay so if you say any sequence of symbols okay the you would get the free tree so to speak if any sequence of symbols was equivalent correct okay but so in that case what you would see is every node of the tree would just branch out and every node will be an event that just says add symbols to things in this case it's a more constrained setup and i don't understand what this is doing um [Music] so every event is the application of an axiom what is that what's that it's the token deduplicated version what does that mean i mean usually if you take an individual combinator like an s and a k out of its natural habitat it's pretty meaningless like like an element of a string i mean a string that is the big difference between the wolf model rules where the atoms of space have an identity even if extracted from their from their environment so to speak whereas the truth of a string system is that these characters and the string should be labeled by their position just as in a turing machine you say the head is at a certain position on the tape now in the combinator case it is more complicated because you have to label every s and k by its tree part number i believe yes that's why the combinator and operator systems are treated like the the internal workings are yet different again okay but when you do that if you are deduplicating across all levels of the tree and across all part numbers what the heck are you doing i mean it's just you're not de-duplicating across all levels of the tree you're only duplicating at the same level of the tree hence why there are multiple g's here and multiple why okay wait a minute that's just your choice of deduplication semantics yes yeah sure i mean so actually i ultimately i want to replace this deduplicate tokens option with with something that you know a bit like with event selection functions that has several built-in deduplication semantics although in general there's a question of is do is deduplication context independent because that's not i mean deduplication okay the most course deduplication just says if you're an x and there's another x you know conflate them right the next level up would say if you're an x embedded in f of x don't deduplicate relative to one embedded in g of x right so in a sense the duplication what you need i mean it reminds me it's a bizarre way of these of these uh things for doing you know like xml editing and so on where you depend on the you know where instead of just saying replace this with this you say if there is this set of parent elements in the in the tree then do this otherwise don't right right yeah you basically yeah you need to know everything that's further above you in the tree in general in the combinator case it's a tree because the actual representation of the thing itself is uh well in both the operator case and the combinator case right the operator case is also expression trees right but okay so you're doing bi-level deduplication yeah correct that's correct so in a sense if i were to write out the expression trees you are sliding it's as if you have them on rails and you're sliding tokens around on those rails but you're never crossing down to a different level of of rail yep exactly oh my gosh just imagine people in in traditional pure mathematics trying to describe that operation so like here you can you can see it directly right so there's a y here and there's a y here why are they two why they're two different y's this y is contained inside the outer g but not the energy whereas this y is contained inside the outer g and the energy all right just as a little exercise if you were to describe this pure mathematically you have at your disposal traditional pure mathematics category theory type theory what on earth would you say to describe this i mean this is easy to describe in terms of symbolic operations in terms of structural symbolic thinking so to speak but how would you describe something like this this context dependent you know in other words you're making equivalence classes what are they are they are they some level of some type hierarchy that that's being i don't know what the heck it is is that a rhetorical question no it was a question for you because i don't have any idea you just have a d you have a deduplication semantics that depe that depends on the recursion history of each operand but well the recursion depth and in more general yes i understand that and if you're actually dealing with a symbolic language that's a meaningful thing to do but i'm asking the question if you i mean when you look at these things in type theory there is a somewhat reasonable representation of these nested thingies yep first statement yep whereas category theory doesn't really have much to say about these nested thingies is that true statement i don't really understand what that means i mean type theory and category theory are different completely i'm asking in type theory it is fairly natural to talk about these recursive kinds of structures yes but in category theory i don't think that it is why do you say that well what do they correspond to i mean in terms of the when you recurse down like this i don't understand the statement i mean type theory is a model of computation category theory is not i understand that okay fair enough but i'm the question that i'm asking is if you were to try and describe what you just described as a piece of pure mathematics so to speak standard pure mathematics non-symbolic structural or pure mathematics i'm just curious what does pure mathematics i mean you know is there is there some you know because these are things which are somewhat natural to discuss in a symbolic structural way and i'm just wondering whether there's a way to get there from you know does this connect to some version of i don't know what uh you know some kind of interpretation of set theory where you've got things inside things and i don't know i i'm just that was what i was just curious about is there any obvious thing that that that this connects to or is it something that is not well not easily formulated and so this traditional pure mathematical terms i don't even understand the question i mean is it anything that anything that involves a notion of recursion you can express this idea in right so yeah of course set theory class theory type theory all permit recursion um fair enough i i okay i i i think i was making a point which is which is which has gone splat the um no but but uh um i'm just i'm just trying to understand in the case of group theory here for example i'm just i'm just struggling to understand if you were thinking about this in terms of universal algebra for example is there an interpretation of this token granularized oh boy yeah of the sort of token granularized version of of algebra sure it corresponds to a different d looping operation on a loop space ah okay this was the kind of thing i was fishing for okay what is the loop space well they're the primary objects of study in universal algebra wait a minute universal algebra in the maybe i'm 100 years out of date but in the whitehead version of universal algebra what uh you know what one meant was just there are operators they take some number of operands and you are writing down expressions that involve just a a symbolically specified operator f for example and then you're discussing for example proof uses internally it's what multi-way operator systems use it's what our entire theorem improving stack uses which the thing i just described well equational logic is logic that describes universal algebraic theories yes i understand but then what is a loop space well so a loop is the kind of is the minimal case of a universal algebraic theory it's the kind of fundamental object of study in universal algebra how is it defined well it's it's the algebraic structure that's defined by the equality predicate oh i see okay i i mean i remember that there are axioms of loop theory and i that which i never paid much attention to but those that what you're saying is those are the axioms that describe function application equality is that right effectively yeah and then in the the same construction that would give you a you know a stone space for a boolean algebra or heightening algebra or something you there's a analogous procedure that lets you construct a topological space that preserves the structure of the loop and that gives you a loop space and then there's a way that you can so-called d-loop that space which gives you back to something axiomatic from this topological structure and there are multiple ways of de-looping um and i haven't proved it formally but i'm pretty sure that these different representations of multi-way operator systems essentially correspond to different d-looping operations um why is it called a loop space by the way i don't know there is some reason i but i've forgotten what it was okay fair enough but i mean in when you talk about looping and de-looping and you talk about the space so you've got this thing we can represent it symbolically as you know f of x's and y's you know some operator f and then variables x and y and so on and so on and so on what is the way that you make a space out of that like i said the same way you do it in for a stone space you you you treat the symbolic operators as defining enclosure um or containment relations between open sets so you get a topology oh this is this is poking at the top of my reading level here but i suspect it's closely related to the things you've been doing recently with top bosses and so on but we're not probably going to have time to get into that but but can you explain that for a second so you're saying interpret the variables as open sets so when you have okay you have an equational theory uh well actually let's do an explicit example right you have an equational theory like this this defines an order relation okay it's complicated by the fact that okay you you have these kind of you have these loops that appear here but there's a there's a way you've got nothing to do with loop spaces they're just loops yes i i believe they have nothing to do with loop spaces okay you could get rid of them so in effect what you're what you're saying is we've got some collection of expressions and they're partially ordered by this operation yep right um and so that gives so now interpret each one of those expressions as an open set and we we define a rule that says one open set is contained in another if the corresponding expression appears below the other one in this partial order okay so if you do this for the case of a boolean algebra you get there's a you get a particular kind of topological space of stone space uh if you do this for the case of a general kind of universal algebraic theory you get a loop space so by the way the the stone thing there's a what is the stone representation theorem or something which whose main consequences that i know are rather straightforward facts about you know models of of boolean algebra being only two to the n sized and things like that but if you did this for a boolean algebra what what does this picture look like for boolean algebra uh we can find out uh yeah why don't we find out it's it's it's i think i might have put boolean algebra as an example in multi-way operator systems documentation but i can't remember if i was clever i would have put it in examples but i didn't um okay but that's all right we can you just have to have an equivalent i mean you just have to have the the relations i mean you otherwise you're just nanding everything together and then you're reducing it by certain relations um i think that's an overly complicated i would recommend using my pure nand one wouldn't that be easier yeah i think that's easier maybe maybe you want to use if you wanted to to combine a bit more i would recommend you use the one which has the version of those axioms of mine that has um that has commutativity as well yeah promising uh oh gosh well okay if it's more difficult if you have multiple relations just use one relationship no no it's fine i was trying to work out is there an easy way i can do this programmatically but there probably is but i can't be bothered um oh wait no i need formal symbols this would have been a lot easier you need to generate some variables why do you need the unique why don't you just have it hang out of the module uh i guess you could couldn't you it would generally forget the uniques just just to have them be module variables yeah fair point okay let me let me try that to begin with wait this is going to be this is going to get ugly i suspect um but i don't know what what should we use well yeah you can have a dot b i mean and that that should or b dot c x dot y whatever you want to use let me just check that that works okay yes yeah and it's going to look like the free graph for a long time in fact it might look like the free graph for 100 steps right because the shortest non-trivial proof in your axiom system is long that's by the way an interesting fact in its own right that it um okay so let me understand what this is doing so this is applying the axioms to make equivalence expressions and now you're claiming so what does stone's this whole stone business say about this now that we didn't have a two-way version of this suppose it was just a one-way implicative thing yeah then we have some dag and that dag gives us an order relation yep um and so what you could do is just say let's associate every expression with a with an open set and let's say that one open set is contained in another if and only if the one corresponding you know the corresponding expressions precede each other um so then the whole dag defines the open set structure of some topological space yep and that's a stone space and what what's what happens in that space well then and then it allows you to translate you know these algebraic questions about pulling algebra into topological questions about the space and what can you say about that space i don't know okay and i mean the loop spaces what can you say about them i mean do they have any known sort of topological features and things they certainly do and they become but i i don't i'm not an expert on this particular area i know that it becomes the only thing i know is that it becomes relevant um so whether you have a boolean algebra or a heightening algebra uh changes well for obvious reasons i guess it changes the distributivity properties of the topology of the space you construct and i think the same is true with loop spaces what is the distributivity of topology so the the the distributive law that you end up with for a heightening algebra gives you effectively a distributive law for open sets in the corresponding space what is the distributive law for open sets distributive for what intersection union or what yeah okay so distributive law of the underlying algebra implies and with the definition you've given that's not surprising implies a distributive law for intersection union right right and so then it becomes relevant to say whether you're dealing whether you're doing a constructive or non-constructive logic as to what you know whether you have a boolean algebra or a heightening algebra and that so you end up with spaces which have slightly different uh where intersections and unions behave sorry finite intersections and finite unions behave differently right all right we probably are coming to the end of our time here and uh what we got through like some fraction of a topic and a topic here yeah we have we had at least two other really interesting topics that um we can cover another time um right and uh uh i did just want to say like so so there are various reasons for wanting to investigate these global multi-wave systems wanting to understand the relationship between space like hypersurface structure and branchial space structure is one of them but another thing that i'm kind of taking with and the reason i was looking at these combinators and operator system things is i kind of want to know what would a theorem prover look like uh if it didn't have glob you know if it didn't use global states so the the implications that are used in the proof of global but the actual specifications of the intermediate steps are tokenized um that's something i kind of wanted to experiment with at some point well in fact segways to a comment on a livestream from bob saying that tree needs pruning yeah you need a theorem prover a fair improver just you know eventually is going to go down a single path in that tree although i don't know is there a visualization of the operation of the theorem prover that shows what alternative parts it is considering i mean we could construct one that that wouldn't i mean but that's not actually how the theorem prover works it's not searching path systems no i know it's doing this completion thing um but in principle we could do that you know as like an as a sort of pedagogical thing we could show that right the the issue is here so you say go ahead just to explain why i think that's exciting so um suppose that you have some big complicated expression you know that has lots of nesting of g's and things in it for instance and somewhere inside that expression you end up proving you have two expressions and they're basically the same but inside one deep nested inside one of them there's a g x y and deep nested inside the other there's a g word and you prove they're the same now in an ordinary theorem prover that wouldn't constitute a proof that g x y equals g y x but for a local one it would so it lets you so that the class of lemmas that you can extract from a proof of a given size is i i worked out should be exponentially larger for a global theorem improvement than it would be for a global one okay let me paraphrase that statement okay so for combinators for example you've got giant combinator trees in the normal way of representing a combinator tree every sub even if there are identical subtrees they are not they're not those common sub-expressions are not uh are not unified so to speak and in the same way you're saying if you can prove some result about a deeply nested tree it is not useful unless you are only proving results based on the root of the tree right right and so the question is is tree-like separation which you i think have not been a big fan of but anyway i still like the concept of tree-like separation of things which are you know different levels of nesting so to speak in the tree the question is it is a fact about algebra that things are in some sense homogeneous with respect to tree-like separation just like if we've got a list and there is a two a's at different places on the list those a's are considered the same what you're commenting on is that uh a a thing that's f of sub tree sub tree is the same independent of what those sub trees you know there can be relations about that like f of sub 3x sub tree y being the same as f sub tree y sub tree x could be the same independent of what those sub trees are and where that appears right right actually so in a sense ordinary theorem are using basically a multi-way operator system structure which doesn't exploit this tree-like homogeneity right whereas if you had a global theorem prover you could just say well let me take one g-token 1x token one white token and prove something about this one g token this one x token this one y token and ignore everything else um sorry let me understand that because isn't it the case that the downward tree is still covered that is the thing i just described just by virtue of variables f of x y equals f of y x those variables x and y can be arbitrary trees what's not covered is that that f x y could itself appear as a sub element of another tree no exactly it's where the so the case i'm talking about is the case where the theorem you want to prove is an equivalence between deeply nested subtrees right so so in a sense hacks like hyper module paramodulation all those kinds of things are ways of extracting little pieces of of of uh of expressions and you know gluing them together in different ways and what i think you might be getting towards is the generalization of that that of those hacks something like that yeah right because i mean what what you're saying is instead of those hacks take pieces of expressions and say how to you know how to sort of grind them up and combine them and so on but what you're saying is if i've got two trees and i want to yeah the question is how does one tree read on another tree and it could read by saying you go down to a sub-tree to look at how this tree applies to that tree yeah i i claim what you've got is a which would be very nice because i mean right now in theorem proving technology they're you know i don't even know does our their improvement use any of this power modulation stuff yes it uses i mean so power modulation is the thing that allows you to effectively interconvert between um equalities and pattern rules in internally isn't that something like what we're talking about here yes yeah this is a generalized version of that okay fine all right but as you kind of point out it's it's a version of that of like rule orientation or paramodulation but which applies to subtrees as uh you know in addition to just whole trees this seems like a very interesting idea jonathan i mean it seems like something where uh i mean in the history of of theorem proving like people invented these things in the 1970s and 80s and things these various kinds of you know grind up expression hackery type things yeah well so okay so so then what's what's mounting here is a collection of kind of more principled approaches to theorem proving that there's what you're developing basically and then this is one you've also got the energy heuristic have you done the have you used the energy heuristic yet uh it depends what you mean by i mean we used it in a bunch of the zx calculus stuff i mean that's the closest to used in anger as far as i know right but have we but like there's no version of find equation or proof that uses the energy heuristic no i've been meaning to put one into the function repository at some point um is it easy to do should be um yeah should be don't you have to get into the insides of the of the of the theorem improver to be able to do it no because so there's um okay so without wanting to go too into the weeds internally we use it but there's a call to this world meister c code that we use for doing fast there are two things we use it for doing fast confluence checking and fast critical pair analysis and so we can call critical pair analysis basically on demand and so all i have to do is just modify the the weight that rather than calling that function repeatedly we only call it when the energy heuristic says that we should call it um it doesn't actually require any modification of the of the back end c code just changes how so so it's a choice of what the ranking of what counts as a plausible critical pair to try is right exactly exactly so so the ordinary approach is some kind of greedy algorithm where you just say give me all the critical pairs and i'll resolve all of them i'll keep going until we reach confluence here what we want to do is say give me some critical pairs and i'll only resolve the ones which which lie on branches that have high calls or connectivity um right but i mean the the usual whatever unfailing completion type thing involves doing all of the critical pairs right yes and so this is in part a ranking so in a sense it's a ranking algorithm for critical pairs that prioritizes some over others and unlike in global basis there's all these crazy things like you know reverse lexicographic degree order et cetera et cetera et cetera and this is is there an analog of that in the theorem proving world well yeah i mean so so there are people i mean as you well know the um brook berger's algorithm for grobner basis reduction and the clintonics completion algorithm are the same they're one they're both instances of this more general scheme um and so the same like you know reverse like graphic orderings that work for group and a basis reduction in book burger's algorithm you can also apply for kinetics completion but at some level these orderings of effectively what deductions to apply when are a little bit arbitrary and what we're trying to devise is one which is not so arbitrary right but i mean in a sense it's the same heuristic as the subdivide space when space has higher curvature because that is what makes the curvature it's as a way to do pdes this is the analog of that in essentially in bronchial space yeah it's a it's an adaptive mesh refinement for theorems exactly yes but otherwise thought of as just as one of them is adaptive mesh refinement in physical spaces adaptive mesh refinement and branchial space right right it's interesting but so so that's one meta approach then you're dealing with the fact that the element the entities in their improving are tree-like rather than being uh list-like so to speak and then the third thing is this proof-to-proof transformation stuff that you've been talking about yeah right that's less well-developed i want to i need to invest more time in that because it is interesting i mean but so the concept there is just as the state of uh just as an expression is a symbolic thing on which you can do transformations so is a proof right so if i take we need to add some more notable theorems i'm going to make a note of suggesting that as a project and well i mean the problem i had so i think all these are ones that i added the problem is that it's quite hard to find universal algebraic theorems if you if you sort of like no i understand but particular groups are very common but but theorems about groups in general are kind of rare um anyway so the underlying idea is that like a proof graph is you know it's some combinatorial structure a bit like a zx diagram or a bit like a generalized you know tagged wolf or model system onto which you can then apply replacement operations yep so there are things like you know uh if you have two substitution lemmas and they're adjacent we can merge them because essentially by applying genson's cut elimination lemma in reverse um similarly if you know if there's a substitution number and then we can we can split it there are cases in which we can split it into two by applying genomes called elimination remember and there are similar things we can do like if you have a critical pair and it has two ancestors then you can um there are cases in which you can merge the ancestors and things like that um so there's a collection of uh purely combinatorial rules that you can apply to these proof graphs that give you equivalent proofs of the same proposition and the idea is to do theorem so you can construct a multi-way system of that and then the idea is to do theorem proving on that multi-way system so you're proving equivalences between proofs and this gives you ways of doing sort of right but the question of i mean the thing which is interesting which is not done by traditional theorem provers is the go to the simplest one i mean that is to given given two proofs what you're saying is you may have a way to establish the path between those proofs right and it will be already interesting to look at how far is it between two proofs because you might very well find that there are distinct proofs that have a big topological hole in the space and proof space in between those proofs and you might detect that by seeing that it's a long way from this proof to that proof right right and in effect that would be telling us that the if the higher inductive type has you know if we take its homotopic interpretation that that then that current that the corresponding type space has a non-trivial genus yes we would be we'd be directly probing right which is something that homotopy type theory has not been able to look at right well certainly not concretely um do they know even theoretically what genuses of of types of those things alike yeah people have constructed higher inductive types that whose spaces have non-trivial genus but they're very artificial constructions they're certainly not the kinds of things you would expect you know to see in actual meta mathematics um right i mean look the main thing that i think is interesting is this notion of destructive interference between proofs that is you've got two proofs where in an intermediate stage they are utterly incompatible yeah which i think is the analog of photons go through two slits and um i'm not quite sure it's totally it's similar to that type of situation i think um yeah there's some kind of branchial event horizon yes but i mean so so but that's one thing is to understand the structure of proof space which seems highly interesting to me the other thing would be to actually take a proof and simplify it i mean norma gill has been trying to take my billion algebra thing you know generated automatically generated proof i think he's gotten it down by some 20 30 or something by by i think essentially by by hand feeding certain you know guiding a theorem prover about what to do but the question would be is there a principal way i mean it reminds me a lot of things like compressing neural nets and things like this where you're basically asking the question can you make an equivalent well or program simplification i mean okay you know to be fair we have never succeeded in doing program simplification we've tried it for for decades and we've never really got very far in it actually it makes me wonder i mean we have got a whole bunch of rules for program simplification have we ever tried running though i guess i guess we don't have a find you know a simplify based on find equation of proof what would that look like i mean find equation of proof just says is it all equivalent to true well so this is what we did this is precisely what we did for the zx calculus case right this is how the the circuit simplifier works and the idea here would be to apply the same you know exactly the same code but for proof graphs rather than for zx diagrams using gradient decentralize it doing something more fancy it's not using gradient descent no it's it's um it's you it's not using anything really apart from this causal edge heuristic but it must no but it must pick a way down a way to get smaller i mean okay in that sense it's using gradient it's just it's picking the one that it's picking whatever minimizes the vertex count yeah right but that's not obviously correct i mean it might be the case you have to go over a hill to get down the hill well sure yeah you can always get stuck in local minima right but so that's a different issue but but it could very well be that you can get a long way just by essentially following gradient descent in this proof space in this proof simplification space i mean there's no evidence is there any evidence okay in fact the very claim that very thing we've just been talking about about the homotopic structure of proof space is going to tell you where the gradient percent is going to work because if if proof space is very uniform and there aren't any big holes in it and things then you can expect the gradient descent stands a chance if the thing is full of holes then gradient descent is going to get stuck well the homology if the space is only one case in which gradient descent can fail right we also don't know anything about the geometry um that's true that's partly good but but on the other hand by by studying proof to proof transformations we could reconstruct your geometry as well absolutely i mean in a sense what what you would be doing then is to make a graph in which every node of the graph is a proof and you can say so for example you can you have a bronze shield proof graph so to speak i don't know is it a branch i'm not sure what it is it's a nearest neighbor proof graph given the space given a space of proofs you can look at which proofs are you know what are the neighboring proofs in proof space based on these transformations i would think yep yeah that will be interesting to know and um what's your guess about the structure of that space what what kind of a space is it i'm i await data i don't know i i i'm curious to know just how dependent it is on the underlying theory um it doesn't give a damn about this well yeah i bet all it cares about is if it cares about anything it cares about um things about the way that substitution dilemmas combine and things like that yeah i mean because so far if we look at our proof graphs you know can you pick that what was that one that's a group group theory i don't have a clue that that's good theory can you identify you know looking at the you know looking at the proof graph can you say anything about what the underlying theory is so you can infer some things by how many critical parallels there are because there are some theories that are closer to confluence than others um but apart from that i have no way of telling right which are closer to decidability effectively yeah well i mean they're all yeah yeah i guess that's one way to think about it right i i have to disappear because we actually got another live stream at a completely different topic coming up in about 15 minutes um and uh well very interesting don't forget to save that notebook jonathan okay we should put it in our collection uh what do you want to call it local local multi-wave dash 01 or something i shall add it to the archive yeah please um very interesting look forward to seeing all those functions um uh somewhere okay we'll see everybody at uh another time

Channel: Wolfram

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Id: uWLWm4espfQ

Length: 156min 50sec (9410 seconds)

Published: Tue Oct 26 2021