The Problem of Distributed Consensus

Distributed Consensus with Cellular Automata & Related Systems Research Conference

In preparation for a convention entitled “Distributed Consensus with Cellular Automata & Related Systems” that we’re organizing with NKN (= “New Type of Community”) I made a decision to discover the issue of distributed consensus utilizing strategies from A New Kind of Science (sure, NKN “rhymes” with NKS) in addition to from the Wolfram Physics Project.

A Easy Instance

&#10005
BlockRandom[SeedRandom[77]; 
 Module[{pts = 
    RandomPointConfiguration[HardcorePointProcess[0.09, 2, 2], 
      Rectangle[{0, 0}, {40, 40}]]["Points"], clrs}, 
  clrs = Desk[
    RandomChoice[{.65, .35} -> {Hue[0.15, 0.72, 1], Hue[
       0.98, 1, 0.8200000000000001]}], Size[pts]]; 
  Graphics[{EdgeForm[Gray], 
    Desk[Style[Disk[pts[[n]]], clrs[[n]]], ]}]]]

Take into account a set of “nodes”, every considered one of two attainable colours. We need to decide the bulk or “consensus” shade of the nodes, i.e. which shade is the extra frequent among the many nodes.

One apparent technique to search out this “majority” shade is simply sequentially to go to every node, and tally up all the colours. But it surely’s probably far more environment friendly if we will use a distributed algorithm, the place we’re working computations in parallel throughout the assorted nodes.

One attainable algorithm works as follows. First join every node to some variety of neighbors. For now, we’ll simply choose the neighbors based on the spatial format of the nodes:

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ConsensusState[points_, colors_, nn_ : 5] := 
 NearestNeighborGraph[points, nn, DirectedEdges -> True, 
  DistanceFunction -> EuclideanDistance, 
  VertexStyle -> MapThread[Rule, {points, colors}], 
  VertexSize -> 0.75, EdgeStyle -> \!\(\*
TagBox[
StyleBox["Gray",
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)];

BlockRandom[SeedRandom[77]; 
 Module[{pts = 
    RandomPointConfiguration[HardcorePointProcess[0.09, 2, 2], 
      Rectangle[{0, 0}, {40, 40}]]["Points"], clrs},
  clrs = 
   Desk[
    RandomChoice[{.65, .35} -> {Hue[0.15, 0.72, 1], Hue[
       0.98, 1, 0.8200000000000001]}], Size[pts]]; 
  ConsensusState[pts, clrs]]]

The algorithm works in a sequence of steps, at every step updating the colour of every node to be regardless of the “majority shade” of its neighbors is. Within the case proven, this process converges after a number of steps to make all nodes have the “majority shade” (which right here is yellow)—or in impact “agree” on what the bulk shade is:

&#10005
ConsensusState[points_, colors_, nn_:5] := NearestNeighborGraph[points,nn,DirectedEdges->True, DistanceFunction->EuclideanDistance,VertexStyle -> MapThread[Rule, {points, colors}], VertexSize -> 0.75, EdgeStyle -> Grey]
NodeDependencies[points_, nn_:5]:= Desk[n-> Flatten[Map[Position[points, #]&,VertexOutComponent[NearestNeighborGraph[points,nn, DirectedEdges -> True, DistanceFunction->EuclideanDistance], factors[[n]], {1}]]], n, Vary[Length[points]]]
SynchronousStepNewColors[dependencies_, colors_]:= 
Flatten[Map[With[neighbors = Sort[Counts[Part[colors, Last[#]]], Better],
If[DuplicateFreeQ[Values[neighbors]],
First[Keys[neighbors]], 
colours[[First[#]]]]]&, dependencies]]
GraphicsGrid[Partition[BlockRandom[SeedRandom[77];Module[{pts = RandomPointConfiguration[HardcorePointProcess[0.09, 2, 2], Rectangle[{0, 0}, {40, 40}]]["Points"],clrslist, highlights},
clrslist=NestList[SynchronousStepNewColors[NodeDependencies[pts], #]&, Desk[RandomChoice[{.65,.35}->{Yellow,Red}], Size[pts]], 7];
MapIndexed[With[{colors = #}, Graph[ConsensusState[pts, colors],ImageSize->150]] &,clrslist]
]],4],ImageSize-> 600]

It is a easy instance of a distributed consensus algorithm in motion. The problem we’ll focus on right here is to search out essentially the most environment friendly and strong such algorithms.

The Background

In any decentralized system with computer systems, individuals, databases, measuring units or the rest one can find yourself with totally different values or outcomes at totally different “nodes”. However for all types of causes one usually needs to agree on a single “consensus” worth, that one can for instance use to “decide and go on to the following step”.

Blockchains are one instance of techniques that want this type of consensus to “end every block”. Conventional blockchains obtain consensus by what quantities to a centralized mechanism (although there are a number of “decentralized” copies of the blockchain that’s produced).

However there at the moment are beginning to be distributed analogs of blockchains that want distributed consensus algorithms. And the principle inspiration for the algorithms being developed are cellular automata (and to a lesser extent spin techniques in statistical mechanics).

One concern is to make the algorithm as environment friendly as attainable. One other is to make it as strong as attainable, for instance with respect to random noise—or malicious errors—launched at or between nodes.

The quantity of random noise will be regarded as one thing like a temperature. And a minimum of in sure circumstances there is usually a “part transition” in order that under a sure “temperature” there will be zero impact on the consensus output—implying robustness to a sure stage of noise.

A few of what occurs will be studied utilizing strategies from normal equilibrium statistical physics. However normally one has to take account of the time dependence or evolution of the system, resulting in one thing like a probabilistic mobile automaton (intently associated to directed percolation, dynamic spin techniques, and so forth.).

As I’ll discuss below, within the early days of computing, there was nice curiosity in synthesizing dependable techniques out of unreliable elements. And by the Nineteen Sixties there was research first of neural nets after which of mobile automata with probabilistic components. And a few shocking outcomes had been obtained that confirmed that mobile automata might be arrange that will be strong with respect to a sure nonzero stage of noise.

One function of mobile automata is that their components are all assumed to be organized in a particular array, and to be up to date in parallel “on the identical time” in a sequence of steps. For a lot of sensible purposes, nevertheless, one as a substitute needs components which can be related in some form of graph (which will even be dynamic), and which can be on the whole up to date asynchronously, in no explicit order.

The straightforward instance we gave above is a graph cellular automaton: the connections between components are outlined by a graph, however the updates are all executed synchronously at every step. Previously, it’s been troublesome to investigate the extra common setup the place there isn’t any inflexible notion of both area or time. However that is precisely the setup in our new Physics Project, and so there’s now the potential to make use of its formalism and outcomes (in addition to instinct imported from physics) to make additional progress.

Deterministic Mobile Automata

To begin getting some instinct for the issue of distributed consensus, let’s take into account the next quite simple setup. Now we have a line of cells, every with considered one of two attainable colours. Then we replace the colours of those cells in a sequence of steps, based mostly on a neighborhood rule which is dependent upon neighboring cells. This method is a one-dimensional mobile automaton—of the sort that I started studying more than 40 years ago.

We think about that the preliminary situation includes a fraction p of crimson cells. We would like all of the cells to show crimson if p > , and all of them to show yellow if p < . The obvious rule that may obtain this could simply substitute every cell by the bulk shade in its neighborhood (rule 232 in my numbering scheme):

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RulePlot[CellularAutomaton[232], 
 ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
   1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange]

Right here’s what rule 232 does beginning with 70% crimson cells in a “random configuration”:

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BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  CellularAutomaton[232, RandomChoice[{.7, .3} -> {1, 0}, 120], 20], 
  Mesh -> True, 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange]]

As we will see, it manages to realize slightly “native consensus”, however in the end it’s not profitable at reaching a “international consensus” through which all cells are the identical shade.

And we would think about that there’d be no rule for a 1D deterministic mobile automaton that will result in international consensus (or have the ability to clear up the “density classification downside” of deciding whether or not the density of preliminary crimson cells is above or under 50%). But it surely seems that this isn’t true. And for instance in 1978 the next “radius 3” rule (operating on size-7 neighborhoods) was constructed (and we’ll name it the “GKL rule”):

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{l3_, _, l1_, c_, r1_, _, r3_} :> 
 If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0]

Right here’s what this rule does with 60% crimson cells:

&#10005
BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  CellularAutomaton[{FromDigits[
     Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
       If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 3}, 
   RandomChoice[{.6, .4} -> {1, 0}, 300], 60], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

And right here’s what it does with 40% crimson cells:

&#10005
BlockRandom[SeedRandom[569]; 
 ArrayPlot[
  CellularAutomaton[{FromDigits[
     Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
       If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 3}, 
   RandomChoice[{.4, .6} -> {1, 0}, 300], 60], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

In each these circumstances, the rule efficiently achieves “international consensus”. And actually one can show that this rule will all the time do that, a minimum of after sufficiently many steps. Right here’s a plot of how the density evolves as a operate of time for various preliminary densities:

&#10005
information=ParallelTable[If[p==.5,Nothing,MeanAround/@Transpose[Table[Mean/@CellularAutomaton[{FromDigits[Tuples[{1,0},7]/. {l3_,_,l1_,c_,r1_,_,r3_}:>If[If[c==0,r1+r3,l1+l3]+c>=2,1,0],2],2,3},RandomChoice[{p,1-p}->{1,0},5000],200],200]]],{p,.3,.7,.05}];
ListLinePlot[MapThread[Callout[#1, Row[{Style["p",Italic],"=",#2}]]&,information, Circumstances[Range[.3, .7, .05], Besides[0.5]]]]

And what we see is that there’s what appears like a part transition: for preliminary density p < 0.5, the ultimate density is strictly 0, whereas for preliminary density p > 0.5, it’s as a substitute precisely 1.

What occurs exactly at p = 0.5? In a way the mobile automaton “can’t make up its thoughts” and on an infinite line it generates an infinite nested sequence of domains that alternate between 0 and 1:

&#10005
BlockRandom[SeedRandom[24425]; 
 ArrayPlot[
  CellularAutomaton[{FromDigits[
     Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
       If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 3}, 
   RandomChoice[{.5, .5} -> {1, 0}, 1000], 500], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

This nested construction is typical of what’s seen in critical phenomena in statistical physics, and in reality mobile automata like this are the very easiest examples of “true” phase transitions that I do know. (Like different part transitions, these don’t turn into “sharp” besides in infinite techniques. In typical statistical mechanics one doesn’t get part transitions in 1D, however that’s a consequence of the idea of microscopic reversibility, which doesn’t apply to mobile automata like this.)

So what different mobile automaton guidelines obtain consensus like this? There are not any radius-1 guidelines that work. And if one searches all 232 radius-2 guidelines (as I did for A New Kind of Science), the perfect one finds are a handful of examples that obtain “approximate consensus” within the sense that almost all, although not all, of the cells go to the “majority worth” (that is the r = 2 rule 4196304428, for p = 0.6):

&#10005
BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  CellularAutomaton[{4196304428, 2, 2}, 
   RandomChoice[{.6, .4} -> {1, 0}, 500], 200], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

By the best way, amongst radius-1 guidelines, there’s rule 184 (usually used as a primary mannequin of highway site visitors circulate), which doesn’t obtain consensus on “total density”, however does achieve this with respect to left- and right-moving stripes, right here with the nested sample generated when p = 0.5:

&#10005
BlockRandom[SeedRandom[567]; 
 ArrayPlot[CellularAutomaton[184, RandomChoice[{1, 0}, 400], 180], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

What about “attaining consensus sooner”? Right here’s a comparability of our authentic GKL rule with one other radius-3 rule (found by genetic programming strategies) whose common consensus time is shorter:

&#10005
Column[BlockRandom[SeedRandom[24125]; 
    ArrayPlot[
     CellularAutomaton[#, RandomChoice[{.48, .52} -> {1, 0}, 800], 
      180], 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False, 
     ImageSize -> ]] & /@ {{339789091192587366278221041213531750560, 2,
     3}, {339841014953466429132970652455676805280, 2, 3}}]

It’s not recognized on the whole what the “quickest” radius-3 rule is. The 2 guidelines above have the function that they “do their job” in a reasonably “simple-looking” means. However there are additionally guidelines like the next that do their job in a “extra ornate” means:

&#10005
Column[BlockRandom[SeedRandom[24125]; 
    ArrayPlot[
     CellularAutomaton[#, RandomChoice[{.48, .52} -> {1, 0}, 800], 
      240], ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False, 
     ImageSize -> ]] & /@ {{337607298446901146542393000444934784552, 2,
     3}, {338557163619953682141694933300561896488, 2, 
    3}, {313421633154342960352882914658469183496, 2, 3}}]

Human-engineered guidelines (like the primary one above) nearly inevitably work in easier and extra “comprehensible” methods. However expertise elsewhere (corresponding to with optimal sorting networks) means that optimum guidelines will usually be ones that don’t look easy of their conduct, and that may’t realistically be constructed by normal engineering strategies, and primarily simply need to be discovered “experimentally” by looking out the computational universe of attainable guidelines.

A notable function of significantly the sooner guidelines we checked out is that they present a small variety of kinds of very distinct “domains” with particular partitions or boundaries between them. And in some ways such partitions will be regarded as being like localized structures, “defects” or “particles”. However for our functions right here what tends to be vital is whether or not these particles transfer round, and whether or not they annihilate one another to depart a uniform “consensus” ultimate state.

Within the easy majority rule it’s inevitable that there are static area partitions:

&#10005
BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  CellularAutomaton[232, RandomChoice[{.6, .4} -> {1, 0}, 300], 60], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange]]

The reason being that as quickly as a website is bigger than the mobile automaton neighborhood, a cell on the boundary of the area will inevitably see a balanced variety of cells of every shade on the 2 sides of the boundary. So the cell itself will act as a “tie breaker”, and can all the time resolve to remain its personal shade—thereby making the area boundary keep as it’s.

So what if we have now a longer-range rule, that samples extra distant cells? With range-2 (i.e. a 5-cell neighborhood) “flawed domains” with widths under 4 disappear:

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majrn[n_] := 
 FromDigits[If[Total[#] > n/2, 1, 0] & /@ Tuples[{1, 0}, n], 2]

BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  CellularAutomaton[{majrn[5], 2, {{-2}, {-1}, {0}, {1}, {2}}}, 
   RandomChoice[{.6, .4} -> {1, 0}, 300], 60], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange]]

Issues work a bit higher if the cells being sampled aren’t adjoining, however are for instance within the sample :

&#10005
majrn[n_] := 
 FromDigits[If[Total[#] > n/2, 1, 0] & /@ Tuples[{1, 0}, n], 2]

BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  CellularAutomaton[{majrn[5], 2, {{-3}, {-1}, {0}, {2}, {4}}}, 
   RandomChoice[{.6, .4} -> {1, 0}, 300], 60], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange]]

However no finite-size sampling with the pure majority rule will take away all domains. What concerning the GKL rule? This rule really solely samples 5 cells, however its “extremities” are at distance 3. So can we “enhance” it by having it pattern extra distant cells?

Right here’s a comparability of some circumstances (the primary one is the unique):

&#10005
GraphicsGrid[
 Partition[
  Labeled[BlockRandom[SeedRandom[567]; 
      ArrayPlot[
       CellularAutomaton[{4177065992, 2, List /@ #}, 
        RandomChoice[{.6, .4} -> {1, 0}, 300], 100], 
       ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
         1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange, 
       ImageSize -> 300]], Model[#, 11]] & /@ {{-3, -1, 0, 1, 
     3}, {-5, -1, 0, 1, 5}, {-3, -1, 0, 1, 5}, {-3, -1, 0, 2, 4}}, 2]]

Right here we’ve solely mentioned mobile automata with two attainable colours for every cell. We may additionally take into account guidelines that contain different “helper” colours that both disappear earlier than the ultimate state is reached, or outline further consensus states.

However Does It All the time Work?

We’ve seen that there are 1D mobile automata that—a minimum of within the examples we’ve checked out—obtain “majority consensus”. However given a specific rule, will it all the time attain consensus, or are there exceptions?

As a primary method to get a well-defined model of that query, we will take into account finite cellular automata, say with a complete of n cells, and cyclic boundary situations. There are a complete of twon attainable configurations on this case, and we will signify all attainable paths of evolution of the mobile automaton utilizing a state transition graph.

Right here’s the graph for the GKL rule we mentioned above, for the case n = 5. Every node within the graph is coloured based on whether or not its “red-cell fraction” is above or under . And what we see on this case is “excellent density classification” or “excellent consensus”, with all states accurately resulting in all-red or all-yellow states:

&#10005
With[{n = 5}, 
 Graph[# -> 
     CellularAutomaton[{339789091192587366278221041213531750560, 2, 
        3}][#] & /@ Tuples[{1, 0}, n], 
  VertexStyle -> (# -> 
       If[Mean[#] > 1/2, Hue[0.98, 1, 0.8200000000000001], Hue[
        0.15, 0.72, 1]] & /@ Tuples[{1, 0}, n]), 
  GraphLayout -> {"VertexLayout" -> "LayeredDigraphEmbedding", 
    "PackingLayout" -> "LayeredLeft"}, EdgeStyle -> \!\(\*
TagBox[
StyleBox[
InterpretationBox[
RowBox[{"CloudGet", "[", "\"\<http://wolfr.am/VtlQl86f\>\"", "]"}],
Hue[0.75, 0, 0.35],
Editable->False,
Selectable->False],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)]]

However as quickly as we glance, for instance, at n = 7, we instantly see an issue:

&#10005
With[{n = 7}, 
 Graph[# -> 
     CellularAutomaton[{339789091192587366278221041213531750560, 2, 
        3}][#] & /@ Tuples[{1, 0}, n], 
  VertexStyle -> (# -> 
       If[Mean[#] > 1/2, Hue[0.98, 1, 0.8200000000000001], Hue[
        0.15, 0.72, 1]] & /@ Tuples[{1, 0}, n]), VertexSize -> .4, 
  GraphLayout -> {"VertexLayout" -> "LayeredDigraphEmbedding", 
    "PackingLayout" -> "Layered"}, EdgeStyle -> \!\(\*
TagBox[
StyleBox[
InterpretationBox[
RowBox[{"CloudGet", "[", "\"\<http://wolfr.am/VtlQl86f\>\"", "]"}],
Hue[0.75, 0, 0.35],
Editable->False,
Selectable->False],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)]]

The states

&#10005
ArrayPlot[
   CellularAutomaton[{FromDigits[
      Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
        If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 3}, #,
     4], ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
     1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False, 
   Mesh -> True, MeshStyle -> Orange, ImageSize -> Tiny] & /@ {{0, 0, 
   1, 0, 0, 1, 1}, {0, 0, 1, 1, 0, 1, 1}}

and their cyclic variations “get caught” and don’t obtain consensus. At measurement 11 there’s one other concern: now a number of states that ought to have achieved “consensus 1” really go to “consensus 0”:

&#10005
With[{size11 = 
   Graph[# -> 
       CellularAutomaton[{339789091192587366278221041213531750560, 2, 
          3}][#] & /@ Tuples[{1, 0}, 11], 
    VertexStyle -> (# -> 
         If[Mean[#] > 1/2, Hue[0.98, 1, 0.8200000000000001], Hue[
          0.15, 0.72, 1]] & /@ Tuples[{1, 0}, 11]), VertexSize -> .4, 
    EdgeStyle -> \!\(\*
TagBox[
StyleBox[
InterpretationBox[
RowBox[{"CloudGet", "[", "\"\<http://wolfr.am/VtlQl86f\>\"", "]"}],
Hue[0.75, 0, 0.35],
Editable->False,
Selectable->False],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)]}, 
 Grid[{{Show[Graph[# , AspectRatio -> 1/3], 
       ImageSize -> ] &[
     WeaklyConnectedGraphComponents[size11][[1]]], 
    With[{stuck = 
       Catenate[
        EdgeList /@ Drop[WeaklyConnectedGraphComponents[size11], 2]]},
      Present[Graph[stuck, 
       VertexStyle -> (# -> 
            If[Mean[#] > 1/2, Hue[0.98, 1, 0.8200000000000001], Hue[
             0.15, 0.72, 1]] & /@ (First /@ caught)), 
       EdgeStyle -> \!\(\*
TagBox[
StyleBox[
InterpretationBox[
RowBox[{"CloudGet", "[", "\"\<http://wolfr.am/VtlQl86f\>\"", "]"}],
Hue[0.75, 0, 0.35],
Editable->False,
Selectable->False],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\), 
       GraphLayout -> {"VertexLayout" -> "LayeredDigraphEmbedding", 
         "PackingLayout" -> "NestedGrid"}], 
      ImageSize -> ]]}, {Present[
       Graph[# , AspectRatio -> 1/3], 
       ImageSize -> ] &[
     WeaklyConnectedGraphComponents[size11][[2]]], SpanFromAbove}}, 
  Alignment -> ]]

The states that “get to the flawed consensus” right here all turn into cyclic variations of the next

&#10005
ArrayPlot[
   CellularAutomaton[{FromDigits[
      Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
        If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 3}, #,
     7], ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
     1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False, 
   Mesh -> True, MeshStyle -> Orange, ImageSize -> Tiny] & /@ {{1, 0, 
   1, 1, 0, 0, 0, 1, 1, 0, 1}, {0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0}}

the place within the first case there are 6 cells and 5 , but the ultimate state is all , and within the second case it’s the opposite means round.

And it seems that there’s really a common downside: one can show that there’s no rule that may completely obtain “majority consensus” on a finite array with cyclic boundary situations.

What about on an infinite array? Right here it’s attainable to realize “excellent majority consensus” for all however a set of “particular preliminary situations” with measure 0. An instance of such a “particular preliminary situation” is an infinite repetition of both of the 2 blocks proven above. These preliminary situations—as a substitute of going to consensus—will simply stay fastened with time.

If preliminary situations are generated “at random”, with the worth of every cell being chosen based on sure fastened chances, then there’s successfully zero chance of getting one of many “exception” preliminary situations. And although the “tapers” is likely to be arbitrarily lengthy, there’s no likelihood of not ultimately reaching a consensus state.

However this conclusion is dependent upon the concept that preliminary situations are actually generated “at random”. If, for instance, they had been generated by a particular program, then although the preliminary situations may appear “statistically random” with respect to sure assessments, it doesn’t imply that they received’t give particular weight to the “distinctive” preliminary situations.

Past One Dimension

In a single dimension one can clarify the truth that sure configurations “get caught” and don’t obtain consensus by saying that in 1D results can’t “get round one another”. However in 2D there isn’t any such constraint.

So then what concerning the “pure 2D majority rule” (totalistic code 56):

&#10005
RulePlot[CellularAutomaton[<|"Dimension" -> 2, "Neighborhood" -> 5, 
   "TotalisticCode" -> 56|>], 
 ColorFunction -> (Mix[{[email protected][0.15, 0.72, 1], 
      [email protected][0.98, 1, 0.8200000000000001]}, #] &), 
 MeshStyle -> Orange]

Beginning say from 30% 1s we once more see that issues get caught:

&#10005
BlockRandom[SeedRandom[23424]; 
 ArrayPlot[#, 
    ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
      1 -> Hue[0.98, 1, 0.8200000000000001]}, ImageSize -> 120, 
    Body -> False, Mesh -> True, MeshStyle -> Orange] & /@ 
  CellularAutomaton[<|"Dimension" -> 2, "Neighborhood" -> 5, 
    "TotalisticCode" -> 56|>, 
   RandomChoice[{.3, .7} -> {1, 0}, {30, 30}], {{0, 4}, All}]]

Right here is the corresponding evolution proven in 3D, with time happening:

&#10005
BlockRandom[SeedRandom[23424]; 
 ArrayPlot3D[#, 
    ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
      1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange] &@
  CellularAutomaton[<|"Dimension" -> 2, "Neighborhood" -> 5, 
    "TotalisticCode" -> 56|>, 
   RandomChoice[{.3, .7} -> {1, 0}, {30, 30}], {{0, 10}, All}]]

However right here is one other rule (9-neighbor totalistic code 976):

&#10005
GraphicsGrid[
 Partition[
  BlockRandom[SeedRandom[23424]; 
   ArrayPlot[#, 
      ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
        1 -> Hue[0.98, 1, 0.8200000000000001]}, ImageSize -> 80, 
      Body -> False] & /@ (CellularAutomaton[<|"Dimension" -> 2, 
       "Neighborhood" -> 9, "TotalisticCode" -> 976|>, 
      RandomChoice[{.45, .55} -> {1, 0}, {30, 30}], {{0, 17}, All}])],
   7]]

And now what we see is that on this case blob-like domains of the “minority shade” get left over, however regularly get smaller. We will see the phenomenon in 3D:

&#10005
BlockRandom[SeedRandom[23424]; 
 ArrayPlot3D[#, 
    ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
      1 -> Hue[0.98, 1, 0.8200000000000001]}] &@(CellularAutomaton[<|
     "Dimension" -> 2, "Neighborhood" -> 9, "TotalisticCode" -> 976|>,
     RandomChoice[{.45, .55} -> {1, 0}, {50, 50}], {{0, 40}, All}])]

Taking a look at a spacetime slice within the heart, and letting extra distant cells “recede into the fog”, we see what appears like “diffusive” conduct, with area partitions in impact executing random walks that eventually annihilate:

&#10005
BlockRandom[SeedRandom[23424]; 
 ArrayPlot[
  Mean /@ Transpose[
    MapIndexed[#1*(1.6^-Last[#2]) &, 
     CellularAutomaton[<|"Dimension" -> 2, "TotalisticCode" -> 976, 
       "Neighborhood" -> 9|>, 
      RandomChoice[{.45, .55} -> {1, 0}, {220, 40}], 80], {-1}], 
    2 <-> 3], 
  ColorFunction -> (Mix[{[email protected][0.15, 0.72, 1], 
       [email protected][0.98, 1, 0.8200000000000001]}, #] &), 
  Body -> False]]

The rule we simply noticed is near the bulk rule on a 9-cell 3×3 area, aside from totals 4 and 5, that are taken to offer 1 and 0 reasonably than 0 and 1. If we use the pure majority rule on the three×3 area it will get caught:

&#10005
GraphicsGrid[
 Partition[
  BlockRandom[SeedRandom[23424]; 
   ArrayPlot[#, 
      ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
        1 -> Hue[0.98, 1, 0.8200000000000001]}, ImageSize -> 80, 
      Body -> False] & /@ (CellularAutomaton[<|"Dimension" -> 2, 
       "Neighborhood" -> 9, "TotalisticCode" -> 992|>, 
      RandomChoice[{.45, .55} -> {1, 0}, {30, 30}], {{0, 8}, All}])], 
  7]]

But it surely seems to be simple to search out 2D majority guidelines that don’t get caught. In actual fact, mainly any majority rule that samples cells in an uneven means will work.

For example, take into account a rule that samples the next cells in every 3×3 neighborhood:

&#10005
majplot[offsets_] := 
 ArrayPlot[
  Reverse[Transpose[SparseArray[(2 + offsets) -> 1, {3, 3}]]], 
  Mesh -> True, ImageSize -> 40]

majplot[{{0, 1}, {1, 0}, {0, 0}}]

Right here is the 3D evolution of this rule ranging from 45% 1s:

&#10005
BlockRandom[SeedRandom[23424]; 
 ArrayPlot3D[#, 
    ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
      1 -> Hue[
       0.98, 1, 0.8200000000000001]}] &@(CellularAutomaton[{{{_, 
        a_, _}, {_, b_, c_}, {_, _, _}} :> If[a + b + c >= 2, 1, 0]}, 
    RandomChoice[{.45, .55} -> {1, 0}, {50, 50}], {{0, 80}, All}])]

And here’s what a spacetime slice appears like:

&#10005
BlockRandom[SeedRandom[23424]; 
 ArrayPlot[
  Mean /@ Transpose[
    MapIndexed[#1*(1.6^-Last[#2]) &, 
     CellularAutomaton[{{{_, a_, _}, {_, b_, c_}, {_, _, _}} :> 
        If[a + b + c >= 2, 1, 0]}, 
      RandomChoice[{.45, .55} -> {1, 0}, {220, 40}], 60], {-1}], 
    2 <-> 3], 
  ColorFunction -> (Mix[{[email protected][0.15, 0.72, 1], 
       [email protected][0.98, 1, 0.8200000000000001]}, #] &), 
  Body -> False]]

The conduct as a operate of the preliminary density exhibits a transparent transition at 50%:

&#10005
Row[ParallelTable[
  BlockRandom[SeedRandom[23424]; 
   Labeled[ArrayPlot[
     Mean /@ Transpose[
       MapIndexed[#1*(1.6^-Last[#2]) &, 
        CellularAutomaton[{{{_, a_, _}, {_, b_, c_}, {_, _, _}} :> 
           If[a + b + c >= 2, 1, 0]}, 
         RandomChoice[{p, 1 - p} -> {1, 0}, {60, 40}], 60], {-1}], 
       2 <-> 3], ImageSize -> 72, 
     ColorFunction -> (Mix[{[email protected][0.15, 0.72, 1], 
          [email protected][0.98, 1, 0.8200000000000001]}, #] &), 
     Body -> False], Model[PercentForm[p], 11]]], {p, .3, .7, .05}]]

Listed here are outcomes for various samplings of cells within the 3×3 neighborhood; all efficiently obtain consensus:

&#10005
MajorityRule[offsets_] := 
 With[{n = Length[offsets]}, {FromDigits[
    If[# > n/2, 1, 0] & /@ Reverse[Range[0, n]], 2], {2, 1}, offsets}]

majrow[oo_] := 
 Row[BlockRandom[
   SeedRandom[23424]; {majplot[oo], 
    ArrayPlot3D[#, 
       ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
         1 -> Hue[0.98, 1, 0.8200000000000001]}, 
       ImageSize -> 220] &@(CellularAutomaton[MajorityRule[oo], 
       RandomChoice[{.45, .55} -> {1, 0}, {50, 50}], {{0, 60}, All}]),
     ArrayPlot[
     Mean /@ 
      Transpose[
       MapIndexed[#1*(1.6^-Last[#2]) &, 
        CellularAutomaton[MajorityRule[oo], 
         RandomChoice[{.45, .55} -> {1, 0}, {100, 60}], 60], {-1}], 
       2  3], 
     ColorFunction -> (Mix[{[email protected][0.15, 0.72, 1], 
          [email protected][0.98, 1, 0.8200000000000001]}, #] &), 
     Body -> False, ImageSize -> 300]}], Spacer[25]]

majrow[{{1, -1}, {1, 1}, {0, 0}}]
&#10005
majrow[{{0, -1}, {1, 1}, {0, 0}}]
&#10005
majrow[{{0, -1}, {1, 1}, {-1, 0}}]
&#10005
majrow[{{0, 1}, {-1, 0}, {1, 1}, {1, 0}, {0, -1}}]

With our authentic 5-cell “symmetrical” neighborhood we will get very comparable conduct by setting issues up just like the 1D GKL rule:

&#10005
{{{_, a_, _}, {b_, c_, d_}, { _, e_, _}} :> 
  If[If[c == 0, a + b, d + e] + c >= 2, 1, 0]}
&#10005
{BlockRandom[SeedRandom[23424]; 
  ArrayPlot3D[#, 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[
        0.98, 1, 0.8200000000000001]}] &@(CellularAutomaton[{{{_, 
         a_, _}, {b_, c_, d_}, { _, e_, _}} :> 
       If[If[c == 0, a + b, d + e] + c >= 2, 1, 0]}, 
     RandomChoice[{.45, .55} -> {1, 0}, {50, 50}], {{0, 80}, All}])], 
 BlockRandom[SeedRandom[23424]; 
  ArrayPlot[
   Mean /@ Transpose[
     MapIndexed[#1*(1.6^-Last[#2]) &, 
      CellularAutomaton[{{{_, a_, _}, {b_, c_, d_}, { _, e_, _}} :> 
         If[If[c == 0, a + b, d + e] + c >= 2, 1, 0]}, 
       RandomChoice[{.45, .55} -> {1, 0}, {100, 40}], 60], {-1}], 
     2 <-> 3], 
   ColorFunction -> (Mix[{[email protected][0.15, 0.72, 1], 
        [email protected][0.98, 1, 0.8200000000000001]}, #] &), 
   Body -> False]]}

Mobile Automata with Noise

Thus far we’ve assumed that when it’s began, the evolution of the mobile automaton is completely deterministic. However what if there’s some “noise” within the evolution—say if the values of cells are randomly flipped with some probability? Right here’s what occurs with the straightforward majority rule in 1D on this case:

&#10005
BlockRandom[SeedRandom[34646]; 
 ArrayPlot[
  NestList[MapAt[1 - # &, CellularAutomaton[232][#], 
     Listing /@ RandomInteger[{1, 400}, 5]] &, 
   RandomChoice[{.4, .6} -> {1, 0}, 400], 100], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

What concerning the GKL rule? At low ranges of noise the rule will usually “combat it off” and nonetheless obtain consensus:

&#10005
BlockRandom[SeedRandom[32546]; 
 ArrayPlot[
  NestList[MapAt[1 - # &, 
     CellularAutomaton[{FromDigits[
         Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
           If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 
        3}][#], Listing /@ RandomInteger[{1, 400}, 5]] &, 
   RandomChoice[{.4, .6} -> {1, 0}, 400], 120], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

However ultimately the extent of noise turns into too nice, and consensus is often misplaced:

&#10005
BlockRandom[SeedRandom[32546]; 
 ArrayPlot[
  NestList[MapAt[1 - # &, 
     CellularAutomaton[{FromDigits[
         Tuples[{1, 0}, 7] /. {l3_, _, l1_, c_, r1_, _, r3_} :> 
           If[If[c == 0, r1 + r3, l1 + l3] + c >= 2, 1, 0], 2], 2, 
        3}][#], Listing /@ RandomInteger[{1, 400}, 30]] &, 
   RandomChoice[{.4, .6} -> {1, 0}, 400], 150], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

Normally, the presence of “noise” turns our system from an bizarre mobile automaton right into a probabilistic mobile automaton. (And this, in flip, is equal to what’s typically known as directed percolation, or to a spin system that’s taken to evolve in time with random updates based on guidelines with sure weightings. It’s additionally associated to what’s typically been known as an “interacting particle system”—through which for instance boundaries of areas observe one thing like an array of random walks, that annihilate after they meet. )

Let’s discuss in a bit extra element concerning the total conduct of the GKL rule. When there’s no noise, it exhibits a pointy transition from ultimate state 0 to ultimate state 1 when the preliminary density goes from under 0.5 to above 0.5. However what occurs after we add noise? We will summarize the consequence by a traditional physics-style part diagram:

&#10005
(*res2=With[{length=500,steps=1000},
BlockRandom[SeedRandom[625];
ParallelTable[Mean[Nest[MapAt[1-#&,CellularAutomaton[{\
339789091192587366278221041213531750560,2,3}][#],Listing/@RandomInteger[{\
1,length},Floor[length*\[Epsilon]]]]&,RandomChoice[{p,1-p}\[Rule]{1,0}\
,size],steps]],{p, 0, 1,.001}, {\[Epsilon], 0, 0.15,.001}]]];*)


res2 = Import[
   "https://www.wolframcloud.com/obj/sw-writings/DistributedConsensus/\
Data/gklnoise-01.wxf"];

ListDensityPlot[res2, DataRange -> {{ 0, 0.15}, { 0, 1}}, 
 ColorFunction -> (Blend[{RGBColor[1., 0.9279999999999999, 0.28], 
      RGBColor[0.8200000000000001, 0., 0.0984000000000001]}, #] &), 
 AspectRatio -> .7, FrameLabel -> "noise stage", "preliminary density"]

This diagram exhibits the ultimate density produced by the rule as a operate of the preliminary density and the noise stage. At zero noise stage, there’s a reasonably sharp transition as a operate of preliminary density. (It’s not completely sharp as a result of this diagram was generated by sampling a finite variety of preliminary situations in a finite area.) And because the noise stage will increase, the sharp transition appears to outlive for some time—till ultimately a essential noise stage is reached at which it disappears.

Is there a rigorous method to analyze what’s occurring? Properly, not but. And actually for a very long time it was thought that within the presence of noise any 1D system like this could essentially be ergodic, within the sense that it will ultimately go to all attainable states, and definitely not evolve from totally different preliminary densities to totally different ultimate states.

However within the Eighties a complicated cellular automaton was constructed that it was attainable to show wouldn’t present such conduct. The system was put collectively for the aim of “doing dependable computation even within the presence of noise” and was arrange utilizing reasonably elaborate software-engineering-like methodology. However in the end it was only a 1D mobile automaton, albeit with an astronomically difficult rule. And the essential level was that as much as some nonzero stage of noise, the system may reliably carry out a computation—corresponding to attaining majority consensus.

However does one actually need a system with such difficult underlying guidelines to do that? Undoubtedly not. And the state of affairs jogs my memory of what occurred with the issue of bizarre computation universality in cellular automata. Again within the Nineteen Fifties it appeared one may obtain this with a really difficult setup, constructed in an engineering-like means. However now we all know that truly even one of many very easiest conceivable 1D mobile automata—rule 110—is already computation universal. And actually the Principle of Computational Equivalence implies that at any time when we see conduct that’s not clearly easy, we will anticipate computation universality.

After all, it doesn’t look like we should always must have computation universality to get distributed consensus—although the Precept of Computational Equivalence means that computation universality is “low cost” so would possibly in impact “come alongside without cost” with guidelines that produce other mandatory properties. (And by the best way, this isn’t a trivial concern, as a result of when techniques are able to common computation there’s the potential for them to “do one thing one couldn’t predict”, together with, for instance, escape of some laptop safety constraint one thinks one’s outlined.)

However figuring out that there’s a really difficult mobile automaton that achieves distributed consensus even within the presence of noise makes one marvel what the only mobile automaton which may do that is likely to be. And based mostly on my earlier expertise, I might anticipate it’ll be quite simple—just like the GKL rule—although it could be very troublesome to show this.

It is likely to be helpful to make a number of remarks about the entire concern of “noise”. In a way when one says there’s “noise” in a system one’s saying that the system is “open”, and there’s one thing coming from “exterior” it that one can’t predict. However as an “approximation” one can think about simply having some pseudorandom generator of noise—just like the rule 30 cellular automaton. After which one as soon as once more has a “closed” system, to which one can instantly apply pondering based mostly for instance on the Precept of Computational Equivalence.

However what about “really unpredictable noise”? To say that that is current is to say that there are totally different paths of historical past that the system may observe, and one doesn’t know which one shall be adopted in any explicit case. Knowledgeable by our Physics Project, we will signify these potentialities by defining a multiway graph, through which there’s a department at any time when two totally different states are generated, relying on the noise.

However along with branches, there will also be merges within the multiway graph. And within the reasonably trivial case of a mobile automaton with the id rule, permitting each attainable particular person cell to be flipped by noise, we get the multiway graph:

&#10005
getStateGraphics[state_] := 
  Framed[
   Style[
    ArrayPlot[{state}, 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, 
     MeshStyle -> Orange], Hue[0.62, 1, 0.48]], 
   Background -> Directive[Opacity[0.2], Hue[0.62, 0.45, 0.87]], 
   FrameMargins -> {{2, 2}, {0, 0}}, RoundingRadius -> 0, 
   FrameStyle -> Directive[Opacity[0.5], Hue[0.62, 0.52, 0.82]]];

getStateRenderingFunction[] := 
  Inset[getStateGraphics[ToExpression[#2]], #1, 
    Heart, ] &;

flipbits[list_] := 
  Desk[list -> MapAt[1 - # &, list, i], i, Size[list]];

SimpleGraph[
 Flatten[
  NestList[Flatten[flipbits /@ (Last /@ #)] &, flipbits[{0, 0, 0}], 
   5]], VertexShapeFunction -> getStateRenderingFunction[], 
 VertexSize -> 1, EdgeStyle -> Hue[0.75, 0, 0.35], 
 PerformanceGoal -> "High quality"]

Right here’s what occurs if we apply the bulk mobile automata rule (rule 232) after every “noise flip” (and go a complete of simply 2 steps):

&#10005
getStateGraphics[state_] := 
  Framed[
   Style[
    ArrayPlot[{state}, 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, 
     MeshStyle -> Orange], Hue[0.62, 1, 0.48]], 
   Background -> Directive[Opacity[0.2], Hue[0.62, 0.45, 0.87]], 
   FrameMargins -> {{2, 2}, {0, 0}}, RoundingRadius -> 0, 
   FrameStyle -> Directive[Opacity[0.5], Hue[0.62, 0.52, 0.82]]];

getStateRenderingFunction[] := 
  Inset[getStateGraphics[ToExpression[#2]], #1, 
    Heart, ] &;

flipbitsCA[ru_, list_] := 
 Desk[list -> ru[MapAt[1 - # &, list, i]], i, Size[list]]

SimpleGraph[
 With[{g = 
    Graph[
     Flatten[
      NestList[
       Flatten[
         flipbitsCA[CellularAutomaton[232], #] & /@ (Final /@ #)] &, 
       flipbitsCA[CellularAutomaton[232], {0, 1, 1, 1, 0}], 3]]]}, 
  Graph[g, VertexShapeFunction -> getStateRenderingFunction[], 
   AspectRatio -> 1/2, VertexSize -> 1.6, 
   EdgeStyle -> Hue[0.75, 0, 0.35], PerformanceGoal -> "High quality"]]]

Going extra steps, with thicker edges representing extra updating occasions connecting the identical states, we get:

&#10005
getStateGraphics[state_] := 
  Framed[
   Style[
    ArrayPlot[{state}, 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, 
     MeshStyle -> Orange], Hue[0.62, 1, 0.48]], 
   Background -> Directive[Opacity[0.2], Hue[0.62, 0.45, 0.87]], 
   FrameMargins -> {{2, 2}, {0, 0}}, RoundingRadius -> 0, 
   FrameStyle -> Directive[Opacity[0.5], Hue[0.62, 0.52, 0.82]]];

getStateRenderingFunction[] := 
  Inset[getStateGraphics[ToExpression[#2]], #1, 
    Heart, ] &;

flipbitsCA[ru_, list_] := 
 Desk[list -> ru[MapAt[1 - # &, list, i]], i, Size[list]]

With[{g = 
   Graph[
    Flatten[
     NestList[
      Flatten[
        flipbitsCA[CellularAutomaton[232], #] & /@ (Final /@ #)] &, 
      flipbitsCA[CellularAutomaton[232], {0, 1, 1, 1, 0}], 4]]]}, 
 SimpleGraph[g, VertexShapeFunction -> getStateRenderingFunction[], 
  VertexSize -> 1.25, PerformanceGoal -> "High quality", 
  EdgeStyle -> 
   Thread[
    EdgeList[
      SimpleGraph[
       g]] -> (Directive[Hue[0.75, 0, 0.35], 
         Thickness[1.5*Counts[EdgeList[g]][#]/Size[EdgeList[g]]], 
         Arrowheads[
          7.5*Counts[EdgeList[g]][#]/Size[EdgeList[g]]]] & /@ 
       EdgeList[SimpleGraph[g]])]]]

There are a number of delicate limits to be taken. The scale of the mobile automaton is being taken to infinity. The variety of steps can also be being taken to infinity (although slower). And by saying that there’s solely a sure “density of noise” we’re successfully taking limits on the relative weightings of edges.

To have a system which achieves consensus even within the presence of noise, solely explicit attractors should survive in these limits. However fairly what sort of underlying rule is critical for this we don’t know—although my guess is that it’ll in the end be surprisingly easy.

Will computation universality “come alongside for the journey”? I don’t know, however I wouldn’t be shocked if it did. Although it’s price understanding that the definition of computation universality in a multiway system like that is considerably delicate. (I not too long ago mentioned it within the context of multiway Turing machines, however there are nonetheless extra points when one’s fascinated with chances and “probabilistic weightings” of various paths.)

“Purposeful Assaults” on a Mobile Automaton

We’ve simply talked concerning the results of “random noise” on consensus in a mobile automaton. However what about “noise” (or “errors”) which can be “purposefully launched”? Is there a sample of some probably small variety of errors that can, for instance, flip the consensus consequence?

One model of this query—harking back to adversarial examples in neural networks—is simply to ask what adjustments should be made to an preliminary situation to “flip its consequence”. Or, put one other means: let’s say one has a system (just like the GKL rule) that mainly achieves appropriate consensus for nearly all randomly chosen preliminary situations. Now we ask the query of whether or not there’s a systematic method to tweak a given randomly chosen preliminary situation to make it “result in the flawed reply”. (One can consider this as a bit like asking whether or not one can discover a nonce that can make a cryptographic hash come out in a specific means.)

For sure, there are numerous subtleties to this query. What will we imply by “random preliminary situations”? Presumably a periodic state wouldn’t qualify. What sorts of “tweaks” can we make?

One thing that may conceivably occur is that there’s a sure conduct with “bizarre” preliminary situations, however there’s some particular “seed” that—if it happens—will produce unbounded (“tumor-style”) progress that ultimately takes over the system, as on this easy instance from rule 122:

&#10005
BlockRandom[SeedRandom[244234]; 
 ArrayPlot[
  CellularAutomaton[122, 
   ReplacePart[Append[Riffle[RandomInteger[1, 101], 0], 0], 140 -> 1],
    40], Body -> False]]

If as a substitute of simply “attacking” the preliminary situations one permits the opportunity of, say, altering the worth of a specific cell on each step, it’s straightforward to finish up, for instance, with “immovable lumps” that in impact forestall “full consensus”:

&#10005
BlockRandom[SeedRandom[3257]; 
 ArrayPlot[
  NestList[MapAt[RandomChoice[{0, 1} -> {# &, 1 - # &}], 
     CellularAutomaton[{339789091192587366278221041213531750560, 2, 
        3}][#], Listing /@ Vary[195, 205]] &, 
   RandomChoice[{.4, .6} -> {1, 0}, 400], 120], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, Body -> False]]

However what if one considers making adjustments at a small variety of locations within the ongoing evolution—say in impact including slightly “clever noise” at places rigorously computed from the precise sample of evolution? Will the system all the time have the ability to “heal itself” from such “Byzantine tampering”, in impact “correcting few-bit errors”? Or is there some explicit “vulnerability” that may be exploited to “corrupt” the ultimate outcomes with only a few rigorously chosen adjustments?

One can consider the 2 consensus ultimate states as being attractors, whose basins of attraction embody all preliminary situations above or under density . Alternatively, one can consider the mobile automaton as “fixing the classification downside” of “recognizing the preliminary density”. And maybe there’s some method to lengthen the mobile automaton to a neural web with steady weights, after which use machine studying strategies to iteratively discover minimal locations the place weights will be modified.

Graph Mobile Automata

In an bizarre mobile automaton, values are assigned to cells specified by a particular grid. However as a generalization one can enable the cells to lie on the nodes of a graph—after which to take the neighbors on the graph to outline the neighbors for use within the rule:

&#10005
RandomGraph[{20, 40}, EdgeStyle -> Gray, 
 VertexStyle -> 
  Table[i -> (RandomInteger[] /. {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}), {i, 20}], 
 VertexSize -> .5]

There’s one rapid concern right here. Within the primary definition of a regular mobile automaton, the rule “takes its arguments” in a particular order. But when one’s coping with an bizarre graph (versus, for instance, an ordered hypergraph), all one is aware of is what nodes are related to a given node—with no rapid ordering outlined.

And this means constraints on the kind of mobile automaton rule we will use. One can consider establishing a “geodesic ball” round every node within the graph. Successive “shells” comprise nodes which can be successive graph distances away from a given node. However the mobile automaton rule can’t distinguish which “place” a given cell is at inside a specific shell; all it could actually do is depend the full variety of cells in every shell which have a given worth.

If the graph is vertex transitive, in order that the graph construction round each node within the graph is similar (as for a Cayley graph), then the mobile automaton rule can mainly comprise a set desk of outcomes that rely solely on the variety of cells of every worth in every shell. However for a common graph the rule for the mobile automaton should enable for arbitrary numbers of cells in every shell.

And one case the place this occurs to be simple to do is for the straightforward majority rule. So right here’s an instance of this rule utilized to “geodesic shells of radius 1” within the graph above:

&#10005
SynchronousStepNewColors[dependencies_, colors_] := 
 Flatten[
  Map[With[neighbors = 
       Sort[Counts[Part[colors, Last[#]]], Better],
     If[DuplicateFreeQ[Values[neighbors]],
      First[Keys[neighbors]], 
      colours[[First[#]]]]] &, dependencies]]

GraphMajorityCA[graph0_, p_, steps_, radius_ : 1] := 
 BlockRandom[SeedRandom[14];
  With[{graph = IndexGraph[graph0]}, 
   With[{init = 
      Association[# -> RandomChoice[{1 - p, p} -> {1, 0}] & /@ 
        VertexList[graph]]}, 
    Module[{dependencies},
     dependencies = 
      Table[
       n -> 
        VertexOutComponent[graph, VertexList[graph][[n]], 
         radius], {n, VertexCount[graph]}];
     Graph[EdgeList[graph], 
        VertexStyle -> 
         MapThread[
          Rule[#1, 
            Switch[#2, 0, Hue[0.15, 0.72, 1], 1, Hue[
             0.98, 1, 0.8200000000000001]]] &, {Keys[init], #}],  
        EdgeStyle -> Grey, VertexSize -> .5] & /@ 
      NestList[SynchronousStepNewColors[dependencies, #] &, 
       Values[init], steps]
     ]]]]

Rotate[#, 90 Degree] & /@ (Graph[#, ImageSize -> 190] & /@ 
   GraphMajorityCA[\!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 
           19, 20}, {Null, 
SparseArray[
            Automatic, {20, 20}, 0, {
             1, {{0, 2, 5, 7, 9, 13, 20, 25, 29, 33, 36, 41, 43, 46, 
               48, 54, 59, 63, 68, 72, 80}, {{5}, {7}, {11}, {16}, {
               18}, {11}, {13}, {16}, {18}, {1}, {6}, {9}, {20}, {
               5}, {13}, {15}, {16}, {17}, {19}, {20}, {1}, {9}, {
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               19}}}, Pattern}]}, {EdgeStyle -> {
GrayLevel[0.5]}, VertexSize -> {0.5}, 
            VertexStyle -> {
             18 -> Hue[0.98, 1, 0.8200000000000001], 6 -> 
              Hue[0.98, 1, 0.8200000000000001], 15 -> 
              Hue[0.15, 0.72, 1], 1 -> Hue[0.15, 0.72, 1], 20 -> 
              Hue[0.15, 0.72, 1], 9 -> 
              Hue[0.98, 1, 0.8200000000000001], 13 -> 
              Hue[0.98, 1, 0.8200000000000001], 2 -> 
              Hue[0.98, 1, 0.8200000000000001], 8 -> 
              Hue[0.15, 0.72, 1], 3 -> 
              Hue[0.98, 1, 0.8200000000000001], 4 -> 
              Hue[0.98, 1, 0.8200000000000001], 11 -> 
              Hue[0.98, 1, 0.8200000000000001], 10 -> 
              Hue[0.15, 0.72, 1], 19 -> 
              Hue[0.98, 1, 0.8200000000000001], 7 -> 
              Hue[0.15, 0.72, 1], 16 -> Hue[0.15, 0.72, 1], 17 -> 
              Hue[0.98, 1, 0.8200000000000001], 14 -> 
              Hue[0.15, 0.72, 1], 12 -> Hue[0.15, 0.72, 1], 5 -> 
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               DiskBox[17, 0.09587857253283466]}, 
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MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->{
        "NetworkGraphics", FrontEnd`GraphicsHighlightColor -> 
         Hue[0.8, 1., 0.6]},
FormatType->TraditionalForm,
FrameTicks->None]\), .3, 3])

So what globally occurs with this rule? For the graph

&#10005
Graph[ResourceFunction["TorusGraph"][{3, 3}], VertexStyle -> White, 
 VertexSize -> .2, EdgeStyle -> \!\(\*
TagBox[
StyleBox[
InterpretationBox[
RowBox[{"CloudGet", "[", "\"\<http://wolfr.am/VtlQl86f\>\"", "]"}],
Hue[0.75, 0, 0.35],
Editable->False,
Selectable->False],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)]

right here’s what the state transition graph appears like when the rule is utilized, whereas above nodes are coloured based on which worth is within the majority:

&#10005
SynchronousStepNewColors[dependencies_, colors_] := 
 Flatten[
  Map[With[neighbors = 
       Sort[Counts[Part[colors, Last[#]]], Better],
     If[DuplicateFreeQ[Values[neighbors]],
      First[Keys[neighbors]], 
      colours[[First[#]]]]] &, dependencies]]

GraphMajorityCASTG[graph0_, radius_ : 1] :=
 
 With[{graph = IndexGraph[graph0]},
  Module[{dependencies},
   dependencies = 
    Table[
     n -> 
      VertexOutComponent[graph, VertexList[graph][[n]], radius], {n, 
      VertexCount[graph]}];
   # -> SynchronousStepNewColors[dependencies, #] & /@ 
    Tuples[{1, 0}, VertexCount[graph]]
   ]]

With[{g = 
   GraphMajorityCASTG[ResourceFunction["TorusGraph"][{3, 3}], 1]}, 
 Graph[g, 
  VertexStyle -> (# -> 
       If[Mean[#] > 1/2, Hue[0.98, 1, 0.8200000000000001], Hue[
        0.15, 0.72, 1]] & /@ VertexList[g]), EdgeStyle -> \!\(\*
TagBox[
StyleBox[
InterpretationBox[
ButtonBox[
TooltipBox[
GraphicsBox[{
{GrayLevel[0], RectangleBox[{0, 0}]}, 
{GrayLevel[0], RectangleBox[{1, -1}]}, 
{Hue[0.75, 0, 0.35], RectangleBox[{0, -1}, {2, 1}]}},
AspectRatio->1,
DefaultBaseStyle->"ColorSwatchGraphics",
Body->True,
FrameStyle->Hue[0., 0., 0.23333333333333334`],
FrameTicks->None,
ImageSize->Computerized, 12.879,
PlotRangePadding->None],
StyleBox[
RowBox[{"Hue", "[", 
RowBox[{"0.75`", ",", "0", ",", "0.35`"}], "]"}], NumberMarks -> 
           False]],
Look->None,
BaseStyle->{},
BaselinePosition->Baseline,
ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
If[
Not[
AbsoluteCurrentValue["Deployed"]], 
            SelectionMove[Typeset`box$, All, Expression]; 
            FrontEnd`Personal`$ColorSelectorInitialAlpha = 1; 
            FrontEnd`Personal`$ColorSelectorInitialColor = 
             Hue[0.75, 0, 0.35]; 
            FrontEnd`Personal`$ColorSelectorUseMakeBoxes = True; 
            MathLink`CallFrontEnd[
FrontEnd`AttachCell[Typeset`box$, 
FrontEndResource["HueColorValueSelector"], {0, }, 
               Left, Prime, 
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                "SelectionDeparture", "ParentChanged", 
                 "EvaluatorQuit"}]]]],
DefaultBaseStyle->{},
Evaluator->Computerized,
Methodology->"Preemptive"],
Hue[0.75, 0, 0.35],
Editable->False,
Selectable->False],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)]]

Little question there are common outcomes that may be proved concerning the “success fee” for almost all mobile automaton rule on graphs. However experiments are inclined to counsel that the rule does significantly better on graphs than it does on common arrays.

Presumably there are graph-theoretic options of the underlying graph that have an effect on the efficiency. Increased connectivity presumably helps, not least as a result of it tends to keep away from “bridges” the place colours will be balanced on “all sides” of a specific node. Lack of symmetry additionally in all probability tends to inhibit the looks of cycles. And on the whole one can consider the “spreading of consensus” as being a minimum of somewhat like a percolation process.

For bizarre mobile automata, it’s clear what it means to ask concerning the “infinite-size restrict”. However for graphs it’s solely instantly clear when one’s coping with some readily extensible household of graphs (like grids or torus graphs or varied Cayley graphs). And for arbitrary “random graphs” the outcomes will in all probability rely considerably on the graph distribution used.

In our Physics Challenge we have now been involved with massive graphs that may be “grown” based on native guidelines. We anticipate such graphs usually to indicate sure “statistical regularities” within the “continuum restrict”. In our undertaking, we characterize the construction of those graphs by searching for instance on the growth rates of volumes of geodesic balls, and figuring out issues like dimension and curvature from them. So what’s going to occur if we run a majority rule mobile automaton on a big graph that has sure “geometrical” properties?

Primarily we have to ask what the “continuum restrict” of the bulk rule mobile automaton is. The grids utilized in bizarre mobile automata are too particular for them to realize any form of generic such restrict. However on “geometrizable” graphs, it’s extra cheap to anticipate such a continuum restrict.

We will attempt contemplating a 1D instance. The preliminary values are then simply given by a steady operate of place:

&#10005
With[{if = (SeedRandom[69774]; 
    Interpolation[RandomReal[{-10, 10}, 10], 
     InterpolationOrder -> 6])}, 
 Plot[if[x], {x, 1, 10}, Filling -> Axis, 
  ColorFunctionScaling -> False, 
  ColorFunction -> (If[#2 > 0, Hue[0.98, 1, 0.8200000000000001], Hue[
      0.15, 0.72, 1]] &), Body -> True, AspectRatio -> 1/3, 
  FrameTicks -> None]]

The “consensus consequence” on this case ought to be a continuing operate whose worth is successfully the signal of the integral of this operate. However what sort of integro-differential-algebraic equation can reproduce the time evolution isn’t clear.

Going again to majority mobile automata on graphs, it’s price noting that if the sides of the graph will be assigned each constructive and destructive weights, then the system is successfully like a synchronous model of a neural web. The analog of this on a daily grid (which is structurally like a spin glass) is then recognized to indicate varied options of computational irreducibility.

As an alternative of desirous about underlying graphs with weighted edges, we will take into account mobile automaton guidelines that don’t simply contain pure nearest-neighbor majority. For instance, we may take into account guidelines which have totally different weights for geodesic shells of various radii (very like the activation-inhibition cellular automata used to mannequin issues like organic pigmentation patterns).

However is it actually true that solely totalistic guidelines based mostly on geodesic shells can be utilized for graph mobile automata? To do greater than this requires in impact defining “instructions” within the graph. However our Physics Challenge has offered a wide range of mechanisms for doing simply this—and this in precept for establishing non-totalistic graph mobile automata.

Asynchronous Updating

An vital function of mobile automata is the idea that each one cell values are up to date “concurrently” or “synchronously” in a particular collection of steps. However in sensible examples of distributed consensus one’s usually coping with values which can be as a substitute up to date asynchronously. In impact, what one needs is to “break down” the synchronous updating of an bizarre mobile automaton right into a sequence of updates of particular person cells, with the order of those updates not being specified by any explicit rule.</P

So an apparent first query is: “Does it really matter in what order these particular person updates are executed?” And typically it doesn’t. Right here’s an instance. As an alternative of an bizarre mobile automaton, take into account a block mobile automaton through which at every step pairs of values adjoining cells are changed by new values:

&#10005
RulePlot[SubstitutionSystem[{{0, 0} -> {0, 0}, {1, 0} -> {0, 1}, {0, 
     1} -> {0, 1}, {1, 1} -> {1, 1}}], 
 ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
   1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange]

For synchronous updating, we would apply these guidelines in a scientific “brick-like” sample. However to review asynchronous updating, let’s simply apply these guidelines in random positions at every step. Listed here are a number of examples of what can occur:

&#10005
BlockRandom[SeedRandom[235234]; 
 With[{i = RandomInteger[1, 30]}, 
  Desk[ArrayPlot[
    NestList[
     First[
       Sort[{Flatten[
          MapAt[Sort, Partition[#, 2], 
           Union[List /@ RandomInteger[{1, Length[#]/2}, 20]]]], 
         RotateRight[
          Flatten[
           MapAt[Sort, Partition[RotateLeft[#], 2], 
            Union[
             List /@ RandomInteger[{1, Length[#]/2}, 20]]]]]}]] &, i, 
     63], ImageSize -> 150, Computerized, 
    ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
      1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, 
    MeshStyle -> Orange], 4]]]

And the notable function is that although the precise evolution in every case is totally different, the ultimate result’s all the time the identical—on this case simply comparable to having all sorted earlier than .

It doesn’t work this manner for all guidelines, however for this rule, whatever the intermediate states which can be produced, there’s all the time eventual consistency within the ultimate consequence.

Because it seems, this type of phenomenon is essential in our Physics Challenge. And certainly the generalization that we name “causal invariance” is what leads, for instance, to relativistic invariance. However from the formalism of the Physics Challenge we additionally get a common method to asynchronous evolution: hint all attainable “replace histories” utilizing a multiway graph.

Right here is the multiway graph for the straightforward sorting rule above:

&#10005
getStateGraphics[state_] := 
  Framed[
   Style[
    ArrayPlot[{state}, 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, 
     MeshStyle -> Orange], Hue[0.62, 1, 0.48]], 
   Background -> Directive[Opacity[0.2], Hue[0.62, 0.45, 0.87]], 
   FrameMargins -> {{2, 2}, {0, 0}}, RoundingRadius -> 0, 
   FrameStyle -> Directive[Opacity[0.5], Hue[0.62, 0.52, 0.82]]];

getStateRenderingFunction[] := 
  Inset[getStateGraphics[ToExpression[#2]], #1, 
    Heart, ] &;

ResourceFunction[
  "MultiwaySystem"][{{1, 0} -> {0, 1}}, {{1, 0, 1, 1, 0, 0, 
   1}}, 8, "StatesGraph", 
 "StateRenderingFunction" -> getStateRenderingFunction[], 
 VertexSize -> 1.75, PerformanceGoal -> "High quality"]

As anticipated, all attainable replace histories ultimately converge to the identical ultimate state.

So what concerning the majority rule mobile automaton? It doesn’t all the time present eventual consistency, as this instance exhibits:

&#10005
randomOrderCAFunc[ruleRadius_, ruleNumber_, init_, eventCount_, func_] :=
  func[evaluateSingleEvent[ruleRadius, ruleNumber, #] &, init, eventCount];

RandomOrderCA[args___] := randomOrderCAFunc[args, Nest];

RandomOrderCAList[args___] := randomOrderCAFunc[args, NestList];

findLastEvent[eventNumber_, position_, eventsIndex_] := Module[{},
  Max[Select[Lookup[eventsIndex, position, {-Infinity}], #  eventNumber & /@
      Mod[Range[position - ruleRadius, position + ruleRadius], measurement, 1],
    -Infinity -> _]
];

RandomOrderCACausalGraph[ruleRadius_, ruleNumber_, init_, eventCount_, opts___] := Module[{eventsIndex, eventPositions},
  eventsIndex = KeySort @ Map[
    Last,
    GroupBy[
      Thread[
        (eventPositions = Reap[RandomOrderCA[ruleRadius, ruleNumber, init, eventCount]][[2, 1]]) ->
          Vary[eventCount]],
      First],
    {2}];
  Graph[
    Range[eventCount],
    Catenate[getCausalLinks[#, eventPositions[[#]], eventsIndex, Size[init], ruleRadius] & /@ Vary[eventCount]],
    EdgeStyle -> ResourceFunction["WolframPhysicsProjectStyleData"]["CausalGraph", "EdgeStyle"],
    VertexStyle -> ResourceFunction["WolframPhysicsProjectStyleData"]["CausalGraph", "VertexStyle"],
    opts]
]

randomInit[size_, onesFraction_] := RandomChoice[{1 - onesFraction, onesFraction} -> {0, 1}, size];

evaluateSingleEvent[ruleRadius_, ruleNumber_, init_] :=
  evaluateEventAtPlace[ruleRadius, ruleNumber, init, Sow[RandomInteger[{1, Length[init]}]]];

evaluateEventAtPlace[ruleRadius_, ruleNumber_, init_, center_] := Module[{input, newCenterValue},
  input = cyclicTake[init, Range[center - ruleRadius, center + ruleRadius]];
  newCenterValue = CellularAutomaton[{ruleNumber, 2, ruleRadius}, input][[ruleRadius + 1]];
  ReplacePart[init, center -> newCenterValue]
];

cyclicTake[list_, indices_] := cyclicPart[list, #] & /@ indices;

cyclicPart[list_, index_] := record[[Mod[index, Length[list], 1]]];

getStateGraphics[state_] := 
  Framed[
   Style[
    ArrayPlot[{state}, 
     ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
       1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, 
     MeshStyle -> Orange], Hue[0.62, 1, 0.48]], 
   Background -> Directive[Opacity[0.2], Hue[0.62, 0.45, 0.87]], 
   FrameMargins -> {{2, 2}, {0, 0}}, RoundingRadius -> 0, 
   FrameStyle -> Directive[Opacity[0.5], Hue[0.62, 0.52, 0.82]]];

getStateRenderingFunction[] := 
  Inset[getStateGraphics[List @@ #2], #1, 
    Heart, ] &;

SeedRandom[643767 + 5]; SimpleGraph[
 NestGraph[
  Table[state @@ evaluateEventAtPlace[1, 232, List @@ #, c], ] &, state @@ randomInit[9, 0.7], 20, 
  VertexShapeFunction -> getStateRenderingFunction[],  
  PerformanceGoal -> "High quality", VertexSize -> 1, 
  ResourceFunction["WolframPhysicsProjectStyleData"]["StatesGraph", 
   "Options"]]]

So which means that on the whole it issues in what order asynchronous updates are executed, or, in impact, what “path of historical past” is taken. However to get a way of typical conduct, we will take into account random sequences of updates. Right here’s an instance of what one will get, doing one replace per step:

&#10005
caEvaluateCompiled = 
  FunctionCompile[
   Function[{Typed[rule, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[rad, "MachineInteger"], 
     Typed[init, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[eventCount, "Integer64"]}, 
    Module[{state, position, substate, rulePart, newCellValue},
     
     state = init;
     Do[
      position = RandomInteger[{1, Length[state]}];
      substate = 
       state[[
        Mod[#, Length[state], 1] & /@ 
         Vary[position - rad, position + rad]]];
      rulePart = Fold[2 #1 + #2 &, 0, substate] + 1;
      newCellValue = rule[[rulePart]];
      state[[position]] = newCellValue;
      , eventCount];
     state
     ]]];

Needs to be compiled for all machine targets:

caEvaluateCompiled = CompiledCodeFunction[
Association["Signature" -> TypeSpecifier[{
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64", 
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64"} -> 
      "PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]]], "Enter" -> Compile`Program[{}, 
Function[{
Typed[rule, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[rad, "MachineInteger"], 
Typed[init, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[eventCount, "Integer64"]}, 
Module[{state, position, substate, rulePart, newCellValue}, state = init; 
        Do[position = RandomInteger[{1, 
Length[state]}]; substate = Half[state, 
Map[Mod[#, 
Length[state], 1]& , 
Vary[position - rad, position + rad]]]; 
          rulePart = Fold[2 # + #2& , 0, substate] + 1; 
          newCellValue = Half[rule, rulePart]; 
          Half[state, position] = newCellValue; Null, eventCount]; state]]], 
    "ErrorFunction" -> Computerized, "InitializationName" -> 
    "Initialization_667648a7_ecf6_4a9d_97c7_f8cc07f5743e", "ExpressionName" -> 
    "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> 
    "Predominant", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, 
    "CompiledIR" -> Affiliation["MacOSX-x86-64" -> ByteArray[CompressedData["
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95quYlkc+H/bqQ28
"]]], "orcInstance" -> 140373448223744, "orcModuleId" -> 1, 
    "targetMachineId" -> 140373447970304], 5481287872, 5481287664, 5481287744,
    5481283584, 
   "{\"PackedArray\"[\"Integer64\", TypeFramework`TypeLiteral[1, \
\"Integer64\"]], \"Integer64\", \"PackedArray\"[\"Integer64\", \
TypeFramework`TypeLiteral[1, \"Integer64\"]], \"Integer64\"} -> \"PackedArray\
\"[\"Integer64\", TypeFramework`TypeLiteral[1, \"Integer64\"]]"];

RandomAsynchronousCellularAutomaton[{rn_, 2, r_}, init_, {t_, ct_}] := 
 NestList[
  caEvaluateCompiled[Reverse[IntegerDigits[rn, 2, 2^(2 r + 1)]], r, #, ct] &, 
  init, t]

BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  RandomAsynchronousCellularAutomaton[{232, 2, 1}, 
   RandomChoice[{.7, .3} -> {1, 0}, 400], {100, 1}], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange, 
  Body -> None]]

And right here’s the corresponding consequence if we do 20 updates per step:

&#10005
caEvaluateCompiled = 
  FunctionCompile[
   Function[{Typed[rule, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[rad, "MachineInteger"], 
     Typed[init, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[eventCount, "Integer64"]}, 
    Module[{state, position, substate, rulePart, newCellValue},
     
     state = init;
     Do[
      position = RandomInteger[{1, Length[state]}];
      substate = 
       state[[
        Mod[#, Length[state], 1] & /@ 
         Vary[position - rad, position + rad]]];
      rulePart = Fold[2 #1 + #2 &, 0, substate] + 1;
      newCellValue = rule[[rulePart]];
      state[[position]] = newCellValue;
      , eventCount];
     state
     ]]];

Needs to be compiled for all machine targets:

caEvaluateCompiled = CompiledCodeFunction[
Association["Signature" -> TypeSpecifier[{
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64", 
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64"} -> 
      "PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]]], "Enter" -> Compile`Program[{}, 
Function[{
Typed[rule, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[rad, "MachineInteger"], 
Typed[init, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[eventCount, "Integer64"]}, 
Module[{state, position, substate, rulePart, newCellValue}, state = init; 
        Do[position = RandomInteger[{1, 
Length[state]}]; substate = Half[state, 
Map[Mod[#, 
Length[state], 1]& , 
Vary[position - rad, position + rad]]]; 
          rulePart = Fold[2 # + #2& , 0, substate] + 1; 
          newCellValue = Half[rule, rulePart]; 
          Half[state, position] = newCellValue; Null, eventCount]; state]]], 
    "ErrorFunction" -> Computerized, "InitializationName" -> 
    "Initialization_667648a7_ecf6_4a9d_97c7_f8cc07f5743e", "ExpressionName" -> 
    "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> 
    "Predominant", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, 
    "CompiledIR" -> Affiliation["MacOSX-x86-64" -> ByteArray[CompressedData["
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"]]], "orcInstance" -> 140373448223744, "orcModuleId" -> 1, 
    "targetMachineId" -> 140373447970304], 5481287872, 5481287664, 5481287744,
    5481283584, 
   "{\"PackedArray\"[\"Integer64\", TypeFramework`TypeLiteral[1, \
\"Integer64\"]], \"Integer64\", \"PackedArray\"[\"Integer64\", \
TypeFramework`TypeLiteral[1, \"Integer64\"]], \"Integer64\"} -> \"PackedArray\
\"[\"Integer64\", TypeFramework`TypeLiteral[1, \"Integer64\"]]"];

RandomAsynchronousCellularAutomaton[{rn_, 2, r_}, init_, {t_, ct_}] := 
 NestList[
  caEvaluateCompiled[Reverse[IntegerDigits[rn, 2, 2^(2 r + 1)]], r, #, ct] &, 
  init, t]

BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  RandomAsynchronousCellularAutomaton[{232, 2, 1}, 
   RandomChoice[{.7, .3} -> {1, 0}, 400], {100, 20}], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange, 
  Body -> None]]

We would have thought that asynchronous updating would add sufficient randomness to “break ties” and stop issues getting caught. However in truth it’s not arduous to see that the outcomes are on this case ultimately no totally different from synchronous updating.

What about for one thing just like the GKL rule? Listed here are asynchronous outcomes for it, now with 50 updates per step (with initially 60% ).

&#10005
caEvaluateCompiled = 
  FunctionCompile[
   Function[{Typed[rule, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[rad, "MachineInteger"], 
     Typed[init, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[eventCount, "Integer64"]}, 
    Module[{state, position, substate, rulePart, newCellValue},
     
     state = init;
     Do[
      position = RandomInteger[{1, Length[state]}];
      substate = 
       state[[
        Mod[#, Length[state], 1] & /@ 
         Vary[position - rad, position + rad]]];
      rulePart = Fold[2 #1 + #2 &, 0, substate] + 1;
      newCellValue = rule[[rulePart]];
      state[[position]] = newCellValue;
      , eventCount];
     state
     ]]];

Needs to be compiled for all machine targets:

caEvaluateCompiled = CompiledCodeFunction[
Association["Signature" -> TypeSpecifier[{
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64", 
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64"} -> 
      "PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]]], "Enter" -> Compile`Program[{}, 
Function[{
Typed[rule, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[rad, "MachineInteger"], 
Typed[init, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[eventCount, "Integer64"]}, 
Module[{state, position, substate, rulePart, newCellValue}, state = init; 
        Do[position = RandomInteger[{1, 
Length[state]}]; substate = Half[state, 
Map[Mod[#, 
Length[state], 1]& , 
Vary[position - rad, position + rad]]]; 
          rulePart = Fold[2 # + #2& , 0, substate] + 1; 
          newCellValue = Half[rule, rulePart]; 
          Half[state, position] = newCellValue; Null, eventCount]; state]]], 
    "ErrorFunction" -> Computerized, "InitializationName" -> 
    "Initialization_667648a7_ecf6_4a9d_97c7_f8cc07f5743e", "ExpressionName" -> 
    "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> 
    "Predominant", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, 
    "CompiledIR" -> Affiliation["MacOSX-x86-64" -> ByteArray[CompressedData["
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\"Integer64\"]], \"Integer64\", \"PackedArray\"[\"Integer64\", \
TypeFramework`TypeLiteral[1, \"Integer64\"]], \"Integer64\"} -> \"PackedArray\
\"[\"Integer64\", TypeFramework`TypeLiteral[1, \"Integer64\"]]"];

RandomAsynchronousCellularAutomaton[{rn_, 2, r_}, init_, {t_, ct_}] := 
 NestList[
  caEvaluateCompiled[Reverse[IntegerDigits[rn, 2, 2^(2 r + 1)]], r, #, ct] &, 
  init, t]

BlockRandom[SeedRandom[567]; 
 ArrayPlot[
  RandomAsynchronousCellularAutomaton[{\
339789091192587366278221041213531750560, 2, 3}, 
   RandomChoice[{.6, .4} -> {1, 0}, 400], {150, 50}], 
  ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
    1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange, 
  Body -> None]]

And in contrast to what we noticed above for synchronous updating after we added noise, the change to asynchronous updating appears to utterly destroy “majority consensus” for this rule. Observe that utilizing extra updatings per step doesn’t enhance the consequence (which ought to ultimately be not ):

&#10005
caEvaluateCompiled = 
  FunctionCompile[
   Function[{Typed[rule, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[rad, "MachineInteger"], 
     Typed[init, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[eventCount, "Integer64"]}, 
    Module[{state, position, substate, rulePart, newCellValue},
     
     state = init;
     Do[
      position = RandomInteger[{1, Length[state]}];
      substate = 
       state[[
        Mod[#, Length[state], 1] & /@ 
         Vary[position - rad, position + rad]]];
      rulePart = Fold[2 #1 + #2 &, 0, substate] + 1;
      newCellValue = rule[[rulePart]];
      state[[position]] = newCellValue;
      , eventCount];
     state
     ]]];

Needs to be compiled for all machine targets:

caEvaluateCompiled = CompiledCodeFunction[
Association["Signature" -> TypeSpecifier[{
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64", 
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64"} -> 
      "PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]]], "Enter" -> Compile`Program[{}, 
Function[{
Typed[rule, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[rad, "MachineInteger"], 
Typed[init, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[eventCount, "Integer64"]}, 
Module[{state, position, substate, rulePart, newCellValue}, state = init; 
        Do[position = RandomInteger[{1, 
Length[state]}]; substate = Half[state, 
Map[Mod[#, 
Length[state], 1]& , 
Vary[position - rad, position + rad]]]; 
          rulePart = Fold[2 # + #2& , 0, substate] + 1; 
          newCellValue = Half[rule, rulePart]; 
          Half[state, position] = newCellValue; Null, eventCount]; state]]], 
    "ErrorFunction" -> Computerized, "InitializationName" -> 
    "Initialization_ebf3942d_117d_4dd2_a02b_59c5c41b2c75", "ExpressionName" -> 
    "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> 
    "Predominant", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, 
    "CompiledIR" -> Affiliation["MacOSX-x86-64" -> ByteArray[CompressedData["
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73j/yfF4/+x5PHnK4eE5h/fvSUdR//1/gIavxg==
"]]], "orcInstance" -> 
    140546848939520, "orcModuleId" -> 1, "targetMachineId" -> 
    140546849609216], 5154796080, 5154795872, 5154795952, 5154791424, 
   "{\"PackedArray\"[\"Integer64\", TypeFramework`TypeLiteral[1, \
\"Integer64\"]], \"Integer64\", \"PackedArray\"[\"Integer64\", \
TypeFramework`TypeLiteral[1, \"Integer64\"]], \"Integer64\"} -> \"PackedArray\
\"[\"Integer64\", TypeFramework`TypeLiteral[1, \"Integer64\"]]"];

RandomAsynchronousCellularAutomaton[{rn_, 2, r_}, init_, {t_, ct_}] := 
 NestList[
  caEvaluateCompiled[Reverse[IntegerDigits[rn, 2, 2^(2 r + 1)]], r, #, ct] &, 
  init, t]

Grid[Partition[
  Labeled[
     BlockRandom[SeedRandom[567]; 
      ArrayPlot[
       RandomAsynchronousCellularAutomaton[{\
339789091192587366278221041213531750560, 2, 3}, 
        RandomChoice[{.6, .4} -> {1, 0}, 400], {150, #}], 
       ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
         1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange, 
       Body -> None, ImageSize -> {300, 150}]], Model[#, 11], 
     Spacings -> {0, -0.5}] & /@ {100, 1000, 10000, 10^5}, 2], 
 Spacings -> {0, 0}]

So what occurs if we seek for guidelines that obtain consensus asynchronously? Within the nearest-neighbor case, the straightforward majority rule does greatest, though it’s mainly no good.

Listed here are outcomes for a number of guidelines discovered by looking out one million range-2 guidelines:

&#10005
caEvaluateCompiled = 
  FunctionCompile[
   Function[{Typed[rule, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[rad, "MachineInteger"], 
     Typed[init, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[eventCount, "Integer64"]}, 
    Module[{state, position, substate, rulePart, newCellValue},
     
     state = init;
     Do[
      position = RandomInteger[{1, Length[state]}];
      substate = 
       state[[
        Mod[#, Length[state], 1] & /@ 
         Vary[position - rad, position + rad]]];
      rulePart = Fold[2 #1 + #2 &, 0, substate] + 1;
      newCellValue = rule[[rulePart]];
      state[[position]] = newCellValue;
      , eventCount];
     state
     ]]];

Needs to be compiled for all machine targets:

caEvaluateCompiled = CompiledCodeFunction[
Association["Signature" -> TypeSpecifier[{
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64", 
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64"} -> 
      "PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]]], "Enter" -> Compile`Program[{}, 
Function[{
Typed[rule, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[rad, "MachineInteger"], 
Typed[init, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[eventCount, "Integer64"]}, 
Module[{state, position, substate, rulePart, newCellValue}, state = init; 
        Do[position = RandomInteger[{1, 
Length[state]}]; substate = Half[state, 
Map[Mod[#, 
Length[state], 1]& , 
Vary[position - rad, position + rad]]]; 
          rulePart = Fold[2 # + #2& , 0, substate] + 1; 
          newCellValue = Half[rule, rulePart]; 
          Half[state, position] = newCellValue; Null, eventCount]; state]]], 
    "ErrorFunction" -> Computerized, "InitializationName" -> 
    "Initialization_667648a7_ecf6_4a9d_97c7_f8cc07f5743e", "ExpressionName" -> 
    "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> 
    "Predominant", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, 
    "CompiledIR" -> Affiliation["MacOSX-x86-64" -> ByteArray[CompressedData["
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"]]], "orcInstance" -> 140373448223744, "orcModuleId" -> 1, 
    "targetMachineId" -> 140373447970304], 5481287872, 5481287664, 5481287744,
    5481283584, 
   "{\"PackedArray\"[\"Integer64\", TypeFramework`TypeLiteral[1, \
\"Integer64\"]], \"Integer64\", \"PackedArray\"[\"Integer64\", \
TypeFramework`TypeLiteral[1, \"Integer64\"]], \"Integer64\"} -> \"PackedArray\
\"[\"Integer64\", TypeFramework`TypeLiteral[1, \"Integer64\"]]"];

RandomAsynchronousCellularAutomaton[{rn_, 2, r_}, init_, {t_, ct_}] := 
 NestList[
  caEvaluateCompiled[Reverse[IntegerDigits[rn, 2, 2^(2 r + 1)]], r, #, ct] &, 
  init, t]

GraphicsGrid[
 ParallelTable[
    BlockRandom[SeedRandom[567]; 
     ArrayPlot[
      RandomAsynchronousCellularAutomaton[{#, 2, 2}, 
       RandomChoice[{p, 1 - p} -> {1, 0}, 300], {200, 400}], 
      ColorRules -> {0 -> Hue[0.15, 0.72, 1], 
        1 -> Hue[0.98, 1, 0.8200000000000001]}, 
      Body -> None]], {p, .3, .7, .1}] & /@ {4272826020, 4242057736, 
   4265795970, 3663321792, 3953790018, 4020250803}]

These guidelines had been the highest performers when it comes to having “closest-to-majority-consensus” common conduct. Within the photos right here, a mean of two updates per cell is being executed between successive rows.

If we plot the ultimate density as proven in these photos in opposition to preliminary density, listed below are the outcomes for the primary 3 guidelines (with rule numbers 4272826020, 4242057736, and 4265795970):

&#10005
caEvaluateCompiled = 
  FunctionCompile[
   Function[{Typed[rule, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[rad, "MachineInteger"], 
     Typed[init, TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
     Typed[eventCount, "Integer64"]}, 
    Module[{state, position, substate, rulePart, newCellValue},
     
     state = init;
     Do[
      position = RandomInteger[{1, Length[state]}];
      substate = 
       state[[
        Mod[#, Length[state], 1] & /@ 
         Vary[position - rad, position + rad]]];
      rulePart = Fold[2 #1 + #2 &, 0, substate] + 1;
      newCellValue = rule[[rulePart]];
      state[[position]] = newCellValue;
      , eventCount];
     state
     ]]];

Needs to be compiled for all machine targets:

caEvaluateCompiled = CompiledCodeFunction[
Association["Signature" -> TypeSpecifier[{
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64", 
"PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]], "Integer64"} -> 
      "PackedArray"["Integer64", 
TypeFramework`TypeLiteral[1, "Integer64"]]], "Enter" -> Compile`Program[{}, 
Function[{
Typed[rule, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[rad, "MachineInteger"], 
Typed[init, 
TypeSpecifier["PackedArray"]["MachineInteger", 1]], 
Typed[eventCount, "Integer64"]}, 
Module[{state, position, substate, rulePart, newCellValue}, state = init; 
        Do[position = RandomInteger[{1, 
Length[state]}]; substate = Half[state, 
Map[Mod[#, 
Length[state], 1]& , 
Vary[position - rad, position + rad]]]; 
          rulePart = Fold[2 # + #2& , 0, substate] + 1; 
          newCellValue = Half[rule, rulePart]; 
          Half[state, position] = newCellValue; Null, eventCount]; state]]], 
    "ErrorFunction" -> Computerized, "InitializationName" -> 
    "Initialization_667648a7_ecf6_4a9d_97c7_f8cc07f5743e", "ExpressionName" -> 
    "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> 
    "Predominant", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, 
    "CompiledIR" -> Affiliation["MacOSX-x86-64" -> ByteArray[CompressedData["
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95quYlkc+H/bqQ28
"]]], "orcInstance" -> 140373448223744, "orcModuleId" -> 1, 
    "targetMachineId" -> 140373447970304], 5481287872, 5481287664, 5481287744,
    5481283584, 
   "{\"PackedArray\"[\"Integer64\", TypeFramework`TypeLiteral[1, \
\"Integer64\"]], \"Integer64\", \"PackedArray\"[\"Integer64\", \
TypeFramework`TypeLiteral[1, \"Integer64\"]], \"Integer64\"} -> \"PackedArray\
\"[\"Integer64\", TypeFramework`TypeLiteral[1, \"Integer64\"]]"];

RandomAsynchronousCellularAutomaton[{rn_, 2, r_}, init_, {t_, ct_}] := 
 NestList[
  caEvaluateCompiled[Reverse[IntegerDigits[rn, 2, 2^(2 r + 1)]], r, #, ct] &, 
  init, t]

GraphicsRow[
 ListLinePlot[
    ParallelTable[
     Mean[
      Table[
       Mean[
        Last[
         RandomAsynchronousCellularAutomaton[{#, 2, 2}, 
          RandomChoice[{p, 1 - p} -> {1, 0}, 300], {200, 400}]]], 20]], {p, 0,
       1, .05}], Filling -> Axis, FillingStyle -> LightOrange, Body -> True, 
    PlotTheme -> "Minimal", ImageSize -> 140] & /@ {4272826020, 4242057736, 
   4265795970}]

In an ideal consensus rule, these can be step features at —and one can anticipate that these outcomes could get nearer to that with bigger numbers of cells and steps.

In an bizarre, synchronous mobile automaton, each cell is in impact up to date at each step, and the graph of “causal relationships” between “updating occasions” types a trivial grid. However in an asynchronous mobile automaton the graph is sparser—with a specific updating occasion being causally related to the earlier occasion that occurred to replace that cell.

However with the setup we have now to this point, this causal graph relies upon solely on which cells are up to date, not on what their colours is likely to be. And with random updates, the causal graph will mainly be like a “random meshing” of the spacetime construction of a system—in order that for instance for a mobile automaton with cyclic boundaries it turns into an approximation to a tube:

&#10005
randomOrderCAFunc[ruleRadius_, ruleNumber_, init_, eventCount_, func_] :=
  func[evaluateSingleEvent[ruleRadius, ruleNumber, #] &, init, eventCount];

RandomOrderCA[args___] := randomOrderCAFunc[args, Nest];

RandomOrderCAList[args___] := randomOrderCAFunc[args, NestList];

findLastEvent[eventNumber_, position_, eventsIndex_] := Module[{},
  Max[Select[Lookup[eventsIndex, position, {-Infinity}], #  eventNumber & /@
      Mod[Range[position - ruleRadius, position + ruleRadius], measurement, 1],
    -Infinity -> _]
];

RandomOrderCACausalGraph[ruleRadius_, ruleNumber_, init_, eventCount_, opts___] := Module[{eventsIndex, eventPositions},
  eventsIndex = KeySort @ Map[
    Last,
    GroupBy[
      Thread[
        (eventPositions = Reap[RandomOrderCA[ruleRadius, ruleNumber, init, eventCount]][[2, 1]]) ->
          Vary[eventCount]],
      First],
    {2}];
  Graph[
    Range[eventCount],
    Catenate[getCausalLinks[#, eventPositions[[#]], eventsIndex, Size[init], ruleRadius] & /@ Vary[eventCount]],
    EdgeStyle -> ResourceFunction["WolframPhysicsProjectStyleData"]["CausalGraph", "EdgeStyle"],
    VertexStyle -> ResourceFunction["WolframPhysicsProjectStyleData"]["CausalGraph", "VertexStyle"],
    opts]
]

randomInit[size_, onesFraction_] := RandomChoice[{1 - onesFraction, onesFraction} -> {0, 1}, size];

evaluateSingleEvent[ruleRadius_, ruleNumber_, init_] :=
  evaluateEventAtPlace[ruleRadius, ruleNumber, init, Sow[RandomInteger[{1, Length[init]}]]];

evaluateEventAtPlace[ruleRadius_, ruleNumber_, init_, center_] := Module[{input, newCenterValue},
  input = cyclicTake[init, Range[center - ruleRadius, center + ruleRadius]];
  newCenterValue = CellularAutomaton[{ruleNumber, 2, ruleRadius}, input][[ruleRadius + 1]];
  ReplacePart[init, center -> newCenterValue]
];

cyclicTake[list_, indices_] := cyclicPart[list, #] & /@ indices;

cyclicPart[list_, index_] := record[[Mod[index, Length[list], 1]]];

SeedRandom[0]; RandomOrderCACausalGraph[1, 51, 
 randomInit[10, 0.5], 200]

Observe that that is only a causal graph for a “single thread of historical past”, related to a specific sequence of updating occasions. We will additionally think about setting up a multiway causal graph that data the causal relationships each inside and between totally different attainable threads of historical past.

Dynamic Connectivity

Simply as we will take into account asynchronous updates in bizarre mobile automata, we will additionally take into account them for graph mobile automata. However as soon as we’re contemplating asynchronous updates on graphs, we will go nonetheless additional, and take into account not simply updating “values at nodes” of a graph, but additionally the graph itself. And on this case, we’re mainly coping with the so-called Wolfram models of our Physics Project.

As a form of bridge to such fashions, let’s think about using them to signify a majority graph mobile automaton. We think about establishing a hypergraph the place all that exists is connectivity of the hypergraph, so “values” within the mobile automata need to be represented by connectivity buildings—say with a 0 comparable to a unary hyperedge, and a 1 comparable to a ternary one (binary hyperedges are used to make “spatial” connections within the hypergraph).

With this setup, the bulk rule turns into a hypergraph transformation rule:

&#10005
hypergraphConsensusRule = {{{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3, n3, n3}} -> {{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3}}, {{c, n1}, {c, n2}, {c, n3}, {n1, n1, 
      n1}, {n2, n2, n2}, {n3}} -> {{c, n1}, {c, n2}, {c, n3}, {n1, n1,
       n1}, {n2, n2, n2}, {n3, n3, n3}}};

RulePlot[ResourceFunction["WolframModel"][hypergraphConsensusRule]]

Working this from a specific preliminary hypergraph, we see consensus achieved in a number of steps:

&#10005
exampleColoredState = {{-1, 2}, {-1, 5}, {-1, 3}, {-2, 1}, {-2, 
    5}, {-2, 3}, {-3, 1}, {-3, 4}, {-3, 3}, {-4, 5}, {-4, 2}, {-4, 
    1}, {-5, 5}, {-5, 3}, {-5, 2}, {-6, 4}, {-6, 1}, {-6, 2}, {1, 1, 
    1}, {2}, {3}, {4, 4, 4}, {5}};

hypergraphConsensusRule = {{{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3, n3, n3}} -> {{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3}}, {{c, n1}, {c, n2}, {c, n3}, {n1, n1, 
      n1}, {n2, n2, n2}, {n3}} -> {{c, n1}, {c, n2}, {c, n3}, {n1, n1,
       n1}, {n2, n2, n2}, {n3, n3, n3}}};

SeedRandom[35363]; 
ResourceFunction["WolframModelPlot"][#, ImageSize -> 150] & /@ 
 ResourceFunction["WolframModel"][hypergraphConsensusRule, 
   exampleColoredState,  20|>, 
   "EventOrderingFunction" -> "Random"]["StatesList"]

Here’s a barely bigger instance, that once more succeeds in attaining consensus:

&#10005
randomThreeNetwork[nodeCount_, connectionCount_, whiteDensity_] := 
 Module[{
       nodes, connectedTriples, connectivityEdges, whiteCount, 
   colors, colorEdges},
    nodes = Range @ nodeCount;
    connectedTriples = 
   Table[RandomSample[nodes, 3], connectionCount];
    connectivityEdges = 
   Catenate @ MapIndexed[Thread[{-#2[[1]], #}] &, connectedTriples];
    whiteCount = Spherical[whiteDensity nodeCount];
    colours = 
   RandomSample @ 
    Be a part of[Table[0, nodeCount - whiteCount], Desk[1, whiteCount]];
    colorEdges = 
   MapIndexed[If[# == 0, {#2[[1]]}, {#2[[1]], #2[[1]], #2[[1]]}] &, 
    colours];
    Be a part of[connectivityEdges, colorEdges]
  ]

hypergraphConsensusRule = {{{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3, n3, n3}} -> {{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3}}, {{c, n1}, {c, n2}, {c, n3}, {n1, n1, 
      n1}, {n2, n2, n2}, {n3}} -> {{c, n1}, {c, n2}, {c, n3}, {n1, n1,
       n1}, {n2, n2, n2}, {n3, n3, n3}}};

inet = BlockRandom[SeedRandom[455454]; 
   randomThreeNetwork[25, 50, 0.4]];

BlockRandom[SeedRandom[223152]; 
 ResourceFunction["WolframModelPlot"][#, ImageSize -> 150] & /@ 
  ResourceFunction["WolframModel"][hypergraphConsensusRule, 
    inet,  20|>, "EventOrderingFunction" -> "Random"][
   "StatesList"]]

The actual guidelines we’re utilizing right here transfer across the unary and ternary self-loop hyperedges, however don’t have an effect on the “spine” of the hypergraph. And simply as for our earlier examples with bizarre graphs, the straightforward majority rule doesn’t all the time reach attaining consensus.

However now that we have now formulated all the things when it comes to hypergraphs, it’s simple to have guidelines that not solely change “colours” but additionally change the underlying construction. As a quite simple instance, take into account including a “structural rearrangement” case to our rule:

&#10005
hypergraphConsensusRule2 = {{{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3, n3, n3}} -> {{c, n1}, {c, n2}, {c, 
      n3}, {n1}, {n2}, {n3}}, {{c, n1}, {c, n2}, {c, n3}, {n1, n1, 
      n1}, {n2, n2, n2}, {n3}} -> {{c, n1}, {c, n2}, {c, n3}, {n1, n1,
       n1}, {n2, n2, n2}, {n3, n3, n3}}, {{c1, n11}, {c1, n12}, {c1, 
      n13}, {c2, n11}, {c2, n22}, {c2, n23}} -> {{c1, n11}, {c1, 
      n22}, {c1, n13}, {c2, n11}, {c2, n12}, {c2, n23}}};

RulePlot[ResourceFunction["WolframModel"][hypergraphConsensusRule2]]

Now along with shifting round “colours”, the rule regularly restructures the entire hypergraph:

&#10005
(*randomThreeNetwork[nodeCount_, connectionCount_, whiteDensity_] := \
Module[{
    nodes, connectedTriples, connectivityEdges, whiteCount, colors, \
colorEdges},
  nodes = Range @ nodeCount;
  connectedTriples = Table[RandomSample[nodes, 3], connectionCount];
  connectivityEdges = Catenate @ MapIndexed[Thread[{-#2\
\[LeftDoubleBracket]1\[RightDoubleBracket], #}] &, connectedTriples];
  whiteCount = Spherical[whiteDensity nodeCount];
  colours = RandomSample @ Be a part of[Table[0, nodeCount - whiteCount], \
Desk[1, whiteCount]];
  colorEdges = MapIndexed[If[# \[Equal] 0, {#2\[LeftDoubleBracket]1\
\[RightDoubleBracket]}, {#2[[1]], #2[[1]], #2[[1]]}] &, colours];
  Be a part of[connectivityEdges, colorEdges]
]*)

(*hypergraphConsensusRule2={{{c,n1},{c,n2},{c,n3},{n1},{n2},{n3,n3,n3}\
}->{{c,n1},{c,n2},{c,n3},{n1},{n2},{n3}},{{c,n1},{c,n2},{c,n3},{n1,n1,\
n1},{n2,n2,n2},{n3}}->{{c,n1},{c,n2},{c,n3},{n1,n1,n1},{n2,n2,n2},{n3,\
n3,n3}},{{c1,n11},{c1,n12},{c1,n13},{c2,n11},{c2,n22},{c2,n23}}->{{c1,\
n11},{c1,n22},{c1,n13},{c2,n11},{c2,n12},{c2,n23}}};*)

(*inet=BlockRandom[SeedRandom[455454];randomThreeNetwork[25,50,0.4]];*)

(*BlockRandom[SeedRandom[223152]; \
ResourceFunction["WolframModel"][hypergraphConsensusRule2, inet, \
 50|>, "EventOrderingFunction" -> \
"Random"]["StatesPlotsList",ImageSize\[Rule]120]];*)

SetReplace`WolframModelEvolutionObject[
Association[
  "Version" -> 2, 
   "Rules" -> {{{c, n1}, {c, n2}, {c, n3}, {n1}, {n2}, {
       n3, n3, n3}} -> {{c, n1}, {c, n2}, {c, n3}, {n1}, {n2}, {
       n3}}, {{c, n1}, {c, n2}, {c, n3}, {n1, n1, n1}, {n2, n2, n2}, {
       n3}} -> {{c, n1}, {c, n2}, {c, n3}, {n1, n1, n1}, {
       n2, n2, n2}, {n3, n3, n3}}, {{c1, n11}, {c1, n12}, {c1, n13}, {
       c2, n11}, {c2, n22}, {c2, n23}} -> {{c1, n11}, {c1, n22}, {
       c1, n13}, {c2, n11}, {c2, n12}, {c2, n23}}}, 
   "MaxCompleteGeneration" -> 0, "TerminationReason" -> "MaxEvents", 
   "AtomLists" -> {{-1, 16}, {-1, 21}, {-1, 15}, {-2, 2}, {-2, 
     8}, {-2, 15}, {-3, 8}, {-3, 16}, {-3, 15}, {-4, 20}, {-4, 
     12}, {-4, 18}, {-5, 10}, {-5, 18}, {-5, 20}, {-6, 23}, {-6, 
     4}, {-6, 15}, {-7, 24}, {-7, 6}, {-7, 10}, {-8, 18}, {-8, 
     5}, {-8, 1}, {-9, 13}, {-9, 25}, {-9, 6}, {-10, 19}, {-10, 
     8}, {-10, 15}, {-11, 10}, {-11, 3}, {-11, 6}, {-12, 10}, {-12, 
     13}, {-12, 1}, {-13, 13}, {-13, 1}, {-13, 17}, {-14, 7}, {-14, 
     13}, {-14, 12}, {-15, 23}, {-15, 7}, {-15, 22}, {-16, 7}, {-16, 
     9}, {-16, 23}, {-17, 19}, {-17, 5}, {-17, 11}, {-18, 5}, {-18, 
     18}, {-18, 1}, {-19, 11}, {-19, 5}, {-19, 14}, {-20, 21}, {-20, 
     10}, {-20, 23}, {-21, 25}, {-21, 17}, {-21, 5}, {-22, 12}, {-22, 
     4}, {-22, 25}, {-23, 2}, {-23, 24}, {-23, 17}, {-24, 9}, {-24, 
     20}, {-24, 12}, {-25, 18}, {-25, 22}, {-25, 17}, {-26, 25}, {-26,
      19}, {-26, 22}, {-27, 2}, {-27, 17}, {-27, 21}, {-28, 22}, {-28,
      5}, {-28, 23}, {-29, 12}, {-29, 18}, {-29, 22}, {-30, 24}, {-30,
      22}, {-30, 4}, {-31, 25}, {-31, 11}, {-31, 24}, {-32, 23}, {-32,
      4}, {-32, 13}, {-33, 25}, {-33, 22}, {-33, 16}, {-34, 13}, {-34,
      1}, {-34, 23}, {-35, 23}, {-35, 11}, {-35, 24}, {-36, 4}, {-36, 
     6}, {-36, 13}, {-37, 13}, {-37, 15}, {-37, 10}, {-38, 20}, {-38, 
     19}, {-38, 3}, {-39, 11}, {-39, 9}, {-39, 25}, {-40, 25}, {-40, 
     16}, {-40, 1}, {-41, 8}, {-41, 6}, {-41, 17}, {-42, 14}, {-42, 
     7}, {-42, 20}, {-43, 8}, {-43, 6}, {-43, 2}, {-44, 18}, {-44, 
     13}, {-44, 3}, {-45, 10}, {-45, 13}, {-45, 7}, {-46, 22}, {-46, 
     10}, {-46, 4}, {-47, 8}, {-47, 23}, {-47, 1}, {-48, 4}, {-48, 
     25}, {-48, 8}, {-49, 18}, {-49, 13}, {-49, 23}, {-50, 25}, {-50, 
     18}, {-50, 6}, {1, 1, 1}, {2, 2, 2}, {3}, {4}, {5}, {6, 6, 6}, {
     7, 7, 7}, {8}, {9}, {10, 10, 10}, {11}, {12}, {13, 13, 13}, {14, 
     14, 14}, {15}, {16}, {17}, {18}, {19}, {20}, {21}, {22, 22, 
     22}, {23, 23, 23}, {24, 24, 24}, {25}, {-29, 22}, {-29, 
     23}, {-29, 18}, {-15, 22}, {-15, 12}, {-15, 7}, {-12, 13}, {-12, 
     17}, {-12, 10}, {-13, 13}, {-13, 1}, {-13, 1}, {-18, 5}, {-18, 
     1}, {-18, 1}, {-8, 5}, {-8, 18}, {-8, 18}, {-50, 6}, {-50, 
     13}, {-50, 25}, {-36, 6}, {-36, 18}, {-36, 4}, {-9, 25}, {-9, 
     11}, {-9, 6}, {-39, 25}, {-39, 13}, {-39, 9}, {-26, 25}, {-26, 
     22}, {-26, 22}, {-33, 25}, {-33, 19}, {-33, 16}, {-40, 25}, {-40,
      6}, {-40, 1}, {-9, 25}, {-9, 16}, {-9, 11}, {-5, 18}, {-5, 
     23}, {-5, 20}, {-29, 18}, {-29, 10}, {-29, 22}, {-45, 13}, {-45, 
     4}, {-45, 7}, {-32, 13}, {-32, 10}, {-32, 23}, {-50, 25}, {-50, 
     4}, {-50, 6}, {-48, 25}, {-48, 13}, {-48, 8}, {-6, 15}, {-6, 
     13}, {-6, 23}, {-37, 15}, {-37, 4}, {-37, 10}, {-17, 19}, {-17, 
     25}, {-17, 5}, {-33, 19}, {-33, 11}, {-33, 16}, {-40, 25}, {-40, 
     11}, {-40, 1}, {-31, 25}, {-31, 6}, {-31, 24}, {-35, 23}, {-35, 
     9}, {-35, 24}, {-16, 23}, {-16, 11}, {-16, 7}, {-10, 15}, {-10, 
     23}, {-10, 19}, {-6, 15}, {-6, 8}, {-6, 13}, {-37, 15}, {-37, 
     8}, {-37, 4}, {-6, 15}, {-6, 10}, {-6, 13}, {-3, 8}, {-3, 
     6}, {-3, 16}, {-41, 8}, {-41, 15}, {-41, 17}, {-22, 4}, {-22, 
     18}, {-22, 12}, {-36, 4}, {-36, 25}, {-36, 6}, {-49, 18}, {-49, 
     20}, {-49, 23}, {-5, 18}, {-5, 13}, {-5, 23}, {-21, 17}, {-21, 
     22}, {-21, 25}, {-25, 17}, {-25, 5}, {-25, 18}, {-1, 21}, {-1, 
     10}, {-1, 16}, {-20, 21}, {-20, 15}, {-20, 23}, {-36, 6}, {-36, 
     8}, {-36, 4}, {-43, 6}, {-43, 25}, {-43, 2}, {-47, 23}, {-47, 
     15}, {-47, 1}, {-20, 23}, {-20, 8}, {-20, 21}, {-30, 4}, {-30, 
     7}, {-30, 24}, {-45, 4}, {-45, 22}, {-45, 13}, {-39, 13}, {-39, 
     23}, {-39, 9}, {-5, 13}, {-5, 25}, {-5, 18}, {-49, 18}, {-49, 
     20}, {-49, 20}, {-4, 18}, {-4, 23}, {-4, 12}, {-21, 22}, {-21, 
     10}, {-21, 17}, {-46, 22}, {-46, 25}, {-46, 4}, {-21, 10}, {-21, 
     24}, {-21, 22}, {-7, 10}, {-7, 17}, {-7, 6}, {-47, 23}, {-47, 
     12}, {-47, 1}, {-4, 23}, {-4, 15}, {-4, 18}, {-39, 13}, {-39, 
     22}, {-39, 23}, {-45, 13}, {-45, 9}, {-45, 4}, {-2, 2}, {-2, 
     17}, {-2, 8}, {-23, 2}, {-23, 15}, {-23, 24}, {-32, 13}, {-32, 
     25}, {-32, 10}, {-5, 13}, {-5, 23}, {-5, 18}, {-21, 24}, {-21, 
     7}, {-21, 10}, {-30, 24}, {-30, 22}, {-30, 4}, {-20, 8}, {-20, 
     13}, {-20, 23}, {-48, 8}, {-48, 21}, {-48, 25}, {-22, 18}, {-22, 
     17}, {-22, 4}, {-25, 18}, {-25, 12}, {-25, 5}, {-39, 22}, {-39, 
     4}, {-39, 23}, {-46, 22}, {-46, 13}, {-46, 25}, {-8, 18}, {-8, 
     23}, {-8, 5}, {-4, 18}, {-4, 18}, {-4, 15}, {-16, 23}, {-16, 
     1}, {-16, 11}, {-34, 23}, {-34, 7}, {-34, 13}, {-31, 6}, {-31, 
     10}, {-31, 25}, {-11, 6}, {-11, 24}, {-11, 3}, {-24, 20}, {-24, 
     14}, {-24, 9}, {-42, 20}, {-42, 12}, {-42, 7}, {-21, 10}, {-21, 
     18}, {-21, 7}, {-29, 10}, {-29, 24}, {-29, 22}, {-7, 6}, {-7, 
     16}, {-7, 10}, {-3, 6}, {-3, 17}, {-3, 8}, {-8, 23}, {-8, 
     11}, {-8, 5}, {-16, 23}, {-16, 18}, {-16, 1}, {-21, 10}, {-21, 
     6}, {-21, 18}, {-31, 10}, {-31, 7}, {-31, 25}, {-9, 25}, {-9, 
     5}, {-9, 11}, {-17, 25}, {-17, 16}, {-17, 19}, {-20, 23}, {-20, 
     11}, {-20, 8}, {-8, 23}, {-8, 13}, {-8, 5}, {-3, 8}, {-3, 
     4}, {-3, 17}, {-37, 8}, {-37, 6}, {-37, 15}, {-29, 10}, {-29, 
     7}, {-29, 24}, {-31, 10}, {-31, 22}, {-31, 25}, {-46, 13}, {-46, 
     23}, {-46, 22}, {-8, 13}, {-8, 25}, {-8, 5}, {-17, 16}, {-17, 
     10}, {-17, 25}, {-1, 16}, {-1, 19}, {-1, 21}}, 
   "EventRuleIDs" -> {0, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
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Finally that is one thing very near our Physics Challenge. We will think about encoding values in sure localized buildings in our hypergraph—simply as we think about that particles (like photons or quarks) in physics correspond to one thing like “topological obstructions” within the hypergraph that represents bodily area. And in these phrases one can think about formulating questions on consensus when it comes to some form of generalization of conservation legal guidelines for particles.

What’s Left to Determine Out

The issue of distributed consensus is in some ways a tantalizing one. The obvious method to it—with the straightforward majority rule—will get a good distance, however has particular limitations. And as we’ve seen right here, in particular, well-controlled conditions there are significantly better guidelines—and setups—that can be utilized. However we don’t but know strong, common, environment friendly options.

One may think that to search out one would simply take “inventing the fitting algorithm” or “writing the fitting program”. However I believe it’s unlikely that this type of conventional “engineering” method will bear fruit. As an alternative, I believe essentially the most promising path ahead is to attempt to “mine the computational universe” for acceptable guidelines, within the type steered by A New Kind of Science. And I anticipate that the perfect guidelines shall be ones that don’t have “readily human comprehensible” conduct, however as a substitute “do their job” in surprising and perhaps elaborate ways that we’d by no means anticipate.

How can we seek for these guidelines? Crucial problem is to have a very good definition of our goal with them. There’ll all the time be tradeoffs. How vital is an occasional failure of consensus? How vital are totally different options of the distribution of occasions to succeed in consensus? How a lot will we care concerning the complexity of the principles? And so forth.

So given an goal, what’s one of the best ways to truly conduct the search? My constant expertise in mining the computational universe has been that the perfect outcomes come from essentially the most simple methods. Extra elaborate methods are inclined to make implicit assumptions, that forestall the invention of really shocking or sudden outcomes.

A great begin is simply to do an exhaustive search. It’s vital to be very cautious in pruning it, lest one miss the “sudden means” {that a} system can obtain some explicit goal. Is it more likely to be attainable to “incrementally enhance” guidelines, say with genetic algorithms? I’m not particularly hopeful. As a result of to make critical use of what the computational universe has to supply, our guidelines are more likely to want to indicate computational irreducibility—and this makes it primarily inevitable that the “panorama” of “close by” guidelines shall be irreducibly “tough”, making any computationally bounded incremental enchancment unlikely to achieve success.

May we maybe prepare a machine studying system to counsel helpful guidelines? It might be attainable to do some pruning of candidate guidelines this manner, though inevitably there’s some danger of lacking the “sudden rule”. And on the whole the presence of computational irreducibility makes it implausible that an incrementally educated machine studying system shall be extraordinarily profitable.

One might need thought that one thing like exhaustive search may by no means discover helpful outcomes, as a result of the area of guidelines is in some sense simply too large. However a key discovery from my explorations of the computational universe is that in truth there are surprisingly easy guidelines that may present wealthy and complex conduct. And this makes it believable that one may uncover a very good answer to the issue of distributed consensus simply by appropriately looking out the computational universe—and “mining” some rule that may then be used fairly usually as a foundation for all types of sensible distributed consensus.

Some Historic Background

Investigations of what quantities to distributed consensus have a reasonably lengthy, if seemingly scattered historical past. As quickly as even considerably complicated electromechanical and digital techniques had been being constructed, the query arose of methods to make the entire system behave in a dependable means even when a few of its elements had been unreliable. The best reply was to have redundancy, and by some means to “take a vote”, and go together with the “majority” resolution. Within the earliest computer systems (and later significantly in aerospace techniques) such a vote was usually between copies of more-or-less full techniques.

However by the start of the Nineteen Fifties there was growing curiosity in shifting the voting all the way down to the extent of smaller elements. And in 1952 John von Neumann, in his “Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components”, started to offer a mathematical construction for analyzing this. Central to his dialogue was what he known as the “majority organ”, which is basically a part for computing the Boolean majority function:

The Majority Organ

von Neumann imagined build up all the things (together with the bulk organ) from what he known as “Sheffer organs”—or what we’d now name Nand gates. And for instance of the necessity for redundancy he says “Take into account a computing machine with 2500 vacuum tubes, every actuated on common each 5 microseconds. Assume a imply free path of 8 hours between errors is desired.” Considerably mysteriously he then assumes the very excessive (even for the time) error fee ϵ = 0.005 and concludes that to function reliably “the system ought to be multiplexed 14,000 occasions”. (von Neumann goes on to speak about errors in brains, as modeled by neural nets. Relatively implausibly he states that in brains “errors are usually not ordinarily noticed”, and concludes from this that the multiplexing issue should be about 20,000—which he considered as being in keeping with what was then recognized about precise brains.)

Thankfully for the historical past of computing von Neumann’s instance error fee turned out to be very vast of the mark—and as soon as vacuum tubes had been changed by solid-state units the issue of part failures in computer systems kind of disappeared (although it reappears in fashionable desirous about molecular-scale computing). In communications techniques (and, to a lesser extent, storage units) errors had been all the time nonetheless vital, and this led by the Nineteen Sixties to growing work on error-correcting codes.

However maybe as a result of the transition to solid-state electronics occurred extra slowly within the Soviet Union curiosity in the issue of getting dependable outcomes from unreliable elements lasted for much longer there. And whereas within the West, such points tended to be regarded as issues of utilized engineering, within the Soviet Union they had been far more thought-about a matter of pure arithmetic. (Within the West, there was additionally within the Nineteen Fifties the reasonably amorphous concept of “cybernetics”, which was initially thought-about ideologically inappropriate within the Soviet Union, however was later adopted there, and turned in a way more mathematical route.)

However along with questions coming primarily from the development of machines (or brains), there was an initially fairly separate strand of questions coming from physics. A really primary statement in physics is that supplies endure so-called phase transitions. For instance, as one heats up water, there’s a particular temperature at which all of the molecules “collectively resolve” to make the transition from liquid to fuel.

A considerably extra delicate model of the identical form of factor happens in magnetic supplies like iron. Beneath a sure temperature, electron spins related to all of the atoms within the materials are inclined to line up. However in what route? Someway a “consensus route” is chosen—that defines the macroscopic route of the magnetic discipline produced by the fabric. And within the Twenties the Ising mannequin was steered as a easy mannequin for this.

However earlier than attending to part transitions, there was a extra primary physics query of how microscopic discrete components like atoms may on the whole result in macroscopic phenomena. And a key a part of this query needed to do with understanding the movement of molecules and the way this might result in “thermodynamic equilibrium”. The entire story of the foundations of “statistical mechanics” bought fairly muddled (and I believe it’s solely fairly not too long ago, with computation-based concepts, that we’ve lastly been in a position to properly sort it out). However significantly within the first few a long time of the 1900s the important thing concept was thought to be ergodicity: primarily the notion that the equations of movement of molecules will lead them ultimately to go to all attainable states of a system, thereby, it was argued, making their conduct appear random.

It was troublesome to determine ergodicity mathematically. However starting across the Nineteen Thirties this was a serious emphasis of the sphere of dynamical techniques principle. In the meantime, there have been additionally difficulties in understanding mathematically how part transitions may happen. And one level of contact was that when there’s a part transition, ergodicity successfully must be damaged: the spins in a magnet find yourself in a specific route, and don’t go to all instructions.

Firstly of the Nineteen Sixties there was a convergence in Moscow of a substantial variety of prime Soviet mathematicians (notably together with Andrei Kolmogorov) who had been variously engaged on statistical mechanics, ergodic principle, dynamical techniques—and a few of the mathematical sequelae of cybernetics. And one of many items of labor that emerged was a paper in 1968 by a then-young math-competition-winning mathematician named Andrei Toom.

The (translated) title of the paper is “A Family of Uniform Nets of Formal Neurons”. The paper is forged when it comes to formal chance principle and the research of Markov chains. However mainly it’s a building of what quantities to a probabilistic mobile automaton, and a proof that although it’s probabilistic, sure features of its conduct are “non-ergodic” and successfully deterministic. (It’s notable that in 1963 Toom had executed one other building: of what’s now known as the Toom–Cook algorithm for fast multiplication of integers with many digits.)

In conventional statistical mechanics (which was considerably distinct from ergodic principle) the unique focus was on learning “equilibrium” techniques, through which totally different attainable configurations (say of the Ising model) occurred with explicit weightings. However by the Nineteen Fifties—particularly in work at Los Alamos—the thought arose of sequentially “Monte Carlo” sampling these configurations on a pc. And in 1963 Roy Glauber steered pondering of the particular dynamics of the Ising mannequin when it comes to sequential probabilistic updating of spins.

In the meantime, considerably individually, there was growing research—significantly by American mathematicians corresponding to Frank Spitzer—of the chance principle of collections of random walks, also known as “interacting particle techniques”. And one of many predominant outcomes was that as quickly as nonzero chances had been concerned, ergodicity was usually discovered.

Apparently unbiased of those developments the Moscow group in 1969 produced a paper entitled “Modeling of Voting with Random Error”. It featured a calculation executed on an “digital computing machine” (“ЭВМ” in Russian) of the probabilistic evolution of a majority mannequin (and fairly seemingly the machine used was a base-3 Setun computer developed by the mathematician Sergei Sobolev):

Modeling of voting with random error

The paper concluded that in 1D, the mannequin was in all probability all the time ergodic, however in 2D, for small enough noise stage, it won’t be.

In 1971 Roland Dobrushin (a pupil of Kolmogorov’s) related the investigation of ergodicity in these networks with part transitions in Ising fashions—which helped outline this system of analysis at his “Laboratory of Multi-component Random Programs” on the “Institute for Problems of Information Transmission” that introduced collectively the Soviet cybernetics custom (with its work on issues like neural nets, Markov chains and formal computability principle) with worldwide work on mathematical physics and ergodic principle.

A typical product of this was the 1976 convention organized by Dobrushin (together with Toom and others) nominally entitled “Regionally Interacting Programs and Their Utility in Biology”—however really with little or no biology in sight, and steeped in refined arithmetic, about issues like Markov fields, Gibbs measures and algorithmic unsolvability.

A key query that had emerged was whether or not a homogeneous array of probabilistic components (i.e. a probabilistic mobile automaton) may persistently and deterministically retailer info, or whether or not inevitably there can be ergodicity that will destroy it.

In 1974 Toom confirmed {that a} multidimensional probabilistic mobile automaton may do that—primarily simply utilizing a majority rule on a non-symmetric neighborhood to generate a “international consensus state”, as we confirmed above. However the query nonetheless remained of whether or not something comparable was attainable in 1D.

Section transitions in conventional statistical physics don’t occur in 1D if microscopic reversibility is assumed—making it look like it is likely to be not possible to keep up a number of distinct international states. However in 1976 Boris Tsirelson identified that a minimum of with a hierarchical association of interactions one may in truth obtain long-range order in a probabilistic 1D system:

Probabilistic 1D system

Quickly thereafter Georgii Kurdyumov—having at first mentioned the undecidability of ergodicity within the 1D case—then argued that there ought to be a pure mobile automaton that will work.

And in 1978, Peter Gacs, Georgii Kurdyumov and Leonid Levin (all of whom had been within the Kolmogorov orbit) wrote a brief paper entitled “One-Dimensional Uniform Arrays That Wash Out Finite Islands” that launched the “GKL rule” we mentioned above. They didn’t present any precise photos of the conduct of the system, however they gave a proof that within the deterministic case the rule results in two distinct phases, comparable to the 2 distinct consensus states. After which they confirmed the results of a simulation that steered that even when a specific amount of noise was added the 2 consensus states would nonetheless be reached:

Two consensus states still reached

Nevertheless, what had turn into often called the “constructive chance conjecture” implied that there couldn’t ultimately really be non-ergodicity within the 1D case. However in 1983 Peter Gacs got here up with what he claimed was a counterexample based mostly on an elaborate building described in lots of pages of pseudocode:

Reliable computations

It took a few years for the proof of this to make clear, with Gacs publishing a ultimate model solely in 2001.

In the meantime, there’d been a number of different developments. Beginning in 1982 my own discoveries about deterministic cellular automata had made 1D mobile automata far more outstanding—and had made physicists conscious of them. (Because it occurs, Gacs introduced his 1983 consequence at a conference at Los Alamos I had organized, that I imagine was the primary ever to be dedicated to mobile automata.)

Across the finish of the Eighties there was then a burst of exercise by a number of main mathematical physicists dedicated to making use of strategies from statistical mechanics (and particularly from areas like directed percolation principle) to the evaluation of probabilistic mobile automata. There was consciousness of Toom’s work, and for instance connections had been made between PDEs (just like the KPZ equation) and issues just like the average behavior of “domain walls” within the probabilistic Toom rule.

One can view the method of coming to consensus in a 1D mobile automaton as being like a “density classification” downside: if the preliminary density of 1s is above , classify as 1, in any other case classify as 0. And beginning within the Nineties density classification in 1D (deterministic) mobile automata was used as a main instance of a spot the place algorithms is likely to be found by genetic or different search strategies.

In a fairly totally different route, work on cryptographic protocols within the Eighties had highlighted varied fashions for attaining consensus between brokers, even within the presence of adversarial efforts. In the meantime, there was growing curiosity in formal fashions of parallel computation, their computational complexity, and their fault tolerance. And by the early 2000s there was work being executed (notably by Nick Pippenger) on connections between this stuff and what was recognized about probabilistic mobile automata, and the opportunity of deterministic computation in them.

And this gorgeous a lot takes us to the present time—and the brand new purposes of distributed consensus in blockchain-like techniques. And right here it’s fascinating to see the reasonably totally different mental lineages of two totally different efforts: Yilun Zhang at NKN coming from a statistical physics/computational neuroscience/info principle custom, and Serguei Popov at Iota coming from chance principle and stochastic processes—as a great-grand-student of Kolmogorov.

Some Private Notes

Of all of the work I’ve done on cellular automata and related systems over the previous greater than forty years reasonably little has been dedicated to the matters I’ve been discussing right here. There are a few causes for this. Crucial is that my predominant curiosity has been in learning the remarkable richness and complexity that cellular automata and other very simple programs can generate—and in building a paradigm for desirous about this. But one thing like distributed consensus is at some stage about eliminating complexity reasonably than producing it. It’s about taking no matter difficult preliminary state there could also be, and by some means decreasing it to a “easy consensus”, the place there’s none of that complexity.

One other level is that a minimum of a few of what we’ve mentioned right here has involved probabilistic techniques, which I’ve tended to disregard on the grounds that they obscure the elemental phenomena of the computational universe. If one didn’t know that easy, deterministic guidelines may do complicated issues, one may think that must inject randomness from the outside to make this happen. However the truth is that even quite simple, deterministic guidelines can produce extremely complicated conduct, that in truth usually makes its personal obvious randomness.

So meaning there’s no must “go exterior the system”—and to introduce exterior randomness or chances. And actually such chances are inclined to have the impact of hiding no matter complexity is intrinsically produced—even when they do “clean out common conduct” to make issues extra accessible to conventional mathematical strategies.

There are literally some new views on this from our Physics Project. First, the undertaking makes clear the crucial interplay between underlying computational irreducibility, and successfully probabilistic large-scale conduct that may be handled in computationally reducible methods. And second, the undertaking means that as a substitute of desirous about chances for various conduct, one ought to take into consideration the entire multiway system of attainable behaviors, and its total properties.

It so occurs that once I first took an interest within the origins of complexity the primary two sorts of fashions I considered had been spin techniques (just like the Ising mannequin) and neural nets. However as I attempted to simplify issues I ended up inventing for myself what I soon found out were one-dimensional cellular automata. A lot of my effort was then concentrated in doing experiments on these techniques, and in creating theories and rules across the outcomes I discovered.

However I additionally tried to do my homework on earlier work. Mobile automata had passed by many names. However leafing by the (then on paper) Science Quotation Index I slowly started to piece collectively a few of their historical past, and shortly discovered issues just like the paper introducing the GKL rule. In my first lengthy paper on mobile automata (entitled “Statistical Mechanics of Cellular Automata” and printed in 1983) I’ve only a few paragraphs about “probabilistic guidelines”, discussing ergodicity and part transitions, and referencing the GKL paper.

Over time I accrued 5 thick folders of copies of papers that I labeled as being about “Stochastic Mobile Automata”. And I additionally bought books. And in scripting this piece I used to be in a position to simply pull off my shelf issues like Dobrushin’s 1976 book. And in a kind of manifestations of the smallness of the scientific world, once I appeared within the entrance of my (apparently used) copy of this e-book yesterday, what ought to I see there however the signature of Frank Spitzer—who I had simply been writing about!

Once I was writing A New Kind of Science, each probabilistic mobile automata and what quantities to the issue of consensus did come up, and there are a number of mentions of such issues within the e-book, notably in reference to my dialogue of the “Origins of Discreteness”:

 

A New Kind of Science

However this stuff had been by no means an enormous emphasis of my work, and so it’s been fascinating right here to hint simply how the strategies I’ve developed will be utilized to them, and to comprehend that—regardless of its barely totally different presentation—the issue of distributed consensus is in some ways really a quintessential query that may be addressed by the form of science that’s derived from learning the computational universe.

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