分布式共识问题澳门新利娱乐网址

澳门新利娱乐网址细胞自动机分布式共识&相关系统研究会议

为一个题为“澳门新利娱乐网址元胞自动机及其相关系统的分布式共识“我们正在组织的NKN(=“新型网络”)我决定探索分布式共识的问题,使用的方法来自澳门新利娱乐网址一种新的科学(是的,NKN与NKS“押韵”)以及从Wolfram物理项目

一个简单的例子

& # 10005

BlockRandom [SeedRandom [77];Module[{pts = RandomPointConfiguration[HardcorePointProcess[0.09, 2, 2], Rectangle[{0,0}, {40,40}] ["Points"], clrs}, clrs = Table[RandomChoice[{. 0, 2, 2],65 .35点}- >{色调(0.15,0.72,1),颜色(0.98、1、0.8200000000000001)}),长度(pts)];图形({EdgeForm(灰色)、表(样式(磁盘(分[[n]]], clr [[n]]], {n,范围(长度[点]]}]}]]]

考虑一个“节点”集合,每个节点都有两种可能的颜色。我们想要确定节点的多数或“一致”颜色,即哪个颜色在节点中更常见。

找到这种“多数”颜色的一个明显的方法就是按顺序访问每个节点,并汇总所有颜色。但是如果我们可以使用分布式算法,在不同的节点上并行运行计算,那么它可能会更有效。

一种可能的算法如下所示。首先将每个节点连接到一定数量的邻居。现在,我们将根据节点的空间布局选择邻居:

& # 10005

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 = Table[RandomChoice[{. 0, 2, 2],65 .35点}- >{色调(0.15,0.72,1),颜色(0.98、1、0.8200000000000001)}),长度(pts)];ConsensusState [pts, clr]]]

该算法按一系列步骤运行,每一步更新每个节点的颜色,使之与其相邻节点的“多数颜色”相同。在本例中,这个过程在几个步骤后收敛,使所有节点都具有“多数色”(这里是黄色),或实际上“同意”什么是多数色:

& # 10005

ConsensusState [points_ colors_ nn_: 5]: = NearestNeighborGraph[点,nn, DirectedEdges - >真的,DistanceFunction - > EuclideanDistance, VertexStyle - > MapThread[规则,{点,颜色}],VertexSize - > 0.75, EdgeStyle - >灰色]NodeDependencies [points_ nn_: 5]: =表[n - >平(地图(位置点,# &,VertexOutComponent [NearestNeighborGraph[点,神经网络,SynchronousStepNewColors[dependenscies_, colors_]:= Flatten[Map[With[{neighbors = Sort[Counts[Part[colors, Last[#]], Greater]}, If[DuplicateFreeQ[Values[neighbors]], First[Keys[neighbors]], colors[[First[#]]]]]&,],},Module[{pts = RandomPointConfiguration[HardcorePointProcess[0.09, 2, 2], Rectangle[{0,0}, {40,40}] ["Points"],clrslist, highlight}, clrslist=NestList[SynchronousStepNewColors[NodeDependencies[pts], #]&, Table[RandomChoice[{.65, #];35} - >{黄、红}],[点]],长度7];MapIndexed[With[{colors = #}, Graph[ConsensusState[pts, colors],ImageSize->150]] &,clrslist]]],4],ImageSize-> 600]

这是一个运行中的分布式共识算法的简单示例。澳门新利娱乐网址我们将在这里讨论的挑战是找到最有效和最健壮的此类算法。

背景

在任何有计算机、人员、数据库、测量设备或其他任何东西的分散系统中,最终可能会在不同的“节点”上产生不同的值或结果。但出于各种原因,人们往往希望就单个“共识”值达成一致,例如,人们可以使用该值“做出决定并继续下一步”。

区块链是需要这种共识才能“完成每个区块”的系统的一个例子。传统的区块链通过一种集中的机制实现共识(即使产生了多个“分散”的区块链副本)。

但现在开始出现需要分布式共识算法的区块链分布式模拟。正在开发的算法的主要灵感是澳门新利娱乐网址元胞自动机(以及统计力学中较小程度上的自旋系统)。

一个问题是使算法尽可能高效。另一种方法是使它尽可能地健壮,例如对于在节点或节点之间引入的随机噪声或恶意错误。

随机噪声的数量可以被认为是类似于温度的东西。至少在某些情况下,可能会有一个“相变”,因此在低于某个“温度”时,对一致性输出的影响可能为零——这意味着对某个噪声水平的鲁棒性。

有些现象可以用标准平衡统计物理学的方法来研究。但在大多数情况下,人们必须考虑系统的时间依赖性或演化,从而导致类似于概率元胞自动机(与定向渗透、动态自旋系统等密切相关)的东西。

正如我将在下面讨论在计算机的早期,人们对用不可靠的部件合成可靠的系统有很大的兴趣。到了20世纪60年代,人们开始研究神经网络,然后研究带有概率元素的细胞自动机。一些令人惊讶的结果表明,细胞自动机可以被建立起来,在一定的非零噪声水平上是稳健的。

元胞自动机的一个特点是,它们的元素都被假定成一个确定的数组,并以一系列步骤“同时”并行更新。然而,对于许多实际应用,人们希望元素以某种图形形式连接(甚至可能是动态的),并且通常是异步更新的,没有特定的顺序。

我们上面给出的简单例子是a图细胞自动机:元素之间的连接由图表定义,但更新都是在每一步同步完成的。在过去,很难分析没有严格的空间或时间概念的更一般的设置。但这正是我们新系统中的设置物理项目因此,现在有可能利用它的形式主义和结果(以及从物理学引入的直觉)来取得进一步的进展。

确定元胞自动机

为了对分布式共识问题有一些直观认识,让我们考虑以下非常简单的设置。澳门新利娱乐网址我们有一排细胞,每个细胞有两种可能的颜色。然后,我们根据一个依赖于邻近细胞的局部规则,按照一系列步骤更新这些细胞的颜色。这个系统是一维的细胞自动机,就像我四十多年前开始学习

我们假设初始条件包含一个分数p红色的细胞。我们希望所有的细胞都变成红色,如果p>,如果它们都变成黄色p<.实现这一点的最明显的规则是将每个细胞替换为其周围的大多数颜色(232规则在我的编号计划中):

& # 10005

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

下面是规则232从“随机配置”中的70%红细胞开始所做的:

& # 10005

BlockRandom[SeedRandom[567];ArrayPlot[CellularaAutomaton[232,RandomChoice[{.7,3}->{1,0},120],20],Mesh->True,ColorRules->{0->色调[0.15,0.72,1],1->色调[0.98,1,0.8200000001],[MeshStyle->橙色]]

正如我们所看到的,它设法达成了一点“局部共识”,但最终它无法达成所有细胞都是相同颜色的“全球共识”。

我们可以想象,对于一个一维确定性细胞自动机来说,没有规则会导致全球共识(或者能够解决决定初始红细胞密度是高于还是低于50%的“密度分类问题”)。但事实证明这不是真的。例如1978年构建了“半径3”规则(适用于7号街区)(我们称之为“GKL规则”):

& # 10005

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

以下是这个规则对60%的红细胞的作用:

& # 10005

BlockRandom [SeedRandom [567];ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0}, 7] /。{l3_, _, l1_、c_ r1_, _, r3_}: >如果[如果[c = = 0, r1 + r3, l1 + l3] + c > = 2, 1, 0), 2), 2、3}({所击垮。6。4}- > {1,0},300),60],ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

这是它对40%的红细胞的作用:

& # 10005

BlockRandom[SeedRandom[569];ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0},7]/.{l3},l3},l1},l1},c},r1},r3}:>如果[If[If[c==0,r1+r3,l1+l3]+c>=2,1,1,0],2,2,3},2,3},随机选择[{.4,.6}->{1,0},300],60],色相[0.15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],假帧[8]

在这两种情况下,该规则都成功地达成了“全球共识”。事实上,我们可以证明这个规则总是这样,至少在足够多的步骤之后。以下是不同初始密度下密度随时间变化的示意图:

& # 10005

data = ParallelTable[如果[p = = 5,没什么,MeanAround / @Transpose[表(意思是/ @CellularAutomaton [{FromDigits(元组[{1,0}7]/。{l3_, _, l1_、c_ r1_, _, r3_}: >如果[如果[c = = 0, r1 + r3, l1 + l3] + c > = 2, 1, 0), 2), 2、3},所击垮[{p, 1 - p} - > {1, 0}, 5000], 200), 200]]], {p。3、7 . 05}];ListLinePlot [MapThread [Callout[# 1,行[{风格(“p”,斜体),“=”,# 2}]]&、{数据,[范围[病例。3, .7, .05],除[0.5]]}]]

我们看到的是,对于初始密度,这看起来像是一个相变p< 0.5,最终密度正好是0,而对于初始密度p>是0.5,正好是1。

具体发生在p= 0.5 ?在某种意义上,元胞自动机“无法做出决定”,在一条无限的直线上,它生成一个在0和1之间交替的无限嵌套序列域:

& # 10005

BlockRandom [SeedRandom [24425];ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0}, 7] /。{l3_, _, l1_、c_ r1_, _, r3_}: >如果[如果[c = = 0, r1 + r3, l1 + l3] + c > = 2, 1, 0), 2), 2、3}({所击垮。5, 5} - > {1, 0}, 1000), 500), ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

这种嵌套结构是典型的见于临界现象在统计物理学中,事实上像这样的细胞自动机是"正确"的最简单的例子相变这我知道。(像其他相变一样,除了在无限系统中,这些相变不会变得“尖锐”。在典型的统计力学中,人们不会在1D中得到相变,但这是微观可逆性假设的结果,这种假设不适用于这样的细胞自动机。)

那么还有什么其他的细胞自动机规则能达成这样的共识呢?没有半径为1的规则。如果一个人搜索所有两个32半径-2规则(就像我做的一种新的科学),最好的一个发现是少数几个实现“近似共识”的例子,即大多数(尽管不是全部)单元格都指向“多数值”(这是r= 2规则4196304428,forp= 0.6):

& # 10005

BlockRandom [SeedRandom [567];ArrayPlot[CellularAutomaton[{4196304428, 2, 2}, RandomChoice[{。6。4}- > {1,0},500),200),ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

顺便说一下,在半径1规则中,有规则184(通常用作道路交通流的基本模型),它在“总体密度”上没有达成共识,但在左向和右向移动的条纹上确实达成了共识,这里有当生成的嵌套模式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]}, Frame -> False]]

那么“更快达成共识”呢?下面是我们原来的GKL规则与另一个平均共识时间更短的radius-3规则(通过遗传编程方法发现)的比较:

& # 10005

列[BlockRandom [SeedRandom [24125];ArrayPlot [CellularAutomaton[#,({所击垮。48, .52} -> {1,0}, 800], 180], ColorRules -> {0 -> Hue[0.15, 0.72, 1], 1 -> Hue[0.98, 1, 0.8200000000000001]}, Frame -> False, ImageSize -> {650, Automatic}] & /@ {{339789091192587366278221041213531750560, 2,3}, {3398410149534664291329676805280, 2,3}}]

一般来说,“最快”半径-3法则是什么还不知道。以上两条规则的特点是它们以一种相当“简单”的方式“完成工作”。但也有如下规则以一种“更华丽”的方式发挥作用:

& # 10005

列[BlockRandom[SeedRandom[24125];阵列图[CellularaAutomaton[#,RandomChoice[{.48,52}->{1,0,800],240],颜色规则->{0->色调[0.15,0.72,1],1->色调[0.98,1,0.8200000000000001],帧->假,图像大小->{650,自动}]&/{33760729844690146542393000449784552,2,3},{338557163696199686868686838383838388,},{313421633154342960352882914658469183496, 2, 3}}]

人类工程规则(就像上面的第一条)几乎不可避免地以更简单和更“可理解”的方式运行。但在其他地方的经验(如与最优排序网络)表明,最优规则通常是那些在行为上看起来不简单的规则,这些规则不能实际地由标准工程方法构建,本质上只需要通过搜索可能规则的计算宇宙来“实验”找到。

我们所研究的早期规则的一个显著特点是,它们显示了少数类型的非常不同的“域”,它们之间有明确的墙或边界。在许多方面,这种墙可以被认为是类似的局部结构“缺陷”或“粒子”。但就我们的目的而言,这里最重要的是这些粒子是否会四处移动,以及它们是否会相互湮灭以留下一个统一的“共识”最终状态。

在简单多数决定原则中,不可避免地存在静态的领域墙:

& # 10005

BlockRandom[SeedRandom[567];ArrayPlot[CellularaAutomaton[232,RandomChoice[{.6,4}->{1,0},300],60],ColorRules->{0->色调[0.15,0.72,1],1->色调[0.98,1,0.8200000001]},MeshStyle->橙色]]

原因是,只要一个域比细胞自动机的邻域大,在域边界上的细胞将不可避免地看到边界两侧各颜色细胞数量的平衡。因此,细胞本身将充当一个“决胜局者”,并将始终决定保持自己的颜色,从而使域边界保持不变。

那么,如果我们有一个更长的范围规则,取样更远的细胞呢?当range-2(即5细胞邻域)宽度低于4的“错误域”消失时:

& # 10005

majrn[n_]:= FromDigits[If[Total[#] > n/ 2,1,0] & /@ Tuples[{1,0}, n], 2] BlockRandom[seerandom [567];ArrayPlot [CellularAutomaton [{majrn[5], 2,{{2},{1},{0},{1},{2}}},{所击垮。6。4}- > {1,0},300),60],ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},MeshStyle - >橙色]]

如果被采样的细胞不是相邻的,但是在模式中,事情会更好一些

& # 10005

majrn[n_]:= FromDigits[If[Total[#] > n/ 2,1,0] & /@ Tuples[{1,0}, n], 2] BlockRandom[seerandom [567];ArrayPlot [CellularAutomaton [{majrn[5], 2,{{3},{1},{0},{2},{4}}},{所击垮。6。4}- > {1,0},300),60],ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},MeshStyle - >橙色]]

但是,使用纯多数规则的有限大小采样不会删除所有域。那么GKL规则呢?这条规则实际上只对5个细胞取样,但其“末端”距离为3。那么,我们能通过让它对更远处的细胞进行取样来“改进”它吗?

下面是几个案例的对比(第一个是原始案例):

& # 10005

GraphicsGrid(分区(标记为[BlockRandom [SeedRandom [567];ArrayPlot[CellularAutomaton[{4177065992, 2, List /@ #}, RandomChoice[{。6。4}- > {1,0},300),100),ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},MeshStyle - >橙色,图象尺寸- > 300]],[# 11风格 ]] & /@ {{- 3 1 0 1 3},{5 1 0 1 5},{3 1 0 1 5},{3 1 0、2、4}},2]]

这里我们只讨论了每个细胞有两种可能颜色的细胞自动机。我们还可以考虑涉及其他“辅助”颜色的规则,这些颜色要么在达到最终状态之前消失,要么定义额外的一致状态。

但它总是有效的吗?

我们已经看到了一维细胞自动机——至少在我们看过的例子中——实现了“多数共识”。但是,给定一个特定的规则,它总是会达成共识吗?还是有例外?

第一个得到这个问题的定义版本的方法,我们可以考虑有限元胞自动机,总共说n单元格和循环边界条件。共有2个n在这种情况下可能的配置,我们可以用a表示元胞自动机的所有可能的进化路径状态转移图

这是我们在上面讨论过的GKL规则的图表n= 5。图中的每个节点都根据其“红细胞分数”是高于还是低于上色.我们在这个例子中看到的是“完美密度分类”或“完美共识”,所有的州都正确地导致了全红或全黄的州:

& # 10005

与[{n = 5},图[# - > CellularAutomaton[{339789091192587366278221041213531750560、2、3 }][#] & /@ 元组({1,0},n), VertexStyle  -> (# -> 如果(意思是[#]> 1/2,色调(0.98、1、0.8200000000000001),色调[0.15,0.72,1]]& / @元组({1,0},n)), GraphLayout - >{“VertexLayout”——>“LayeredDigraphEmbedding”、“PackingLayout”——>“LayeredLeft”},EdgeStyle - > \ !\(\* TagBox[StyleBox[InterpretationBox[RowBox[{"CloudGet", "[", "\"\\"", "]"}], Hue[0.75, 0,0.35], Editable->False, Selectable->False], ShowSpecialCharacters->False, ShowStringCharacters->True, NumberMarks->True], FullForm]\)]]

但当我们看到,例如n=7,我们立即看到一个问题:

& # 10005

与[{n = 7},图[# - > CellularAutomaton[{339789091192587366278221041213531750560、2、3 }][#] & /@ 元组({1,0},n), VertexStyle  -> (# -> 如果(意思是[#]> 1/2,色调(0.98、1、0.8200000000000001),色调[0.15,0.72,1]]& / @元组({1,0},n)), VertexSize——>。4、GraphLayout - >{“VertexLayout”——>“LayeredDigraphEmbedding”、“PackingLayout”——>“分层”},EdgeStyle - > \ !\(\* TagBox[StyleBox[InterpretationBox[RowBox[{"CloudGet", "[", "\"\\"", "]"}], Hue[0.75, 0,0.35], Editable->False, Selectable->False], ShowSpecialCharacters->False, ShowStringCharacters->True, NumberMarks->True], FullForm]\)]]

美国

& # 10005

ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0}, 7] /。{l3_, _, l1_、c_ r1_, _, r3_}: >如果[如果[c = = 0, r1 + r3, l1 + l3] + c > = 2, 1, 0), 2), 2、3}#,4],ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- > False,网- >真的,MeshStyle - >橙色,图象尺寸- >小]& / @ {{0,0,1,0,0,1,1},{0,0,1,1,0,1,1}}

而它们的周期性变化会“卡住”,无法达成共识。在11号协议中还有另一个问题:现在一些本应达成“共识1”的州实际上变成了“共识0”:

& # 10005

(3)2,3}图[{{size11]和[{{size11}图[{{size11}图[{3.11]和[{{3.11]图[{1,0.5,2,3{3}图[{1,0.15,2,3}3}]和/元组[{{1,0.1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3},3},3}]图[{3}]图[[3}3}3}]和/3}]图[[3]图[[3,3,3,3,3}]和/3}图[[3,3,3,3}]图[[3,3,3,3}]和/3}]图[[3\\“,”]“}]、色调[0.75,0,0.35]、可编辑->假、可选->假、ShowSpecialCharacters->假、ShowStringCharacters->真、NumberMark->真、FullForm]\]、网格[{Show[Graph[#,AspectRatio->1/3]、图像大小->{300,自动}]、[WeaklyConnectedgComponents[size11][[1]],带[{Stick=Catenate][EdgeList/@Drop[WeaklyConnectedGraphComponents[size11],2]},显示[Graph[Stick,VertexStyle->(#->如果[Mean[#]>1/2,色调[0.98,1,0.8200000000000001],色调[0.15,0.72,1]&/@(First/@Stick)),EdgeStyle->\!\(\*标记框[StyleBox]解释框[RowBox[{“CloudGet”,“[”,“,”,“\”\\“,”]“}],色调[0.75,0,0.35],可编辑->假,可选择->假],ShowSpecialCharacters->假,ShowStringCharacters->真,NumberMarks->真,FullForm]\),GraphLayout->{“VertexLayout”->“LayeredDigraphEmbedding”,“PackingLayout”->“Nestegrid”},ImageSize->{Automatic,200}]},{Show[Graph[#,AspectRatio->1/3],ImageSize->{300,Automatic}]&[WeaklyConnectedGraphComponents[size11][[2]],SpanFromAbove},对齐->{Center,Center}]]

在这里,“达成错误共识”的状态都是以下的循环变化

& # 10005

ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0},7]/.{l3},l1},c},r1},r3}:>如果[If[c==0,r1+r3,l1+l3]+c>=2,1,0,2,3},{7],颜色规则->0->色调->色调[0.15,0.72,1],色调[0.98,1,0.0000000000001],[0],帧,假网格,真网格大小,橙色/->{{1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1}, {0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0}}

第一种情况是6个细胞和5,但最后的状态就是一切在第二种情况下,情况正好相反。

事实证明,这实际上是一个普遍的问题:我们可以证明,在具有循环边界条件的有限数组上,没有任何规则可以完美地实现“多数共识”。

那么无限数组呢?在这里,除了一组度量为0的“特殊初始条件”之外,对所有条件都有可能实现“完美多数共识”。这种“特殊初始条件”的一个例子是上面所示的两个方块中的任意一个的无限重复。这些初始条件——而不是达成共识——将只是随着时间的推移而保持固定。

如果初始条件是“随机”生成的,每个单元的值是根据一定的固定概率选择的,那么获得一个“异常”初始条件的概率实际上为零。即使“逐渐减少”可能是任意的长,也不可能最终不达成共识。

但这个结论是基于初始条件是“随机”产生的。例如,如果它们是由一个确定的程序生成的,那么尽管初始条件对于某些测试来说似乎是“统计上随机的”,但这并不意味着它们不会给“例外的”初始条件以特殊的权重。

在一维

在一维空间中,我们可以用一维效应无法“绕过彼此”来解释某些构型“卡住”而无法达成共识的事实。但在2D中没有这样的约束。

那么“纯粹的2D多数决定原则”(极权主义的代码56):

& # 10005

RulePlot[CellularAutomaton[<|"Dimension" -> 2, "Neighborhood" -> 5, "TotalisticCode" -> 56|>], ColorFunction -> (Blend[{RGBColor@Hue[0.15, 0.72, 1], RGBColor@Hue[0.98, 1, 0.82000000000000000001]}, #] &), MeshStyle -> Orange]

从30% 1开始,我们再次看到事情卡住了

& # 10005

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

这是随着时间的推移,在3D中显示的相应的演化:

& # 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[{. ArrayPlot3D[#, ColorRules -> {0 -> Hue[0.98, 1, 0.8200000000000001]},3, .7} -> {1,0}, {30,30}], {{0,10}, All}]]

但这里有另一个规则(9个邻居的totalistic代码976):

& # 10005

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

现在我们在这个例子中看到的是状的域“少数颜色”的颜色被留下,但逐渐变小。我们可以在3D中看到这种现象:

& # 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}])]

看着中间的时空切片,让更多遥远的细胞“退隐到雾中”,我们看到了“扩散”行为,域墙实际上在执行最终湮灭的随机漫步

& # 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 - >(混合[{RGBColor@Hue [0.15, 0.72, 1], RGBColor@Hue[0.98、1、0.8200000000000001 ]}, #] &), 框架- >错误]]

我们刚刚看到的规则与9单元格3×3区域的多数规则很接近,除了总数4和5,它们被取为1和0,而不是0和1。如果我们在3×3区域使用纯多数原则,它就会卡住:

& # 10005

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

但事实证明,2D多数决定原则不会被卡住。事实上,基本上任何以不对称方式取样细胞的多数原则都是可行的。

例如,考虑一个规则,它对每个3×3邻居中的以下单元格进行抽样:

& # 10005

majplot[offsets_]:= ArrayPlot[Reverse[Transpose[SparseArray[(2 + offset) -> 1, {3,3}], Mesh -> True, ImageSize -> 40] majplot[{{0,1}, {1,0}, {0,0}}]

下面是这个规则从45% 1开始的3D演变:

& # 10005

BlockRandom[SeedRandom[23424];ArrayPlot3D[#,ColorRules->{0->Hue[0.15,0.72,1],1->Hue[0.98,1,0.8200000001]}和@(CellularaAutomaton[{{u,a{u,a},a},{u},{u,b{u,c},{u,{u,},{u,},},{u,},{u,},}如果[a+b+c>=2,1,1,0]},0],随机选择[.45,55,1,50},{

这就是时空切片的样子:

& # 10005

BlockRandom [SeedRandom [23424];ArrayPlot[是/ @转置[MapIndexed[# 1 *(1.6 ^(去年[2])&、CellularAutomaton [{{{_, _, _}, {_, b_ c _}, {_, _, _}} :> 如果(a + b + c > = 2, 1, 0]},{所击垮。45、55}- > {1,0},{220,40}),60),{1}),2 < - > 3)ColorFunction - >(混合[{RGBColor@Hue [0.15, 0.72, 1], RGBColor@Hue[0.98、1、0.8200000000000001 ]}, #] &), 框架- >错误]]

作为初始密度函数的行为在50%时显示出明显的转变:

& # 10005

行[ParallelTable [BlockRandom [SeedRandom [23424];标记[ArrayPlot[是/ @ ' [MapIndexed[# 1 *(1.6 ^(去年[2])&、CellularAutomaton [{{{_, _, _}, {_, b_ c _}, {_, _, _}} :> 如果(a + b + c > = 2, 1, 0]},所击垮[{p, 1 - p} - >{1, 0},{60, 40}), 60),{1}), 2 < - > 3),图象尺寸- > 72,ColorFunction - >(混合[{RGBColor@Hue [0.15, 0.72, 1], RGBColor@Hue[0.98、1、0.8200000000000001 ]}, #] &),框架- >假],风格(PercentForm [p], 11]]], {p。3、7 . 05}]]

这里是3×3附近不同细胞取样的结果;成功达成共识:

& # 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},所有}]),ArrayPlot[Mean /@ Transpose[MapIndexed[#1*(1.6^- last [#2])] &, CellularAutomaton[MajorityRule[oo], RandomChoice[{.]45、55}- >{1,0},{100年60}),60),{1}),2 < - > 3)ColorFunction - >(混合[{RGBColor@Hue [0.15, 0.72, 1], RGBColor@Hue[0.98、1、0.8200000000000001 ]}, #] &), 框架- > False,图象尺寸- > 300)}),间隔[25]]majrow [{{1 1}, {1}, {0}})

& # 10005

majrow[{0,-1},{1,1},{0,0}]

& # 10005

[{{0, -1}, {1,1}, {-1, 0}}]

& # 10005

majrow [{{0,1}, {1, 0}, {1}, {1, 0}, {0,1}})

和我们原来的五间“对称”的社区我们可以通过设置1D GKL规则得到非常相似的行为:

& # 10005

{{{现代_,_},{b_、c_ d_}, {_, e _, _}} :> 如果[[c = = 0, a + b, d + e] + c > = 2, 1, 0]}

& # 10005

{BlockRandom [SeedRandom [23424];ArrayPlot3D [#, ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)}]@ (CellularAutomaton[{{{现代_,_},{b_、c_ d_}, {_, e _, _}} :> 如果[[c = = 0, a + b, d + e] + c > = 2, 1, 0]},{所击垮。45、55}- >{1,0},{50 50},{80}{0,所有}])],BlockRandom [SeedRandom [23424];ArrayPlot[是/ @转置[MapIndexed[# 1 *(1.6 ^(去年[2])&、CellularAutomaton[{{{现代_,_},{b_、c_ d_}, {_, e _, _}} :> 如果[[c = = 0, a + b, d + e] + c > = 2, 1, 0]},{所击垮。45、55}- > {1,0},{100,40}),60),{1}),2 < - > 3)ColorFunction - >(混合[{RGBColor@Hue [0.15, 0.72, 1], RGBColor@Hue[0.98、1、0.8200000000000001 ]}, #] &), 框架- >假]]}

带有噪声的元胞自动机

到目前为止,我们假设一旦开始,细胞自动机的进化是完全确定的。但是如果进化中有一些“噪音”,比如说细胞的值是以一定概率随机翻转?在这种情况下1D的简单多数决定原则是这样的:

& # 10005

BlockRandom [SeedRandom [34646];ArrayPlot[NestList[MapAt[1 - # &, CellularAutomaton[232][#], List /@ RandomInteger[{1, 400}, 5]] &, RandomChoice[{。4。6}- > {1,0},400),100),ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

那么GKL规则呢?在噪音较低的情况下,这一规则通常会“击退”噪音,但仍能达成共识:

& # 10005

BlockRandom [SeedRandom [32546];ArrayPlot[NestList[MapAt[1 - # &, CellularAutomaton[{FromDigits[Tuples[{1,0}, 7] /。{l3_, _, l1_、c_ r1_, _, r3_}: >如果[如果[c = = 0, r1 + r3, l1 + l3] + c > = 2, 1, 0), 2), 2、3}][#],列表/ @ RandomInteger[{400} 5]] &、[{所击垮。4。6}- > {1,0},400),120),ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

但最终,噪音变得太大,通常会失去共识:

& # 10005

BlockRandom [SeedRandom [32546];ArrayPlot[NestList[MapAt[1 - # &, CellularAutomaton[{FromDigits[Tuples[{1,0}, 7] /。{l3_, _, l1_、c_ r1_, _, r3_}: >如果[如果[c = = 0, r1 + r3, l1 + l3] + c > = 2, 1, 0), 2), 2、3}][#],列表/ @ RandomInteger[{400}, 30]] &、[{所击垮。4。6}- > {1,0},400),150),ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

一般来说,“噪声”的存在将我们的系统从普通的元胞自动机变成了概率元胞自动机。(反过来,这就相当于有时所说的定向渗透,或一个自旋系统,它根据特定权重的规则,随时间随机更新而演化。它还与有时被称为“相互作用粒子系统”的东西有关——例如,区域边界遵循类似于一组随机游动的东西,当它们相遇时就会湮灭。)

让我们更详细地讨论一下GKL规则的总体行为。当没有噪声时,当初始密度从低于0.5到高于0.5时,它显示出从终态0到终态1的急剧转变。但是当我们添加噪音时会发生什么呢?我们可以用一个经典的物理类型的相图来总结这个结果:

& # 10005

(*它=[= 1000}{长度= 500,步骤,BlockRandom [SeedRandom [625];ParallelTable[Mean[Nest[MapAt[1-#&,CellularAutomaton[{\ 339789091192587366278221041213531750560,2,3}][#],List/@RandomInteger[{\ 1,长度},Floor[length*\[Epsilon]]]]&,RandomChoice[{p,1-p}\[Rule]{1,0}\,长度],步骤]],{p, 0,1,},}。}, {\[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。, 0.92799999999999, 0.28], RGBColor[0.8200000000000001, 0。}, #] &), AspectRatio -> .7, FrameLabel -> {"noise level", "initial density"}]

这个图显示了由规则产生的最终密度作为初始密度和噪声水平的函数。在零噪声水平下,作为初始密度的函数会有一个相当尖锐的转变。(它不是完全清晰的,因为这个图是在有限的区域内对有限的初始条件进行抽样生成的。)随着噪音水平的增加,急剧的转变似乎会持续一段时间,直到最终达到一个临界噪音水平,它就消失了。

是否有一种严格的方法来分析发生了什么?哦,还没有。事实上是很长一段时间人们认为在噪声存在的情况下,任何像这样的一维系统都必然是遍历的,从这个意义上说,它最终会访问所有可能的状态,当然不会从不同的初始密度演化到不同的最终状态。

但在20世纪80年代构造了复杂的元胞自动机有可能证明的证据并不会显示出这种行为。这个系统是为了“即使在噪音存在的情况下也能进行可靠的计算”而建立起来的,它使用了相当复杂的类似软件工程的方法。但最终它只是一个一维细胞自动机,尽管有一个天文上复杂的规则。关键的一点是,当噪音达到某个非零水平时,该系统可以可靠地执行计算——比如达成多数人的共识。

但人们真的需要一个具有如此复杂的基本规则的系统来做到这一点吗?毫无疑问。这种情况让我想起了普通人的问题元胞自动机的计算通用性.在20世纪50年代,人们似乎可以通过一个非常复杂的装置,以一种类似工程的方式来实现这一点。但现在我们知道即使是最简单的一维细胞自动机规则110已经是通用的.事实上计算等价原理这意味着当我们看到不明显简单的行为时,我们可以期待计算的普遍性。

当然,我们似乎不应该需要计算通用性来获得分布式共识——尽管计算等价原则表明,计算通用性是“廉价的”,因此实际上可能“免费提供”具有其他必要属性的规则。澳门新利娱乐网址(顺便说一下,这不是一个微不足道的问题,因为当系统能够进行通用计算时,它们就有可能“做一些人们无法预测的事情”,包括,例如,打破一些人们认为自己定义的计算机安全约束。)

但是知道有一个非常复杂的元胞自动机即使在噪音存在的情况下也能实现分布式共识让人想知道最简单的元胞自动机能做到这一点是什么。澳门新利娱乐网址根据我以前的经验,我认为它会非常简单——就像GKL规则——尽管可能很难证明这一点。

对“噪音”的整个问题讲几句话可能是有用的。从某种意义上说,当一个人说一个系统中有“噪音”时,他是在说这个系统是“开放的”,有一些来自“外部”的东西是他无法预测的。但作为一种“近似”,我们可以想象有一些伪随机的噪声发生器,比如规则30元胞自动机.然后我们又有了一个“封闭”的系统,我们可以立即运用基于计算等价原理的思维。

但是“真正不可预测的噪音”呢?说这是存在的意思是说这个系统可以遵循不同的历史轨迹,而我们不知道在任何特定情况下会遵循哪一条。通知我们的物理项目,我们可以通过定义a来表示这些可能性多路图,当产生两种不同的状态时,就会有一个分支,这取决于噪声。

但是除了分支之外,多向图中也可以有合并。在一个具有身份规则的细胞自动机的非常简单的例子中,允许每个可能的单个细胞被噪声翻转,我们得到了多向图:

& # 10005

getStateGraphics[态势]:=陷害[风格[ArrayPlot [{}, ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},网- >真的,MeshStyle - >橙色],色调[0.62、1、0.48]],背景——>指令[[0.2]不透明性,色调[0.62,0.45,0.87]],FrameMargins - > {{2,}, {0}}, RoundingRadius - > 0,框架风格->指令[不透明度[0.5],色调[0.62,0.52,0.82]];getStateRenderingFunction[]:= Inset[getstatergraphics [ToExpression[#2]], #1, Center, {First[#3], Automatic}] &;flipbits[list_]:= Table[list -> MapAt[1 - # &, list, i], {i, Length[list]}];SimpleGraph[Flatten[NestList[Flatten[flipbits /@ (Last /@ #)] &, flipbits[{0,0,0}], 5]], VertexShapeFunction -> getStateRenderingFunction[], VertexSize -> 1, EdgeStyle -> Hue[0.75, 0,0.35], PerformanceGoal -> "Quality"]

以下是我们在每次“噪音翻转”(总共只走2步)后应用多数细胞自动机规则(第232条规则)的结果:

& # 10005

getStateGraphics[state_uzy]:=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]],背景->指令[Opacity[0.2],Hue[0.62,0.45,0.87]],框架边距->{2,2},{0,0},0},0},Orangradius->指令[0.5],色调[0.62,0.52,0.82]];getStateRenderingFunction[]:=Inset[getStateGraphics[ToExpression[#2]],[1,居中,{First[#3],Automatic}]&;flipbitsCA[ru,list][list->ru MapAt[1-#和,list,i],{i,Length[list]}]SimpleGraph[With[{g=Flatte NestList[Flatten[FlipBitMatoton],Raston/]&,flipbitsCA[CellularAutomaton[232],{0,1,1,1,0}],3]]]},图形[g,VertexShapeFunction->getStateRenderingFunction[],AspectRatio->1/2,VertexSize->1.6,EdgeStyle->色调[0.75,0,0.35],性能->质量]]

再走几步,用更厚的边表示更多连接相同状态的更新事件,我们得到:

& # 10005

getStateGraphics[state_uzy]:=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]],背景->指令[Opacity[0.2],Hue[0.62,0.45,0.87]],框架边距->{2,2},{0,0},0},0},Orangradius->指令[0.5],色调[0.62,0.52,0.82]];getStateRenderingFunction[]:=Inset[getStateGraphics[ToExpression[#2]],[1,居中,{First[#3],Automatic}]和;flipbitsCA[ru,list][list->ru MapAt[1-#和,list,i],{i,Length[list]}带有[{g=Graph[Flatte NestList[Flatte flipbitsCA flipbitsCA[flipbitsCA[FlipAtomaton],FlipAtomaton/][CellularAutomaton[232],{0,1,1,1,0}],4]]]},SimpleGraph[g,VertexShapeFunction->getStateRenderingFunction[],VertexSize->1.25,PerformanceGoal->“Quality”,EdgeStyle->Thread[EdgeList[SimpleGraph[g]->(指令[Hue[0.75,0,0.35],厚度[1.5*计数[EdgeList[g]]]]/[g]/[g]长度[g]],箭头[7.5*计数[EdgeList[g]]/长度[EdgeList[g]]]&/@EdgeList[SimpleGraph[g]]]]]]

这里有几个微妙的限制。元胞自动机的大小正变得无穷大。步数也达到了无限(尽管较慢)。通过说只有一定的“噪声密度”,我们有效地限制了边缘的相对权重。

要有一个即使在有噪音的情况下也能达成一致意见的系统,只有在特定的情况下吸引子必须在这些限制下生存。但究竟需要什么样的基本规则,我们还不知道——尽管我猜测它最终会出乎意料地简单。

计算通用性会“顺风顺水”吗?我不知道,但如果真的发生了,我也不会感到惊讶。尽管值得理解的是,在这样的多路系统中计算通用性的定义有些微妙。(我最近在……的背景下讨论过多路图灵机但当人们对不同路径的概率和“概率权重”感兴趣时,还存在更多问题。)

对细胞自动机的“有目的的攻击”

我们刚刚讨论了“随机噪声”对元胞自动机一致性的影响。但是“故意引入”的“噪声”(或“错误”)呢?是否存在一些潜在的小数量错误的模式,例如,会颠覆共识的结果?

这个问题的一个版本让人想起了神经网络中的对抗性例子,它只是问初始条件需要做什么改变才能“翻转其结果”。或者,换言之:假设一个人有一个系统(比如GKL规则)这基本上为几乎所有随机选择的初始条件达成了正确的共识。现在我们问的问题是,是否有一种系统的方法来调整给定的随机选择的初始条件,使其“导致错误的答案”。(人们可以认为这有点像问一个人是否能找到一个能产生错误的临时值密码散列以一种特殊的方式出现。)

不用说,这个问题有许多微妙之处。我们所说的“随机初始条件”是什么意思?大概一个周期状态不符合条件。我们可以做哪些“调整”?

可以想象的是,在“普通”初始条件下存在某种行为,但存在一些特殊的“种子”,如果发生这种行为,将产生无限(“肿瘤式”)生长,最终接管系统,如规则122中的这个简单示例:

& # 10005

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

如果不只是“攻击”初始条件,而是允许在每个步骤中改变特定细胞的值,那么很容易出现“不可移动的块”,实际上阻碍了“完全一致”:

& # 10005

BlockRandom [SeedRandom [3257];ArrayPlot[NestList[MapAt[RandomChoice[{0,1} -> {# &, 1 - # &}], CellularAutomaton[{339789091192587366278221041213531750560, 2,3}][#], List /@ Range[195,205]] &, RandomChoice[{。4。6}- > {1,0},400),120),ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},框架- >错误]]

但是,如果我们考虑在正在进行的进化过程中,在一小部分地方做出改变,比如在根据实际进化模式精心计算的地方添加一点“智能噪音”,那会怎么样呢?系统是否总是能够从这种“拜占庭式的篡改”中“自我修复”,实际上是“纠正几位错误”?或者是否存在某些特定的“漏洞”,可以利用这些漏洞来“破坏”最终结果,只需要一些精心选择的更改?

我们可以把两个一致的终态看作吸引子,吸引子的吸引盆包括所有高于或低于密度的初始条件.或者,可以把元胞自动机看作是“识别初始密度”的“解决分类问题”。也许有一种方法可以将细胞自动机扩展到一个具有连续权值的神经网络,然后使用机器学习方法迭代地找到权值可以改变的最小位置。

图元胞自动机

在普通的元胞自动机中,值被赋给在确定网格中布置的元胞。但一般来说,我们可以允许细胞位于图的节点上,然后取图上的邻居来定义要在规则中使用的邻居:

& # 10005

随机图[{20,40},EdgeStyle ->灰色,VertexStyle ->表[i -> (RandomInteger[] /。{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)}),{我20}],VertexSize - > 5)

这里有一个紧迫的问题。在标准元胞自动机的基本定义中,规则以确定的顺序“取其参数”。但是如果我们处理的是一个普通的图(与之相反,例如,an有序超图),我们只知道哪些节点连接到一个给定的节点——没有立即定义顺序。

这意味着我们可以使用的元胞自动机规则的类型的约束。我们可以考虑围绕图中的每个节点设置一个“测地线球”。连续的“壳”包含节点,这些节点与给定节点的距离是连续的图距离。但是细胞自动机规则不能区分一个给定的细胞在一个特定的壳内处于哪个“位置”;它所能做的就是计算每个壳层中具有给定值的细胞总数。

如果图顶点传递,所以周围的图结构中的每个节点图是相同的(至于凯莱图),然后细胞自动机规则可以包含固定表的结果只取决于细胞的数量的每个值在每个shell。但是对于一般的图,元胞自动机的规则必须允许每个壳层中有任意数量的细胞。

一种情况下,这恰好是简单的多数规则。下面是一个应用于上图中“半径为1的测地壳”的规则示例:

& # 10005

SynchronousStepNewColors [dependencies_ colors_]: =平[地图[[{邻居=排序(计数(部分(颜色,最后[#]]),更大的]},如果[DuplicateFreeQ值(邻居),第一个[键(邻居)],[[第一个颜色 [#]]]]] &, 依赖]]GraphMajorityCA [graph0_、p_ steps_, radius_: 1]: = BlockRandom [SeedRandom [14];With[{init = Association[# -> RandomChoice[{1 - p, p} -> {1,0}] & /@ VertexList[graph]]}, Module[{{dependencies}, dependencies = Table[n -> VertexOutComponent[graph, VertexList[[n]], radius], {n, VertexCount[graph]}];图(EdgeList[图],VertexStyle - > MapThread[规则[# 1、开关(# 2 0色调[0.15,0.72,1],1,色调[0.98、1、0.8200000000000001]]]&、{键(init) #}], EdgeStyle - >灰色,VertexSize - > 5) & / @ NestList [SynchronousStepNewColors依赖性,# &,值(初始化),步骤]]]]][#,90度]& / @旋转(图[#,& /@ GraphMajorityCA[\!\ (\ * GraphicsBox [NamespaceBox[“NetworkGraphics”,DynamicModuleBox[{排版'graph = HoldComplete[图[{1,2,3,4,5,6,7,8,9,10,11,12日,13日,14日,15日,16日,17日,18日,19日,20},{Null, SparseArray[自动,{20、20},0,{1,{{0、2、5、7、9、13、20、25、29日,33岁,36岁,41岁,43岁,46岁,48岁,54岁,59岁,63年,68年,72年,80年},{{5},{7},{11},{16},{18},{11},{13},{16}, {18}, {1}, {6}, {9}, {20}, {5}, {13}, {15}, {16}, {17}, {19}, {20}, {1}, {9}, {13}, {15}, {16}, {10}, {15}, {19}, {20}, {5}, {7}, {12}, {20}, {8}, {15}, {20}, {2}, {3}, {16}, {17}, {20}, {9}, {19}, {3}, {6}, {7}, {17}, {18}, {6}, {7}, {8}, {10}, {18}, {20}, {2}, {4}, {6}, {7}, {11}, {6}, {11}, {14}, {18}, {2}, {4}, {14}, {15}, {17},{6},{8},{12},{20},{5},{6},{8},{9},{10},{11},{15},{19}}},模式}},{EdgeStyle - >{灰度[0.5]},VertexSize - > {0.5}, VertexStyle - >{18 - >色相(0.98、1、0.8200000000000001),6 - >色相(0.98、1、0.8200000000000001),15 - >色相(0.15,0.72,1),1 - >色相(0.15,0.72,1),20 - >色相(0.15,0.72,1),9 - >色相(0.98、1、0.8200000000000001),13 - >色相(0.98、1、0.8200000000000001),2 - >色相(0.98、1、0.8200000000000001),8 - >色相(0.15,0.72,1),3 - >色相(0.98、1、0.8200000000000001),4 - >色相(0.98、1、0.8200000000000001),11 - >色相(0.98、1、0.8200000000000001),10 - >色相(0.15,0.72,1),19 - >色相(0.98、1、0.8200000000000001),7 - >色相(0.15,0.72,1),16 - >色相(0.15,0.72,1),17 - >色相(0.98、1、0.8200000000000001),14 - >色相(0.15,0.72,1),12 - >色相(0.15,0.72,1),5 - >色相[0.15,0.72,1]}}]]},TagBox[GraphicsGroupBox[GraphicsComplexBox[CompressedData[" 1:eJxTTMoPSmViYGAQAWIQ/ f6kr7ggoludhrp7aqlydgefspsgl9fvv2f8lnlkc mvfBvlZItHLWuS/2DGgg/9jtrhs7ftvvM+ptrezlcCgy9vQT9GBy2Ll97u7Y OCaHo7IP/Cbs+mnfZbVWik2F2SGqOd7o0ar/9t8kP65/ vjjl4wi3zxxod5y6vp7 NZUzj4qe5nFo3Jj1qfrrf/v3vhsYxNawO3AEfFuR0nfZ/tTMg6XzfL7Z82q9 dLwv+cG+ZsK3ia8MhBxkHH6bp6f8sQ9p/u2R+eGffUTSNr4jU1/Ym2z+VXtG 8ZJ9ztZLRw4cZnFg2NDoE27D5JBj4p33avNHe1n24FmPsj7bvxN+fM7e86d9 ixWjlEzhJ/tz3z/kCdxkcEiQU3+SM+u5fcT+3BNHM37bP2tYmmyryOug/He9 3TWdr/a919meTv/D5rCNS/Ajo/gXewA1KJH+ "], { {GrayLevel[0.5], Opacity[0.7], Arrowheads[0.], ArrowBox[{{1, 5}, {1, 7}, {2, 11}, {2, 16}, {2, 18}, {3, 11}, {3, 13}, {4, 16}, {4, 18}, {5, 6}, {5, 9}, {5, 20}, {6, 13}, {6, 15}, {6, 16}, {6, 17}, {6, 19}, {6, 20}, {7, 9}, {7, 13}, {7, 15}, {7, 16}, {8, 10}, {8, 15}, {8, 19}, {8, 20}, {9, 12}, {9, 20}, {10, 15}, {10, 20}, {11, 16}, {11, 17}, {11, 20}, {12, 19}, {14, 17}, {14, 18}, {15, 18}, {15, 20}, {17, 18}, {19, 20}}, 0.09587857253283466]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], {Hue[0.15, 0.72, 1], DiskBox[1, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[2, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[3, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[4, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[5, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[6, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[7, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[8, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[9, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[10, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[11, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[12, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[13, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[14, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[15, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[16, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[17, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[18, 0.09587857253283466]}, {Hue[0.98, 1, 0.8200000000000001], DiskBox[19, 0.09587857253283466]}, {Hue[0.15, 0.72, 1], DiskBox[20, 0.09587857253283466]}}}]], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{ "NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None]\), .3, 3])

那么这个规则在全球范围内会发生什么呢?的图

& # 10005

图[ResourceFunction["TorusGraph"][{3,3}], VertexStyle -> White, VertexSize -> .2, EdgeStyle -> \!\(\* TagBox[StyleBox[InterpretationBox[RowBox[{"CloudGet", "[", "\"\\"", "]"}], Hue[0.75, 0,0.35], Editable->False, Selectable->False], ShowSpecialCharacters->False, ShowStringCharacters->True, NumberMarks->True], FullForm]\)]

下面是应用该规则时的状态转移图,而上面的节点是根据哪个值占多数而上色的:

& # 10005

SynchronousStepNewColors[dependencies,colors]:=展平[Map[With[{Neights=Sort[Counts[Part[colors,Last[#]],Greater]},如果[DuplicateFreeQ[Values[Neights]],First[Keys[Neights]],colors[[First[#][Neights]]]]].[[graph0],Radiu 1]:=With[{graph=IndexGraph0]},模块[{dependencies},dependencies=Table[n->VertexOutComponent[graph,VertexList[graph][[n]],radius],{n,VertexCount[graph]};#->SynchronousStepNewColor[dependencies,#->和/@Tuples[{1,0},VertexCount[graph]]]]],带有[{g=GraphMajorityCastyStatg[ResourceFunction[“TorusGraph”[{3,3}],1]},graph[g,VertexStyle]>(#->如果[Mean[/2,色调]>1][0.98,1,0.8200000001],色调[0.15,0.72,1]&/@VertexList[g]),边缘样式->\!\(\*标记框[StyleBox[ExpressionBox[ButtonBox[TooltipBox[GraphicsBox[{{GrayLevel[0],矩形框[{0,0}]},{GrayLevel[0],矩形框[{1,-1}},{Hue[0.75,0,0.35],矩形框[{0,1}],矩形框[{0.75,0,0,0}],默认样式,{1}>“ColorSwatchGraphics”,边框->真,边框样式->色调[0,0,0.23333334`],边框->无,图像大小->{Automatic,12.879},PlotRangePadding->None],样式框[RowBox[{“色调”,“[”,RowBox[{“0.75`,”,“,”,“,”,“,”,“,”,“}],数字标记->假]],外观->无,BaseStyle->{},基线位置->基线,按钮功能:>[{Typeset`box$=EvaluationBox[]},如果[Not[AbsoluteCurrentValue[“Deployed”]],选择移动[Typeset`box$,All,Expression];前端`Private`$$ColorSelectorInitialAlpha=1;前端`Private`$ColorSelectorInitialColor=Hue[0.75,0,0.35];前端`Private`$ColorSelectorUseMakeBoxs=True;MathLink`CallFrontEnd[FrontEnd`AttachCell[Typeset`box$、FrontEndResource[“HueColorValueSelector”]、{0、{Left,Bottom}、{Left,Top}、{ClosingActions->{“SelectionBefore”、“ParentChanged”、“EvaluatorQuit”}]]、DefaultBaseStyle->{}、Evaluator->Automatic、Method->“Preemptive”]、色调[0.75,0,0.35],可编辑->假,可选择->假],ShowSpecialCharacters->False,ShowStringCharacters->True,NumberMarks->True],FullForm]\)]]

毫无疑问,图上多数元胞自动机规则的“成功率”是可以证明的一般结果。但实验表明,该规则在图形上的效果要比在常规数组上好得多。

底层图的图论特征可能会影响性能。更高的连通性可能会有所帮助,尤其是因为它倾向于避免“桥梁”,即颜色可以在特定节点的“所有方面”平衡。缺乏对称性也可能会抑制循环的出现。总的来说,人们可以认为“共识的传播”至少是有点像一个渗透过程

对于普通的细胞自动机来说,问“无限大小限制”意味着什么是很清楚的。但对于图来说,只有在处理一些易于扩展的图族(如网格或环面图或各种Cayley图)时,它才会立即变得清晰。对于任意"随机图“结果可能在很大程度上取决于所使用的图形分布。

在我们的物理项目中,我们关注的是可以根据当地规则“增长”的大型图表。我们期望这样的图常常在“连续极限”中显示某些“统计规律”。在我们的项目中,我们通过查找例子来描述这些图的结构测地线球体积增长率,并从中确定尺寸和曲率之类的东西。那么,如果我们在一个具有某些“几何”属性的大图上运行多数原则元胞自动机,会发生什么呢?

本质上,我们需要问的是多数决定原则元胞自动机的“连续极限”是什么。普通元胞自动机所使用的网格太特殊了,无法达到任何一般的极限。但在“可几何化”的图形上,期望这样的连续极限更合理。

我们可以考虑一维空间的例子。初始值由位置的连续函数给出:

& # 10005

与[{如果= (SeedRandom [69774];插值[RandomReal[{- 10,10}, 10], InterpolationOrder -> 6])}, Plot[if[x], {x, 1,10},填充->轴,ColorFunctionScaling -> False, ColorFunction -> (if[#2 > 0, Hue[0.98, 1,0.82000000000000000001], Hue[0.15, 0.72, 1]] &), Frame -> True, AspectRatio -> 1/3, FrameTicks -> None]]

这种情况下的“一致结果”应该是一个常数函数,其值实际上是该函数积分的符号。但什么样的积分微分代数方程可以再现时间演化尚不清楚。

回到图上的大多数细胞自动机,值得注意的是,如果图的边可以同时被分配正权和负权,那么这个系统实际上就像一个神经网络的同步版本。在一个规则的网格上(在结构上类似于自旋玻璃)则显示计算不可约性的各种特征。

我们可以考虑元胞自动机规则,而不是考虑带有加权边的底层图。例如,我们可以考虑对于不同半径的测地线壳具有不同权重的规则(就像activation-inhibition细胞自动机用来模拟生物色素沉积模式)。

但是,真的只有基于测地线壳层的整体规则才能用于图元胞自动机吗?要做的更多,实际上需要在图表中定义“方向”。但是我们的物理项目已经提供了多种机制来完成这一任务,并且在原则上建立了非totalistic图元胞自动机。

异步更新

细胞自动机的一个重要特征是假设所有的细胞值在一系列确定的步骤中“同时”或“同步”更新。但在分布式共识的实际例子中,人们通常处理的是异步更新的值。澳门新利娱乐网址实际上,我们想要的是将普通细胞自动机的同步更新“分解”为单个细胞的更新序列,这些更新的顺序不受任何特定规则的指定。

所以第一个显而易见的问题是:“这些更新的顺序真的重要吗?”有时也不会。这是一个例子。与普通的元胞自动机不同,考虑一个块元胞自动机,在每个步骤中相邻的值对被新值替换:

& # 10005

RulePlot [SubstitutionSystem [{{0} - > {0}, {1, 0} - > {0,1}, {0,1} - > {0,1}, {1 1} - > {1 1}}], ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},MeshStyle - >橙色]

对于同步更新,我们可以在系统的“类砖”模式中应用这些规则。但是为了学习异步更新,让我们在每一步的随机位置应用这些规则。以下是一些可能发生的情况:

& # 10005

BlockRandom [SeedRandom [235234];With[{i = RandomInteger[1,30]}, Table[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, Automatic},ColorRules -> {0 -> Hue[0.15, 0.72, 1], 1 -> Hue[0.98, 1, 0.8200000000000001]}, Mesh -> True, Mesh style -> Orange], 4]]]

值得注意的是,尽管每种情况下的具体进化是不同的,但在这种情况下,最终的结果总是相同的,只是对应于拥有所有排序之前

并不是所有的规则都是这样的,但是对于这个规则,不管产生的中间状态是什么,最终结果总是具有最终的一致性。

事实证明,这种现象在我们的物理项目中至关重要。事实上,我们称之为"因果不变性例如,是什么导致了相对论不变性。但从物理项目的形式主义,我们也得到了异步进化的一般方法:使用多向图追踪所有可能的“更新历史”。

下面是上面简单排序规则的多路图:

& # 10005

getStateGraphics[态势]:=陷害[风格[ArrayPlot [{}, ColorRules - >{0 - >色相(0.15,0.72,1),1 - >色相(0.98、1、0.8200000000000001)},网- >真的,MeshStyle - >橙色],色调[0.62、1、0.48]],背景——>指令[[0.2]不透明性,色调[0.62,0.45,0.87]],FrameMargins - > {{2,}, {0}}, RoundingRadius - > 0,框架风格->指令[不透明度[0.5],色调[0.62,0.52,0.82]];getStateRenderingFunction[]:= Inset[getstatergraphics [ToExpression[#2]], #1, Center, {First[#3], Automatic}] &;ResourceFunction["MultiwaySystem"][{{1,0} -> {0,1}}, {{1,0,1,1,0,0,1}}, 8, "StatesGraph", "StateRenderingFunction" -> getStateRenderingFunction[], VertexSize -> 1.75, PerformanceGoal -> "Quality"]

正如预期的那样,所有可能的更新历史最终都会收敛到相同的最终状态。

那么多数决定原则细胞自动机呢?它并不总是显示最终的一致性,如下例所示:

& # 10005

randomOrderCAFunc[ruleRadius_, ruleNumber_, init_, eventCount_, func_]:= func[evaluateSingleEvent[ruleRadius, ruleNumber, #] &, init, eventCount];RandomOrderCA[args, Nest]:= randomOrderCAFunc[args, Nest];RandomOrderCAList[args, NestList]:= randomOrderCAFunc[args, NestList];findLastEvent[eventNumber_, position_, eventsIndex_]:= Module[{}, Max[Select[Lookup[eventsIndex, position, {-Infinity}], # < eventNumber &]];getCausalLinks[eventNumber_, position_, eventsIndex_, size_, ruleRadius_]:= Module[{}, DeleteCases[findLastEvent[eventNumber, #, eventsIndex] -> eventNumber & /@ Mod[Range[position - ruleRadius, position + ruleRadius], size, 1], - infinity -> _]];RandomOrderCACausalGraph[ruleRadius_, ruleNumber_, init_, eventCount_, opts___]:= Module[{eventsIndex, eventPositions}, eventsIndex = kesort @ Map[Last, GroupBy[Thread[(eventPositions = Reap[RandomOrderCA[ruleRadius, ruleNumber, init, eventCount]][[2,1]]) -> Range[eventCount]], First], {2}];图[范围[eventCount], Catenate[getCausalLinks[#, eventPositions[[#]], eventsIndex,长度[init], ruleRadius] & /@ Range[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_] := list[[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, Center, {First[#3], Automatic}] &; SeedRandom[643767 + 5]; SimpleGraph[ NestGraph[ Table[state @@ evaluateEventAtPlace[1, 232, List @@ #, c], {c, 1, Length[#]}] &, state @@ randomInit[9, 0.7], 20, VertexShapeFunction -> getStateRenderingFunction[], PerformanceGoal -> "Quality", VertexSize -> 1, ResourceFunction["WolframPhysicsProjectStyleData"]["StatesGraph", "Options"]]]

因此,这意味着一般来说,异步更新的顺序很重要,或者,实际上,采取了什么“历史路径”。但为了了解典型行为,我们可以考虑随机序列的更新。下面是一个例子,说明每一步做一次更新会得到什么:

& # 10005

caEvaluateCompiled=FunctionCompile[Function[{Typed[rule,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],Typed[rad,“MachineInteger”],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]&/@Range[position-rad,position+rad]];rulePart=Fold[2#1+#2&,0,substate]+1;newCellValue=rule[[rulePart]];state[[position]=newCellValue;,eventCount];state]];应为所有机器目标编译:CAEvaluateCompiledCodeFunction=CompiledCodeFunction[Association][“签名”->类型说明符[{“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1,“Integer64”]、“Integer64”、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1]、“Integer64”]]、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1”、“Integer64”]]、“Input”->编译`Program[{}、函数[{Typed[规则、类型说明符][“PackedArray”[“MachineInteger”,1]],键入[rad,“MachineInteger”],[init,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],键入[eventCount,”Integer64]},模块[{state,position,substate,rulePart,newCellValue},state=init;Do[position=RandomInteger[{1,Length[state]}];substate=Part[state,Map[Mod[#,Length[state],1]&范围[position-rad,position+rad]];rulePart=Fold[2#+#2&,0,substate]+1;newCellValue=Part[rule,rulePart];Part[state,position]=newCellValue;Null,eventCount];state]],“ErrorFunction”->自动,“InitializationName”->“Initialization”\U 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doH2oUz7JEf7AHu7hdHbF0aER8YnrfdVftTqv/dJK6vECCuffziUlc//mkvJ 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, Frame -> None]]

这是相应的结果,如果我们每一步做20个更新:

& # 10005

caEvaluateCompiled=FunctionCompile[Function[{Typed[rule,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],Typed[rad,“MachineInteger”],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]&/@Range[position-rad,position+rad]];rulePart=Fold[2#1+#2&,0,substate]+1;newCellValue=rule[[rulePart]];state[[position]=newCellValue;,eventCount];state]];应为所有机器目标编译:CAEvaluateCompiledCodeFunction=CompiledCodeFunction[Association][“签名”->类型说明符[{“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1,“Integer64”]、“Integer64”、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1]、“Integer64”]]、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1”、“Integer64”]]、“Input”->编译`Program[{}、函数[{Typed[规则、类型说明符][“PackedArray”[“MachineInteger”,1]],键入[rad,“MachineInteger”],[init,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],键入[eventCount,”Integer64]},模块[{state,position,substate,rulePart,newCellValue},state=init;Do[position=RandomInteger[{1,Length[state]}];substate=Part[state,Map[Mod[#,Length[state],1]&范围[position-rad,position+rad]];rulePart=Fold[2#+#2&,0,substate]+1;newCellValue=Part[rule,rulePart];Part[state,position]=newCellValue;Null,eventCount];state]],“ErrorFunction”->自动,“InitializationName”->“Initialization”\U 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doH2oUz7JEf7AHu7hdHbF0aER8YnrfdVftTqv/dJK6vECCuffziUlc//mkvJ 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, 20}], ColorRules -> {0 -> Hue[0.15, 0.72, 1], 1 -> Hue[0.98, 1, 0.8200000000000001]}, MeshStyle -> Orange, Frame -> None]]

我们可能会认为异步更新能够添加足够的随机性去“打破联系”并防止游戏卡住。但实际上不难看出,在这种情况下,最终的结果与同步更新没有什么不同。

那么GKL规则呢?这里是它的异步结果,现在每一步有50个更新(最初是60%)).

& # 10005

caEvaluateCompiled=FunctionCompile[Function[{Typed[rule,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],Typed[rad,“MachineInteger”],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]&/@Range[position-rad,position+rad]];rulePart=Fold[2#1+#2&,0,substate]+1;newCellValue=rule[[rulePart]];state[[position]=newCellValue;,eventCount];state]];应为所有机器目标编译:CAEvaluateCompiledCodeFunction=CompiledCodeFunction[Association][“签名”->类型说明符[{“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1,“Integer64”]、“Integer64”、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1]、“Integer64”]]、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1”、“Integer64”]]、“Input”->编译`Program[{}、函数[{Typed[规则、类型说明符][“PackedArray”[“MachineInteger”,1]],键入[rad,“MachineInteger”],[init,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],键入[eventCount,”Integer64]},模块[{state,position,substate,rulePart,newCellValue},state=init;Do[position=RandomInteger[{1,Length[state]}];substate=Part[state,Map[Mod[#,Length[state],1]&范围[position-rad,position+rad]];rulePart=Fold[2#+#2&,0,substate]+1;newCellValue=Part[rule,rulePart];Part[state,position]=newCellValue;Null,eventCount];state]],“ErrorFunction”->自动,“InitializationName”->“Initialization”\U 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doH2oUz7JEf7AHu7hdHbF0aER8YnrfdVftTqv/dJK6vECCuffziUlc//mkvJ 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[{\ 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, Frame -> None]]

与我们在上面看到的同步更新添加杂音不同,对异步更新的更改似乎完全破坏了这一规则的“多数共识”。注意,在每个步骤中使用更多的更新并不会改善结果(最终应该是这样的)):

& # 10005

caevaluatecompiled = functioncompile [function [【键入[规则,打字机[“packedArray”] [“macitorInteger”,1]],键入[rad,“macitorInteger”],键入[init,typespecifier [“packedarray”] [“packedArray”,1],键入[EventCount,“Integer64”]},模块[{状态,位置,子存储,rulepart,newcellvalue},state = init;[position = somandinteger [{1,长度[状态]}];substate = state [[mod [#,长度[状态],1]&/ @范围[位置 -  rad,位置+ rad]];rulepart = fold [2#1 +#2&,0,stalate] + 1;newcellvalue =规则[[rulepart]];州[[位置]] = newcellvalue;,EventCount];状态 ]]];应该编译所有机器目标:caevaluateCompiled = compiledcodefunction [关联[“签名” - >打字机[{“packedArray”[“Integer64”,typeframework`typeliteral [1,“Integer64”],“Integer64”,“PackedArray”[“Integer64”,typeframework`typeliteral [1,“Integer64”]],“Integer64”}  - >“packedArray”[“Integer64”,typeframework`typeliteral [1,“Integer64”]],“输入” - >编译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 = Part[state, Map[Mod[#, Length[state], 1]& , Range[position - rad, position + rad]]]; rulePart = Fold[2 # + #2& , 0, substate] + 1; newCellValue = Part[rule, rulePart]; Part[state, position] = newCellValue; Null, eventCount]; state]]], "ErrorFunction" -> Automatic, "InitializationName" -> "Initialization_ebf3942d_117d_4dd2_a02b_59c5c41b2c75", "ExpressionName" -> "Main_ExprInvocation", "CName" -> "Main_CInvocation", "FunctionName" -> "Main", "SystemID" -> "MacOSX-x86-64", "VersionData" -> {12.3, 0, 0}, "CompiledIR" -> Association["MacOSX-x86-64" -> ByteArray[CompressedData[" 1:eJzVvHlYU0fbMH6ykAQIkCBK2A+LiBZtQGQR1AQQUNGi0rqhSRAQZN83gSQg xFY0qLUIomDRap+quJRFQYggUgVFsWJFAUVFiwoUAQWF38w5CUKfPs/1vr/r +/74uK4kZ+6Zube5tznMOca+4Wt8CQiCsH0QpL1GWxVcIlrgwwJtKoIQnF1q 2hcAgBKA+dLZZss3XAnYPHrxK9XVQk84z5iOIP6gX5VojBBBewb4UEWmOdwG 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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, Frame -> None, ImageSize -> {300, 150}]], Style[#, 11], Spacings -> {0, -0.5}] & /@ {100, 1000, 10000, 10^5}, 2], Spacings -> {0, 0}]

那么,如果我们搜索异步达成共识的规则,会发生什么呢?在最近邻的情况下,简单多数决定原则做得最好,尽管它基本上不好。

以下是通过搜索100万个range-2规则找到的一些规则的结果:

& # 10005

caEvaluateCompiled=FunctionCompile[Function[{Typed[rule,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],Typed[rad,“MachineInteger”],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]&/@Range[position-rad,position+rad]];rulePart=Fold[2#1+#2&,0,substate]+1;newCellValue=rule[[rulePart]];state[[position]=newCellValue;,eventCount];state]];应为所有机器目标编译:CAEvaluateCompiledCodeFunction=CompiledCodeFunction[Association][“签名”->类型说明符[{“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1,“Integer64”]、“Integer64”、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1]、“Integer64”]]、“PackedArray”[“Integer64”、TypeFramework`TypeLiteral[1”、“Integer64”]]、“Input”->编译`Program[{}、函数[{Typed[规则、类型说明符][“PackedArray”[“MachineInteger”,1]],键入[rad,“MachineInteger”],[init,TypeSpecifier[“PackedArray”][“MachineInteger”,1]],键入[eventCount,”Integer64]},模块[{state,position,substate,rulePart,newCellValue},state=init;Do[position=RandomInteger[{1,Length[state]}];substate=Part[state,Map[Mod[#,Length[state],1]&范围[position-rad,position+rad]];rulePart=Fold[2#+#2&,0,substate]+1;newCellValue=Part[rule,rulePart];Part[state,position]=newCellValue;Null,eventCount];state]],“ErrorFunction”->自动,“InitializationName”->“Initialization”\U 667648a7\U ECF6U 4a9d\U 97c7\f8cc07f5743e;“Exprin”;“ExpressionName”->“Main”;“CNName”>”Main_Cinciation、“函数名”->“Main”、“系统ID”->“MacOSX-x86-64”、“VersionData”->{12.3,0,0}”,CompiledIR”->“关联[“MacOSX-x86-64”->”ByteArray[CompressedData[”1.他们的研究成果是一个非常便宜的东西,但他们的研究成果是一个非常便宜的东西,这是一个关于这个问题的研究成果,一个关于QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQB3/EB3/EQQQQQQB3 E3/E3/E3/E8/E8/E8/E8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 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