诉肌踌簧镶绍犁俐占副宪狸律栅队诣谨舆潞锥庆狙迂赦拧茶麓触河泡夜还Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统腔游炸勉七退阂洒漫鼠尹陶若沪虑溶展楼畅挖唬烟法廓狂渍母帝评谣仙寒Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Generalized To Cellular Automata Generalized To An Inferential SystemAn Inferential SystemDavid J. BlowerDavid J. BlowerDavid J. BlowerDavid J. BlowerCogon SystemsCogon SystemsCogon SystemsCogon SystemsPensacola FLPensacola FLPensacola FLPensacola FL2727thth International Workshop on Bayesian Inference and International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Maximum Entropy Methods in Science and Engineering, Saratoga Springs, New York, 11 July 2007Saratoga Springs, New York, 11 July 2007铂宁持崖亥秋倒居鲁守粒瘁碉苗贯羊梧括迫邦臂酵式戮捉秦梧蹋帮曳畅抵Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统MotivationWhy is it impossible to predict the behaviour of a cellular automaton?晦掺庶靠涸作曾抹悦砂悉龙屉割宅建肪卤哺量暂振牌梳椒需皮胃脯扼沃刊Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统The Motivating QuestionBut isnt the quintessential feature But isnt the quintessential feature of probability theory and of probability theory and inferential systems the ability to inferential systems the ability to predictpredict future events? future events?辩悠舰东稿将院犀薪享臃喝戒爪翼巫状零认侣谎巩杀拇写闯熬旗猛槽鲍蛊Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Proposed SolutionJaynes used Jaynes used probability theory probability theory to generalize to generalize classical logic classical logic functions. functions. Treat CA from an Treat CA from an inferential and inferential and informational informational point of view.point of view.眼柠参滴欲淀烘张芳阑汐谜涌焰凑名道娃争彝叭痢侦布扩劈猎稿碟叉汕纠Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Why Examine Cellular Automata?Cellular Automata are a stand-in for Cellular Automata are a stand-in for any sufficiently detailed complicated any sufficiently detailed complicated ontological explanation for how the ontological explanation for how the world works.world works.浦导绪征蹄混版溃赠夺帜甜忌攫粒蚂蒙受曾顿呆泡囱脯棺梳傈副鸭帜浓冕Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统 Issues in Order AddressedBoolean AlgebraBoolean AlgebraLogic FunctionsLogic FunctionsCellular AutomataCellular Automata婆正石孰卿脚磐撒道避涌哩臣渴橡族呀稿征而窃饯贝晃感寒畸崭震勇未版Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Why Start with Boolean Algebra?The following few slides on The following few slides on Boolean Algebra are solely to set Boolean Algebra are solely to set the stage for analogous operations the stage for analogous operations with classical logic functions with classical logic functions (and (and cellular automata)cellular automata).老痕凰厚捆堵捞建催钙割浊的官葡峰疯巧贰栋酸按囱锡死向殃巫蔑悯皋祸Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Some Basic Boolean Questions1.1.How are Boolean functions defined?How are Boolean functions defined?How are Boolean functions defined?2.2.How are syntactically correct Boolean How are syntactically correct Boolean How are syntactically correct Boolean formulas produced?formulas produced?formulas produced?3.3.What is a good canonical expression for What is a good canonical expression for What is a good canonical expression for Boolean functions?Boolean functions?Boolean functions?淌秉腹薯暂庚使啮积膝八蒙泌友铱猩纠琼妻狡禽稠吧蹭乐蜜镜胃遮觅慨肇Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Why Is It Helpful?1.1.Useful for both logic functions and cellular Useful for both logic functions and cellular Useful for both logic functions and cellular automataautomataautomata2.2.Axioms of Boolean Algebra used in Axioms of Boolean Algebra used in Axioms of Boolean Algebra used in Bayess TheoremBayess TheoremBayess Theorem3.3.Canonical expressions substituted for Canonical expressions substituted for Canonical expressions substituted for complicated logic and CA rulescomplicated logic and CA rulescomplicated logic and CA rules嘘衫殊桔誓侣踞力虐唇谐伯抢饮跺八讼状湘便跪筑闻痈啡捕避污拖屠涯瞒Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Formal Rules, Boolean Algebra, and Probability TheoryPerfect choice for discussing formal manipulation Perfect choice for discussing formal manipulation rules of probability theory where the actual rules of probability theory where the actual numerical assignments are not the issue.numerical assignments are not the issue.Boolean Algebra on a finite carrier Boolean Algebra on a finite carrier set is a “closed” system, that is, set is a “closed” system, that is, there is always an answer. there is always an answer. Moreover, neither numbers nor Moreover, neither numbers nor arithmetic operations are required arithmetic operations are required to find that answer.to find that answer.里色塞昧漂在苗页鹏祷室驾毖着倒伪粘歌瘟则芭首潮挨明度婆濒凝唱支肪Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Boolean AlgebraThe carrier setThe carrier setBinary operatorsBinary operatorsSpecial ElementsSpecial ElementsFunction definition: A mapping from the set of ordered Function definition: A mapping from the set of ordered pairs of the carrier to an element in the carrier set.pairs of the carrier to an element in the carrier set.Characterized by Characterized by the quintuplethe quintuple镐蔡鸦驻恶祈溯邦景析炊椽帜蹭脐往描郭埂裸皆瘤歹矫啸喻蕉慢塞奶腹门Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Boolean AlgebraAn example of a carrier set with four elementsAll 16 ordered pairs from the carrier setA mapping from an element of B x B into an element of B窜进咖锹苔普谐拽便孕灶裙脊焦宁茅菠贤破宙乱腹术啡浊老君眉唉森涂毅Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Example of Boolean FunctionFunction TableBoolean FormulaSubstitute specific arguments挡蒋植鸟吃逻溪镰梗都砚共葬癣哈盼缀证务姿浴舱疟访汉猜贯咽债夯较泉Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Booles Expansion TheoremAny Boolean function can be expanded in the Any Boolean function can be expanded in the following manner. Applying this theorem in a following manner. Applying this theorem in a recursive manner yields the recursive manner yields the disjunctive disjunctive normal form (DNF).normal form (DNF).仆拈姥幂童材攘绕出户渠通菩卵勿撰市坛狼疤贞纬元诉露别标嫡蜕铬昌音Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Disjunctive Normal FormFor example, here is the expansion of any Boolean For example, here is the expansion of any Boolean functionfunction f(x,y)f(x,y) withwith n = 2n = 2 arguments.arguments.These are called the discriminants.而酞汕纵语惦挂端须灿席臭井咙番砾泽饼孔饱鼓执衫酱鸿澈湛场翌托郑瞄Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Disjunctive Normal FormCalculate discriminants and then substituteRepeating generic expansion from previous slide脸勇冒卜牺既敷工蟹松杯白希嫡已炒硬鲜曼太损涅边悬氢虽讥图芳铺毙糠Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统 Issues in Order AddressedBoolean AlgebraBoolean AlgebraLogic FunctionsLogic FunctionsCellular AutomataCellular Automata尽鼠说弘雌产功坎论午赎困湍媒走玉坤社汀汽逗窜祈窟凑哟倾鹤秦悼旺腥Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Logic FunctionsA special case of Boolean Algebra沂遁吟荧磨桃须桔确远炭忧啊绽殴栅开睫粳撮拎娟捕毋鸟积庭赣畦神滓歉Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Different NotationFunctions with two arguments written in Functions with two arguments written in generic generic Boolean AlgebraBoolean Algebra notation and notation and then in then in Classical LogicClassical Logic notation. notation.脆典逻合诞希知疥沟堰蛀剂与簇赂象倡伺偿吼请邑罕撬莉辱枉划筷艳漆绑Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Booles Expansion TheoremUsing Booles expansion theorem, the Using Booles expansion theorem, the DNFDNF for any logic function now looks like,for any logic function now looks like,诀震龙顺铸潮淬充皋囤桑蛋岩熬肛货宵沽瑰疤啡乡血边双瘤那估伍磅漳茁Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统The DNF for Logic FunctionsSince the discriminants always take a Since the discriminants always take a functional assignment of either functional assignment of either TRUETRUE or or FALSEFALSE, a typical expansion might look , a typical expansion might look something like this,something like this,The logic function that returns B when arguments A and B given. fThe logic function that returns B when arguments A and B given. f1010 (A,B) on next slide. (A,B) on next slide.吮慨苗挛赦恋蛊藉昧全赌棵王仆同撵寥厕条幅甥侯郑茨梁蕾悟妇息乐服窝Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统All 16 Logic Functions冤泞踌秋呢筒骚片腐庄心语挛函拦巢鹿兵惦雨打稿颊诗退贵诺酪糕期派拦Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Classical SyllogismA is TRUEA is TRUEA implies BA implies BTherefore, B is TRUETherefore, B is TRUE奋响洁柏多鲁凑棱铣彼眺敝告裔蜜屋歌衫菇辫喉篓溃阜蠕淆纸瞎谴荣锌从Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Classical SyllogismRecapitulate Jayness demonstration generalizing Recapitulate Jayness demonstration generalizing classical logic with probability theory classical logic with probability theory (but here I emphasize (but here I emphasize the Boolean Algebra aspects).the Boolean Algebra aspects). * * Chapter 2, pp. 35-36 * Chapter 2, pp. 35-36 皂桌鸳孪痘扶蔓侗袍查央缩帜亭松主索纷瘪颊夯朽呆宪鄙膊毫爽穿吭记绣Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Generalizing Classical LogicBayess TheoremBayess TheoremSubstitute DNF expansion of Implication function A Substitute DNF expansion of Implication function A B BBoolean operations reduce above toBoolean operations reduce above to脑堂臂灼撼产动孽挟利俺矢研褪购贯六搁丫搏马壤锻茎哆衅疤咕油囊佑贰Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Modus PonensSubstitute the shortened Substitute the shortened DNF expansionDNF expansion for the for the implication function in Bayess Theoremimplication function in Bayess TheoremBayess Theorem now looks like thisBayess Theorem now looks like this置亩甄蚕辰么瓤裁夺褐吗她迢幽嫌锅荣眷脂瓷冉笺郑陶刊看悔奴搜模裔逃Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Modus PonensBoolean operations on denominatorBoolean operations on numeratorBayess Theorem持恰趾溃骑织狞班防踪盯省补词醛气顷善零乾土缮逝卿灭岩泰睛宰这尔镣Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Different ApproachNow solve Now solve modus ponensmodus ponens using a using a joint probability table.joint probability table. The answer should be the same.The answer should be the same.批螺郸革刽卜余灼甜川院虽戌爷擒黔怔懊票邓践神萨豫据允枢梁哄涯甫冠Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Joint Probability TableTwo statements Two statements A and BA and B each take on only two values. There each take on only two values. There are four cells in the joint probability table. The model assigning are four cells in the joint probability table. The model assigning these numerical values is the these numerical values is the implication function Aimplication function AB.B.护奇桔刮婉禹康距逝甄氦感乐皿熙蛛园钝旭矿壤橡刚蒜负遍班阜笑介德惋Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Joint Probability TableThe same answer as before. The same answer as before. The model The model MMk k assigns legitimate numerical assigns legitimate numerical values to the joint statements in the four cells values to the joint statements in the four cells of the jpt. The model is the implication function.of the jpt. The model is the implication function.Cell 1Cell 1Cell 3萤缘叉翼哀调鼓获雨饰也畏征戌炮秃鳃肢侍晌蓝晰萍浊翁伊谈走歼涵摘黄Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Probability Theory Generalizes Classical LogicAssume the same logic function, but now Assume the same logic function, but now B is TRUE.B is TRUE. What is the impact on A?What is the impact on A?Here is an “invalid logical argument, but one that is easily solved Here is an “invalid logical argument, but one that is easily solved using probability theory in exactly the same manner as before.using probability theory in exactly the same manner as before.捡技啦甘蛋曹另宿彰爪推泼指晓沸凭彰舍走白荷赣妨恍鬃刚炊树机急康狠Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统The “Invalid” Logical Argument Solved by Probability TheoryUse the numerical assignments from the joint probability table.Probability theory as a generalization of logic returns an answer, while classical Probability theory as a generalization of logic returns an answer, while classical logic refuses to address the issue calling it “undecidable.”logic refuses to address the issue calling it “undecidable.”如甩服说厩薪筏奋缀爸箕蜂蹄浚筐伐氛陨傻箭正搞懒凸吾帆羔节讫就寝蜒Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Placement of 0s in JPTCell 3 indexes joint statement: Cell 3 indexes joint statement: A is TRUEA is TRUE and and B is FALSEB is FALSE.ff13 13 (A, B) has functional assignment of (A, B) has functional assignment of FALSEFALSE if if A is TRUEA is TRUE and and B is FALSEB is FALSE by very definition of by very definition of operator. operator. Therefore, cell 3 Therefore, cell 3 MUST HAVEMUST HAVE a numerical assignment of 0 a numerical assignment of 0 under this model.under this model.胶冈县掂佳恼郑熄乾昂鹃事甫蛮妈柜猫筋烁阉黄益旬氮痪殖酪菊瑞煎创冈Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统 Issues in Order AddressedBoolean AlgebraBoolean AlgebraLogic FunctionsLogic FunctionsCellular AutomataCellular Automata牡挪淋滤察桥岁镊组飞朽轰膝董叶勾箩盗撼雇成惦打皂扯仅文秋诸琅每秋Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Elementary One-Dimensional Cellular AutomataWolframs famous example of a CA proven to Wolframs famous example of a CA proven to be a be a Universal Turing MachineUniversal Turing Machine硅窜存卒证山舷宦篆光骚严妇舆茵炕夺逻供担躺蛋堡笑塔锰拉野咱漓揖蝇Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Elementary One-Dimensional Cellular AutomataFirst few steps of a CA following Rule 110First few steps of a CA following Rule 110卤索凹练历宦释醛赫迹币抛础凝椰奄搏盆疮滑杉葡候侧烬茅沉贱瑟小纤炯Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Many Steps of CA Following Rule 110Interacting Localized Structures. They compute anything that can be computed!Our stand-in for a complicated detailed ontological model of the world.Our stand-in for a complicated detailed ontological model of the world.作市蹬棵编撅柔射耐嘻粘咕烈弛肉等奎谱泥跪遭贾肌懈囊悟瓦鹅吐糙昏温Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Logic Functions with Three VariablesBut But Rule 110Rule 110 is simply a logic function with is simply a logic function with three argumentsthree arguments instead instead of the two arguments as we have examined previously in discussing of the two arguments as we have examined previously in discussing classical logic. There are a total of 256 possible logic functions with classical logic. There are a total of 256 possible logic functions with three variables and three variables and Rule 110 is one of those 256 functionsRule 110 is one of those 256 functions.Here is Here is Rule 110sRule 110s function table for all eight possible settings of the function table for all eight possible settings of the three variables.three variables.Black Black T TWhite White F F个县铝稗湛函簇谰涤免贾寞拼柱吃评宏奢颧庚腐卸腺帝恬痛册艳扣惫晶桥Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统The DNF for Rule 110And, like any logic function, Rule And, like any logic function, Rule 110 can be expanded via the DNF.110 can be expanded via the DNF.The DNF expansion of a three variable logic function, Rule 110.亡涌骤栓窖茂图菌猜苟忻褐正湖柄犬粕抿添竖哆诱羔冤骇罚蚜桥煽奎你蹦Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统The DNF and 0s in the JPTAnd, just as we did when examining modus ponens, we can use the DNF expansion of a logic function to locate the 0s in a joint probability table.We will employ the We will employ the jptjpt as an easier alternative to the formal as an easier alternative to the formal Boolean operations in solving Bayess Theorem applied to CA.Boolean operations in solving Bayess Theorem applied to CA.岳蔷邢垢嘶箔娩靖舶望呜嫉雪古泪了裳审簇婚漾街凰曾之岳支瓤甭娩厚小Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Joint Probability Table with Numerical Assignments Following Rule 110Placement of Placement of 0s dictated 0s dictated by model by model following following logic function logic function f f110110 (A,B,C) (A,B,C)壁唆惰缄削版锣冶蚕浪杯毖宏告辆犊李狄致橙砸谷挨豁憨逝寸撒瓢赡浦轿Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Joint Probability Table with Numerical Assignments Following Rule 110BN+1 cannot be TRUE (black) if AN, BN and CN are also TRUE (black)脆荚振知怀挎证几疹属涉匿胸储胶匠崭矩菩臀芭畜种鬃险阐型逝敦土望巍Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Joint Probability Table with Numerical Assignments Following Rule 110The DNF expansion of Rule The DNF expansion of Rule 110 also tells us where the 110 also tells us where the non-0s must be placed.non-0s must be placed.If functional assignment If functional assignment BBN+1N+1 is is TRUETRUE at these at these five terms, non-zero five terms, non-zero probability is assigned.probability is assigned.Or, Or, 0s0s are placed where are placed where functional assignment isfunctional assignment is FALSEFALSE.也袖叔耸嘲痈攒头刃戈该克驱傈纹庭软瓣喂咋于榆瞧乱严投妄舷狭瘁笺梳Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Bayess Theorem for Rule 110Write out the generic template for updating Write out the generic template for updating a state of knowledge a state of knowledge (Bayess Theorem, of course)(Bayess Theorem, of course) about about the cell to be updated given the colors of the cell to be updated given the colors of three cells at the previous time step and the three cells at the previous time step and the numerical assignment following Rule 110.numerical assignment following Rule 110.事丢避掉状状脏敛妙最谨尧呀唾肆杜牛拟愉宝亿鲁递垢范激剑襄卯讳佑篙Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Updating Color of Cell in CA Using Bayess TheoremLocate and insert values from jpt. Insert the numerical assignment from Cell 12 in the numerator and the numerical assignment from Cells 12 and 4 in the denominator.Bayess Theorem with denominator expanded. Numerical assignment follows from model implementing Rule 110.吾杰帕糕馈叛盂拇呐铝申篙阜猪啼伊风士梯击狱转俯谤噬野础腰冕兰科邹Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统What is the Point?Jaynes generalized logic functions (Boolean functions) by Jaynes generalized logic functions (Boolean functions) by treating them from a probabilistic and inferential standpoint. treating them from a probabilistic and inferential standpoint. CA are composed of Boolean functions, so think of them not CA are composed of Boolean functions, so think of them not from the from the deductivedeductive viewpoint, but rather from an viewpoint, but rather from an informationalinformational standpoint. standpoint.For example, let a wide range of models insert legitimate For example, let a wide range of models insert legitimate numerical values into the cells of a joint probability table for a numerical values into the cells of a joint probability table for a CA. Suppose there is a lack of information about which CA. Suppose there is a lack of information about which model correctly governs the evolution of the CA.model correctly governs the evolution of the CA.Then, as a consequence ensuing for all informational systems, Then, as a consequence ensuing for all informational systems, predictionprediction at least becomes a feasible concept to explore. at least becomes a feasible concept to explore.脑塔爽钦赴库艺椽怠察氰畦捧氖凰满术今晾橙勤辰痕扔矢窿恋梨昨麻犬虹Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Different Models Assign Different Numerical Values“Almost” the “Almost” the model. model. 骚驶崇阵靴恭痘许幂孩咳充僳皂译灿侨螟蔗他羔耙燥立繁兑蛛敬做翘协挨Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统How Do We Predict Future Events in an Inferential System?We use the same formal manipulation We use the same formal manipulation rules we always use in probability theory. rules we always use in probability theory. For example, to update a state of For example, to update a state of knowledge about some future event knowledge about some future event FEFE conditioned on some observed data conditioned on some observed data DD and and involving models assigning numerical involving models assigning numerical values values MMk k 骤脆雪木咽哪矮委戈哼酥凄受腑里急猴扼诧俱遂局媳夏琐栏炎似甘按遗印Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Predicting Future Events in CAThe inferential approach provides a The inferential approach provides a quantitative way for the information quantitative way for the information processor to update its state of knowledge processor to update its state of knowledge about the color of any cell in the cellular about the color of any cell in the cellular automaton. automaton. Here, we have lost information about which Here, we have lost information about which model (rule) is governing the evolution of the CA.model (rule) is governing the evolution of the CA.We must average over the predictions made by 256 modelsWe must average over the predictions made by 256 models戳床蒸衬涝瞩羹酉佯裂困解甸僳邪呀粤了弛另肖鸭褥反蒲荆帜钎扒卑竿品Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Predicting Many Cells of the CAJoint probability table is Joint probability table is impossibly huge! impossibly huge! (And any summation (And any summation is hard too!)is hard too!)The prediction formula for updating arbitrarily many cells of the CA.The prediction formula for updating arbitrarily many cells of the CA.锑醉派婪累四工忱峭步徐接纹吝坪抢觅区锌尸船煽说悬间擎拦刘剿串足谁Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Any Existing Solutions? We have to make the difficult We have to make the difficult transition from transition from micro-eventsmicro-events to to macro-eventsmacro-events in trying to predict the in trying to predict the far future for CA. far future for CA. Historically, Statistical Physics faced Historically, Statistical Physics faced the same problem.the same problem.婚屉钙炙瘴驾讨芳瓤醛径肩系竟玻熊尖门庄坯铃兄疥沪掺拱赣甫聋剥贬况Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Macro-Statements Are Predicted by Inferential Approach* See Jaynes, Chapter 18 , for discussion.Macro-Statements in the form of future frequency counts Macro-Statements in the form of future frequency counts can be predicted conditioned on past frequency counts.can be predicted conditioned on past frequency counts.* *蔗星持果梭陀泳昼坦唬驴宽了例耳哼屹沟叙练吓丢鬃骤坍骑钳在件踪陡墟Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Useful for CA? The same analytical apparatus could be The same analytical apparatus could be applied to predict frequency counts of applied to predict frequency counts of blackblack and and whitewhite cells for each of cells for each of n n cells at cells at some future some future MMthth time step. time step. But is this type of macro-statement one Wolfram would accept But is this type of macro-statement one Wolfram would accept as meaningful? as meaningful? (What has been voluntarily discarded might be (What has been voluntarily discarded might be viewed as very important!)viewed as very important!)舵笔摆奶死炕尊星掠涯盏抢黔策边厩薯姆陕涝割拱肢粮庸财簿轩垛殷起粒Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Can Techniques from Information Geometry Help?qUse concepts from Information Geometry.q Try a- projection from complicated probability distribution at point P in general manifold to computationally tractable distribution at point Q within sub-manifold. (Amari)qMaximum Entropy Principle with measure function m(x) (Jaynes)qMean Field Approximation in Statistical PhysicsqBelief Propagation algorithms合贾酬待险迁学挽父瑞鹃姥肝馅珠玻筐口篙彼逻责垃介虐芳萝嘉酬举把啊Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Sine qua nonSine qua non of predictingIn order to predict, In order to predict, micro-informationmicro-information must be must be voluntarily discarded!voluntarily discarded!讨品后执东擦着濒蚌负邱皿军烯膘睹抛纪圣沽钮直再阂小金租厄赤踊瞅访Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Implications for CAWhat kind of What kind of micro-micro-informationinformation should be should be discarded?discarded?What kind of What kind of macro-macro-statementsstatements make sense for make sense for CA?CA?奏面畜译纸锈勒旭鄂舍稍齐补铸吹掸庶瞥俱附济七综先跺击美咳滇视蒸绥Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统My Main ArgumentqWolfram is pessimistic about being able to predict the behaviour of deductive systems like cellular automata into the far future. qqHe is correct.He is correct.qHowever, (following Jaynes) if CA are treated from the perspective of inference and information, then a limited form of prediction becomes possible.qThis means that we must employ probability theory as the generalization of classical logic, that is, we use inference instead of deduction.qIts a repetition of the old analogy from statistical mechanics. We voluntarily give up information about the micro details of molecular dynamics in exchange for the ability to predict macrovariables.qThe moral: We MUST approach the problem of predicting the consequences of any sufficiently detailed explanatory model of the world through inference and information.槛基受乙剔锡深呆鸯销铜霉溺搭界构挞凄升垮厦汀运墒腮徘夯颖粒丽钮符Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Summary of Technical PointsqBoolean functions f(x1, x2, xn) = f: Bn B for any number of variables.qAbstract, general, closed, do not involve numbers or arithmetic operations, and are easy to comprehend.qTalk focused on n=2 (CL) and n=3 (CA).qPerfect foil for probability theorys formal manipulation rules.厘始弄辣赘汉益岂柠远忱日犊龟喉赠妨读匿哪狞良磋遍赖洽迭抚氮创逸园Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Summary of Technical PointsqqAny such Boolean function can be expanded Any such Boolean function can be expanded into the into the Disjunctive Normal Form.Disjunctive Normal Form.qqA classical logic function is a A classical logic function is a Boolean Boolean functionfunction and can be expressed via the DNF. and can be expressed via the DNF.qqLogic problems (classical syllogisms) can be Logic problems (classical syllogisms) can be solved by solved by probability theoryprobability theory. .qqJoint probability tables illustrate how some Joint probability tables illustrate how some model model assigns numerical values assigns numerical values to joint to joint statements. statements. 究惨贫届略态咏燎纪茎隔隙传盖羌尿不侦讼赋译瀑躬额输耘或胀硼捞傍赴Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Summary of Technical Points qqThe DNF points to where The DNF points to where 0s should be 0s should be placedplaced in a jpt. in a jpt.qqDifferent numerical assignments Different numerical assignments via via Information GeometryInformation Geometry as dictated by as dictated by different models.different models.qqA CA is comprised from A CA is comprised from logic functions of logic functions of n=3 variables.n=3 variables. There is a DNF expansion for There is a DNF expansion for each of the 256 possible functions.each of the 256 possible functions.qqThe DNF points to where 0s should be The DNF points to where 0s should be inserted in a jpt for inserted in a jpt for cellular automata.cellular automata.膀槽寇天喊渡讲祝张腕凯遣剿吼崇幅桥悄痹沥转但绦道夷嫩防萧或圾掘萄Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Summary of Technical PointsqqCA can be CA can be solved and generalizedsolved and generalized using using Bayess Theorem, DNF and the jpt.Bayess Theorem, DNF and the jpt.qqTherefore, all the relative advantages of Therefore, all the relative advantages of inferencing vs. deduction can be brought to inferencing vs. deduction can be brought to bear on the problem of bear on the problem of PREDICTINGPREDICTING the the future behaviour of CA.future behaviour of CA.qqFor example, For example, formal manipulation rulesformal manipulation rules lead lead to the prediction formula.to the prediction formula.椎样荣啄帚踊巴妙鉴揍怪透柔幢貉虱口低食凸楷释孪屿克壬引磺觅谨庙凳Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Summary of Technical PointsqqUse technical notions from Use technical notions from Information Information Geometry.Geometry.qqFind Find computationally tractable computationally tractable distributionsdistributions that are that are “close”“close” to to original complicated distribution.original complicated distribution.qqWhat is an What is an appropriate macro-eventappropriate macro-event for a CA amenable to prediction?for a CA amenable to prediction?蓄亨桃概毡膳仗度纸陷重仲伯赃烷颧狠史氓千景堑筏果膜蟹厩嫉逸戎仍拾Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Final ThoughtCandidate Macro-Event: Wolframs localized structures in Rule 110 CA interacting over time. Isnt it interesting to speculate that such macro-events conspire to produce an unexpected event in the far future?Initial ConditionsSurprising answer奥榆怜暮轩丰发矣翘金蓑斩夺饶敢德料伎讲虐附仗叉探侵埔匡涟吝妈怒不Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统 My Book (in progress) 贮炔孔犀椿东着刻穗详屈铰滓躇糜茁看庚母医铭魂躲昼计疯草笼筛凤灭勺Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统Cellular Automata Gneralized to an Inferential System：元胞自动机广义推理系统