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One-Dimensional CA's"], "Title", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], Cell[TextData[{ StyleBox["Mathematica in Education", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox["\nVol.2 No.3\n(Spring 1993)\n", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Subtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "by\nRichard J. Gaylord\nDepartment of Materials Science\nUniversity of \ Illinois at Urbana-Champaign\ngaylord@ux1.cso.uiuc.edu"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["Introduction "], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[{ StyleBox["A cellular automaton (CA) is a ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["discrete", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox[ " dynamic system of lattice sites\nthat evolve in discrete time steps as \ each site assumes a value from a\nfinite set of values according to ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["local", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["uniform", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox[ " rules determined by the\nvalues of a neighborhood of sites around it.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Stephen Wolfram [Wolfram, 1984] has expressed the view that \"any physical\n\ process can be described by an algorithm and therefore can be represented\nby \ a computational process\" and that \"cellular automata can be regarded as\n\ computers and as models of physical systems\". "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "One cellular automaton which was discussed earlier [Gaylord, 1992], is the\n\ sandpile model which has been used to describe self-organized criticality\n\ (SOC) and catastrophes in complex systems. Now I want to take a look at\nthe \ two most famous cellular automata. In Part I, I'll discuss\none-dimensional \ cellular automata In Part II, I'll turn to the\ntwo-dimensional cellular \ automaton known as the \"Game of Life\"."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["One-Dimensional Cellular Automata "], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "In landmark investigations [Wolfram, 1993] which introduced computation as\n\ a primary tool or methodology for scientific investigation, equal in\n\ importance to both theory and experiment, Wolfram studied the evolution of\n\ one-dimensional cellular automata (1D CA). "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "A 1D CA starts with a row of lattice sites (or cells) each of which has\n\ some value. The row either consists of a single seed site with a non-zero\n\ value, located in the center of the row, or is a disordered state in which\n\ seed sites are randomly distributed in the row. "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "The CA \"evolves\" by producing successive rows of sites, each new row\n\ representing a time step. The value of each site in a new row is determined\n\ by applying a specific rule to the site in the preceding row which lies\n\ directly above the new site. "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "The rule is homogeneous, which means that the same rule is used in\n\ determining the value of each new site, and the rule is local, which means\n\ that it is based the values of a \"neighborhood\" of sites, consisting of the\ \nsite and/or its nearest-neighbor (nn) sites. The rule is applied to all of\n\ the sites in a row simultaneously. For a finite CA, the nn sites of the\n\ first and last sites in a row are taken from the other end of the row (eg.,\n\ the value of the next nn site on the left-hand side of the first site is\n\ given by the last site). The use of this \"wrap-around\" reflecting boundary\n\ condition essentially turns the line into a circle. "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "As the updating of the current state of the CA proceeds and new rows\n\ appear, a pattern emerges. "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "Wolfram recognized that 1D CA's evolve into patterns falling into one of\n\ four general classes (analogies to physical and computing systems are noted\n\ in parentheses): "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "#1 - homogeneous state [analogous to limit points] \n#2 - periodic \ structures [analogous to limit cycles]\n#3 - chaotic pattern [analogous to \ strange attractors]\n#4 - complex, localized structures [possibly analogous \ to universal computers] "], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "The analogies of the behaviors of CA's in the first three classes to the\n\ behaviors of physical systems is quite interesting. Even more intriguing is\n\ the behavior of a CA in the fourth class where there is no simple algorithm\n\ for predicting behavior and it is necessary to actually run the CA\nevolution \ simulation to determine the outcome."], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "The fact that 1D CA's show such great \"complexity\" in their behaviors has\n\ rather profound implications (eg., some people have postulated that the\n\ universe itself, is a cellular automaton). Here, I will develop a program\n\ for creating 1D CA's which can be used to join in the exploration of these\n\ fascinating systems."], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData[" CA Rules"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I can write down the rules that govern the evolution of lattice sites based\n\ on the number of lattice sites that interact and the values that a lattice\n\ site can have. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I'll first calculate the total number of rules for a given set of k and r\n\ values. When 2r+1 sites interact (ie., a given site and its r nn sites on\n\ each side) and each site can have k values (the integer values 0 through\n\ k-1), the total number of possible rules is given by: "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["totalrules[k_, r_] := k^(k^(2r+1))"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Applying this formula to some specific examples:\n\n(1) When each site can \ have two values, 0 or 1, and sites evolve\nindependently of the state of \ their neighbors, there are four possible\nrules,"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["totalrules[2, 0]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ 4\ \>", "\<\ 4\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[" which I can write as"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData[ "{{{1} -> 0, {0} -> 0}, \n {{1} -> 0, {0} -> 1}, \n {{1} -> 1, {0} -> 0}, \n \ {{1} -> 1, {0} -> 1}}"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {{{1} -> 0, {0} -> 0}, {{1} -> 0, {0} -> 1}, {{1} -> 1, {0} -> 0}, {{1} -> 1, {0} -> 1}}\ \>", "\<\ {{{1} -> 0, {0} -> 0}, {{1} -> 0, {0} -> 1}, {{1} -> 1, {0} -> 0}, {{1} -> 1, {0} -> 1}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[{ StyleBox[ "Let me explain the notation being used here. The outermost list is a\n\ nested list consisting of lists of transformation rules for determining the\n\ value of a new site. Each inner list (eg., {{1} -> 1, {0} -> 1}) is called\na \ ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["rule set", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox[ ". The list on the left-hand side of a transformation rule in a\nrule set \ is a tuplet whose elements are the values of the neighborhood of a\nsite (the \ middle element of the tuplet is the value of the site). The\ninteger on the \ right-hand side of the transformation rule is the value of\nthe new site \ created directly beneath the site during a time step of CA\nevolution (eg., \ the transformation rule {0,1,1} -> 0 says that a new site\ncreated beneath a \ site having a value 1 and a left next nn with value 0 and\na right next nn \ with value 1, will have a value 0). ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[{ StyleBox[ "(2) As the number of nearest-neighbor sites involved in the interaction\n\ increases, the number of rules rapidly increases. \nWhen three (2r+1 = 3) \ sites (ie., a given site and its next nn sites on\neach side (", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["v", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], StyleBox["w", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->18], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], StyleBox[ ")) interact and each site can have k = 2 values, there are\nthe following \ possible neighborhood values: ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "{{1, 1, 1}, {1, 1, 0}, {1, 0, 1}, {1, 0, 0}, {0, 1, 1}, \n {0, 1, 0}, {0, \ 0, 1}, {0, 0, 0}}"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Thus, there are eight or (2 2 2) = 2^3 = 8 possible triplets that can be\n\ formed. Each of these triplets can result a site value of 1 or 0. For\n\ example, one rule set is: "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ " {{1,1,1} -> 1, {1,1,0} -> 1, {1,0,1} -> 1, {1,0,0} -> 0, \n {0,1,1} -> 1, \ {0,1,0} -> 0, {0,0,1} -> 0, {0,0,0} -> 1}"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Since each triplet can result in either a 1 or 0, there are a total of \n(2 \ 2 2 2 2 2 2 2) = 2^8 = 256 rule sets."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["totalrules[2, 1]"], "Input", AspectRatioFixed->False, FontFamily->"Courier"], Cell[OutputFormData["\<\ 256\ \>", "\<\ 256\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[{ StyleBox[ "(3) When each site can have two values and interacts with its two nn sites\ \non each side (", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["uv", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], StyleBox["w", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->18], StyleBox["xy", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], StyleBox[ ") there are 2^5 = 32 possible quintuplets and 2^32 =\n4,294,967,296 rules \ sets!", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["totalrules[2, 2]"], "Input", AspectRatioFixed->False, FontFamily->"Courier"], Cell[OutputFormData["\<\ 4294967296\ \>", "\<\ 4294967296\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Courier"]}, Open]], Cell[TextData[ "(4) Increasing the number of values that a site can have beyond two \n\ increases the number of rule sets even more dramatically. When each site\ncan \ have three values and sites interact with their next nn site on each\nside, \ there are 3^(3^3) = 7,625,597,484,987 rules sets."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["totalrules[3, 1] "], "Input", AspectRatioFixed->False, FontFamily->"Courier"], Cell[OutputFormData["\<\ 7625597484987\ \>", "\<\ 7625597484987\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "I would like to have a shorthand notation for expressing a rule set. I can\n\ illustrate how this can be done using example 2. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ " {{1,1,1} -> 1, {1,1,0} -> 1, {1,0,1} -> 1, \n {1,0,0} -> 0, {0,1,1} -> 1, \ {0,1,0} -> 0, \n {0,0,1} -> 0, {0,0,0} -> 1}"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "I take the right-hand sides of the transformation rules in this rule set\n\ and form the binary number 11101001. The base ten equivalent of this binary\n\ number is 233."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["2^^11101001"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ 233\ \>", "\<\ 233\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "So, I can say that the rule number 233 corresponds to the rule set "], "Text",\ Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ " {{1,1,1} -> 1, {1,1,0} -> 1, {1,0,1} -> 1, \n {1,0,0} -> 0, {0,1,1} -> 1, \ {0,1,0} -> 0, \n {0,0,1} -> 0, {0,0,0} -> 1}"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Each of the k^(k^(2r+1)) rules in a 1D CA can be given its own rule number\n\ in this way, using a base k number to represent the right-hand sides of the\n\ transformation rules in the rule set and taking its base 10 equivalent. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["Playing By The Rules"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I now turn to creating a rule set for any of the k^(k^(2r+1)) rules in a 1D \ CA."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I'll start by looking at the rule number 233 example used above. I can\n\ easily create the required rule set by hand:"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData[ "MapThread[Rule, \n {{{1,1,1}, {1,1,0}, {1,0,1}, {1,0,0}, \n \ {0,1,1}, {0,1,0}, {0,0,1},{0,0,0}},\n {1, 1, 1, 0, 1, 0, 0, 1}}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {{1, 1, 1} -> 1, {1, 1, 0} -> 1, {1, 0, 1} -> 1, {1, 0, 0} -> 0, {0, 1, 1} -> 1, {0, 1, 0} -> 0, {0, 0, 1} -> 0, {0, 0, 0} -> 1}\ \>", "\<\ {{1, 1, 1} -> 1, {1, 1, 0} -> 1, {1, 0, 1} -> 1, {1, 0, 0} -> 0, {0, 1, 1} -> 1, {0, 1, 0} -> 0, {0, 0, 1} -> 0, {0, 0, 0} -> 1}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "Now I'll write a general program that generates the rule set for a given\n\ value of k, r and the rule number w."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I first create the list of the number of tuplets that are involved when a\n\ 1D CA has 2r+1 nearest neightbor interactions and k site values. There are\n\ (as usual) several ways to create the tuplets. The following function\n\ appears to be the fastest, by far:"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "tuplets[k_, r_] := \n Reverse[Flatten[Apply[Outer,\n \ Prepend[Table[Range[0,k-1],{(2r+1)}],List]],2r]];"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["Trying the definition out on the k = 2, r = 1 CA,"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["tuplets[2, 1]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData[ "\<\ {{1, 1, 1}, {1, 1, 0}, {1, 0, 1}, {1, 0, 0}, {0, 1, 1}, {0, 1, 0}, {0, 0, 1}, {0, 0, 0}}\ \>", "\<\ {{1, 1, 1}, {1, 1, 0}, {1, 0, 1}, {1, 0, 0}, {0, 1, 1}, {0, 1, 0}, {0, 0, 1}, {0, 0, 0}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[{ StyleBox[ "I next create the list of the right hand sides of the transformation rule\n\ in the rule set. This list is called the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["update value", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" list.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I'll need to convert the rule number w into a list whose elements represent\n\ the number in base k. There is a built-in function, IntegerDigits, which\n\ does this. For example,"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["IntegerDigits[233, 2]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {1, 1, 1, 0, 1, 0, 0, 1}\ \>", "\<\ {1, 1, 1, 0, 1, 0, 0, 1}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "I now have everything I need: a list of the tuplets, a list of update\n\ values, and the MapThread function. However, there's a slight complication.\n\ Consider rule 24 for a k = 2, r = 1 CA."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["IntegerDigits[24, 2]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {1, 1, 0, 0, 0}\ \>", "\<\ {1, 1, 0, 0, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Length[tuplets[2, 1]] == Length[IntegerDigits[24, 2]]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ False\ \>", "\<\ False\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "I can't thread together lists having different lengths (and taking\n\ 2^^10000000 indicates that half of the rule numbers, those below 128, have\n\ less than the prerequisite eight elements for threading with the list of\n\ eight triplets for the k = 2, r = 1 CA). What is needed is to add 0's to\nthe \ front of IntegerDigits[w, k] to bring the list length up to k^(2r+1). I\ncan \ do this using "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "fillcode[k_, r_, w_] :=\n \ Join[Table[0,{k^(2r+1)-Length[#]}],#]&[IntegerDigits[w, k]] "], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["which works for all rule numbers "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["fillcode[2, 1, 24]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {0, 0, 0, 1, 1, 0, 0, 0}\ \>", "\<\ {0, 0, 0, 1, 1, 0, 0, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[CellGroupData[{Cell[TextData["fillcode[2, 1, 233]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {1, 1, 1, 0, 1, 0, 0, 1}\ \>", "\<\ {1, 1, 1, 0, 1, 0, 0, 1}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "(note: I am taking advantage here of the fact that Table returns an empty\n\ list when the iterator is zero or negative). "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Overall then, I can create a rule set with the following formula:"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "ruleset1[k_, r_, w_] :=\n Module[{fillcode1, tuplets1},\n \ fillcode1[x_, y_, z_] :=\n \ Join[Table[0,{x^(2y+1)-Length[#]}],#]&[IntegerDigits[z, x]];\n \ tuplets1[p_, s_] :=\n Reverse[Flatten[Apply[Outer,\n \ Prepend[Table[Range[0,p-1],{(2s+1)}],List]],2s]];\n \ MapThread[Rule, {tuplets1[k, r], fillcode1[k, r, w]}]\n ] \ "], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["or in the form of a totally inscrutable one-liner,"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "ruleset2[k_, r_, w_] :=\n MapThread[Rule, \n \ {Reverse[Flatten[Apply[Outer,\n \ Prepend[Table[Range[0,k-1],{(2r+1)}],List]],2r]],\n \ Join[Table[0,{k^(2r+1)-Length[#]}],#]&[IntegerDigits[w, k]]}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Just to check that this definition works, I'll generate the list of rule\n\ sets for the simplest case of non-interacting sites having values of 1 and\n0 \ which I discussed at the beginning of the previous section."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData[ "{ruleset2[2, 0, 0], ruleset2[2, 0, 1], ruleset2[2, 0, 2], ruleset2[2, 0, \ 3]}"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {{{1} -> 0, {0} -> 0}, {{1} -> 0, {0} -> 1}, {{1} -> 1, {0} -> 0}, {{1} -> 1, {0} -> 1}}\ \>", "\<\ {{{1} -> 0, {0} -> 0}, {{1} -> 0, {0} -> 1}, {{1} -> 1, {0} -> 0}, {{1} -> 1, {0} -> 1}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "(note that if I try to calculate ruleset2[2, 0, 4], I get an error \ message.)"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["ruleset2[2,0,4]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "MapThread::mptc: \n Incompatible dimensions of objects at positions {2, 1\n\ } and {2, 2} of MapThread[Rule, \n {{{1}, {0}}, {1, 0, 0}}]; \ dimensions are {2} and {3}."], "Message", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ MapThread[Rule, {{{1}, {0}}, {1, 0, 0}}]\ \>", "\<\ MapThread[Rule, {{{1}, {0}}, {1, 0, 0}}]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[TextData[ "Now, I'm ready to develop the program for evolving a one-dimensional\n\ cellular automaton."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["The 1D CA Program"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "I'll begin by defining the symbols that appear in the program:\n\nm = number \ of sites per row\nn = maximum number of rows (generations)\nw = rule number \ (this number must be less than k^(k^(2r+1)))\nk = number of values that a \ site can assume \n2r+1 = number of interacting sites"], "Text", Evaluatable->False, AspectRatioFixed->False, FontFamily->"Times"], Cell[TextData[ "I'll first explain the individual parts that go into the program and then\n\ put them together."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[{ StyleBox["The CA starts from ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], StyleBox["row", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSlant->"Italic"], StyleBox[ ", which is either a disordered state in which there\nare randomly \ distributed seed sites ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["row = Table[Random[Integer, {0, k-1}],{m}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["or a single seed site"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "row = ReplacePart[Table[0, {m}],\n Random[Integer, {0, \ k-1}],Ceiling[m/2]] "], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "(note: the seed sites here have randomly determined values. It is also\n\ possible to specify the seed site values). "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["I'll use CArows to evolve the CA."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "CArows = FixedPointList[(Partition[\n \ Join[Take[#,-r],#,Take[#,r]],(2r+1),1]/.ruleset)&, \n row, n]"], "Input",\ AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "FixedPointList applies an anonymous function to the row of seed site(s) to\n\ produce a new row of sites, and then applies the anonymous function to the\n\ new row of sites to produce another row of sites and so on until there is\nno \ change in the values in successive rows or to a maximum of n rows,\nwhichever \ occurs first. The anonymous function"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "(Partition[Join[Take[#,-r],#,Take[#,r]],(2r+1),1]/.ruleset)&"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["performs the following sequence of operations:"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "(1) a reflecting boundary condition is applied to a row of m sites (denoted\n\ here by {...}) to create a list of m+2r values."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["Join[Take[#,-r],#,Take[#,r]]& [{...}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "(2) the above list is partitioned to form a list of m tuplets of size 2r+1\n\ (the elements of successive tuplets will overlap when r > 0) containing the\n\ values of the neighborhoods of each of the m sites in the row."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["Partition[%,(2r+1),1]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "(3) a new row is created using the rule set to determine the value of the\n\ new sites beneath the old sites."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["% /. ruleset"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Putting all of the code fragments together and adding some code to: (1)\n\ print out the number of generations produced by the CA, and (2) create a\n\ two-dimensional RasterArray graphics with randomly chosen RGBColors\n\ representing different site values, I have "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "CAdisordered[w_, k_, r_, m_, n_] :=\n Module[{ruleset, row, CArows},\n \ ruleset =\n MapThread[Rule, \n {Reverse[Flatten[Apply[Outer,\n \ Prepend[Table[Range[0,k-1],{(2r+1)}],List]],2r]],\n \ Join[Table[0,{k^(2r+1)-Length[#]}],#]&[IntegerDigits[w, k]]}];\n row = \ Table[Random[Integer,{0, k-1}],{m}];\n CArows = FixedPointList[(Partition[\ \n Join[Take[#,-r],#,Take[#,r]],(2r+1),1]/.ruleset)&, row, n];\n \ Print[Length[CArows]];\n Show[Graphics[RasterArray[\n \ Reverse[CArows]/.Thread[Range[0, k-1]->\n (Map[RGBColor, \ Table[Random[],{k},{3}]] /.\n RGBColor[{x__}]->RGBColor[x]) \ ]]], AspectRatio->Automatic]\n ]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[CellGroupData[{Cell[TextData["Running the Program"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "We show a few illustrative k = 2, r = 1 1D CA's below."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["GaylordFig.1.eps"], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["SeedRandom[6] \nCAdisordered[78, 2, 1, 50, 15];"], 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Cell[CellGroupData[{Cell[TextData["CA Animation"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "While a static figure can be used to detect CA patterns, it would be nice\n\ to watch the evolutionary process unfolding. To do this, an animation can\nbe \ created. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "In creating an animation using RasterArray graphics, the main thing I have\n\ to take into account is that the rectangles, representing site values, that\n\ re-appear in successive figures in the series that constitute the flip\ncards \ of the animation must be the same size and color. This requires two\nthings: \ "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "(1) each figure must have the same number of rows. This is accomplished by\n\ adding enough rows of 0's to the CArows used in each figure to bring the\n\ total number of lines in the figure up to the Length of the final CA (so I\n\ have to complete the entire CA computation before I create the graphics).\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "(2) the transformation rule for converting site values to RGBColors must be\n\ applied after all of the site values in all of the CA figures in the\n\ animation are calculated and before the graphics command is applied. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData["The CA animation program is: "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "CAdisorderedAnimation[w_, k_, r_, m_, n_] :=\n Module[{ruleset, row},\n \ ruleset = \n MapThread[Rule, \n {Reverse[Flatten[Apply[Outer,\ \n Prepend[Table[Range[0,k-1],{(2r+1)}],List]],2r]],\n \ Join[Table[0,{k^(2r+1)-Length[#]}],#]&[IntegerDigits[w, k]]}];\n row = \ Table[Random[Integer,{0, k-1}],{m}];\n CArows = FixedPointList[(Partition[\ \n Join[Take[#,-r],#,Take[#,r]],(2r+1),1]/.ruleset)&, row, n];\n \ Print[Length[CArows]];\n \ Map[Show[Graphics[RasterArray[#]],AspectRatio->Automatic]&,\n \ Map[Join[Table[Table[0,{m}],{Length[CArows]-#}], \n \ Reverse[Take[CArows,#]]]&,\n \ Range[Length[CArows]] ]/.Thread[Range[0, k-1]->\n (Map[RGBColor, \ Table[Random[],{k},{3}]]/.\n \ RGBColor[{x__}]->RGBColor[x])] ];\n ]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], 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This will\nclose up the graphics cells, leaving only the \ first cell exposed.\nDouble-clicking on the exposed cell starts the \ animation. The speed of the\nanimation can be adjusted using the number keys \ (1 is the slowest speed and\n9 is the fastest speed)).", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["Conclusion"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Happy pattern hunting. We will next look (finally) at various programs for\n\ the Game of Life [unlike politicians, I always fufill my promises\n\ eventually]."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["References"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Wolfram, Stephen. 1984. \"Computer Software in Science and Mathematics\",\n\ Scientific American, 251(Sept.), pp. 188-203"], "SmallText", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Wolfram, Stephen. 1993, \"Wolfram on Cellular Automata and Complexity: A\n\ Reprint Volume\", Addison-Wesley, Reading, Mass. (the original research\n\ articles in this collection, which is due to be published in September\n1993, \ are largely accessible to the non-specialist)."], "SmallText", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Gaylord, Richard J. 1992. \"Catastrophes in Complex Systems\", Mathematica \ in\nEducation, 2(1), pp. 19-23."], "SmallText", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["Algorithmic Housekeeping"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "Below are an improved version of one program (Spreading) and a correction\n\ to another program (Percolation Clustering) that I presented previously."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData[ "Correction to the Percolation Clustering Algorithm:"], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "The clusters in ufinal in Out[1] in the percolation clustering article\n\ [MiE, 2(2), 21-24 (1993)] are supposed to be consecutively numbered but\n\ aren't. This is because the code for relabelling in the clusterLabel\nprogram \ is wrong. "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "relabelrules2 = Thread[new->correctlabels];\nufinal = ul /. relabelrules2;"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["should be replaced by "], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "relabelrules2 = \n Thread[Union[correctlabels]->\n \ Range[Length[Union[correctlabels]]]];\nufinal = ul //. relabelrules1 /. \ relabelrules2;"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "Sorry about that (I might note that the incorrect code was run a number\n\ times without producing an incorrect output; this demonstrates the need to\n\ perform a great many runs to ensure, as much as possible, that a program\n\ works in all cases)."], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"]}, Open]], Cell[CellGroupData[{Cell[TextData["Improved Spreading Algorithm"], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "By changing the way in which the perimeter site with the lowest random\n\ number is selected for the invasion percolation case in the spreading\n\ program [MiE, 1(4), 17-19 (1992)], the program becomes more efficient (and\n\ elegant). This requires a number of changes in the original program and so\nI \ simply give the complete improved program below:"], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[ "spreading[n_Integer,p_Real:-1.0]:= \n Module[\n {cluster = {{0, 0}}, \n \ choices = {{1, 0},{0, 1},{-1, 0},{0, -1}}, \n per = \ Transpose[{Table[Random[],{4}], {{1,0},{0,1},{-1,0},{0,-1}}}], \n \ perTocluster, nn, new, newper, testsite}, \n perTocluster := \n \ (Module[{nn, new, newper},\n AppendTo[cluster, Flatten[#, 1][[2]] \ ];\n per = Complement[per, #];\n nn = Map[Function[x, x + \ Last[cluster]], choices]; \n new = Complement[nn, cluster, \ Transpose[per][[2]] ];\n newper = \ Transpose[{Table[Random[],{Length[new]}],new}];\n per = \ Join[per,newper]])&;\n While[Length[cluster] < n + 1 && Length[per] >0,\n \ testsite = Random[Integer,{1, Length[per]}];\n \ Which[p < 0, perTocluster[First[Sort[per]]],\n \ per[[testsite, 2]] <= p, perTocluster[{per[[testsite]]}], \n \ True, per = Complement[per, {per[[testsite]]}] ] \n ];\n \ Show[Graphics[{RGBColor[0.3,0.3,0.6],\n \ Map[(Rectangle[#-{0.5,0.5},#+{0.5,0.5}])&,cluster]}],\n \ AspectRatio->1, \n PlotRange->Map[({Min[#], \ Max[#]})&,Transpose[cluster]]] \n ]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"]}, Open]]}, Open]]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{28, Automatic}, {Automatic, 1}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V8nh09RAj0000000OXQ<097PXP0AP1Y06`0I@1^0642HSkP 0V97`0000001nR4@0TN2R000000000000000009R?^0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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***********************************************************************)