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(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = input; preserveAspect] (* This notebook generates some examples of cellular automata. Using a standard Mathematica package, the procedures listed below display the evolution of boolean-valued cellular automata. Each automaton is specified as a mapping from {0,1}^3 to {0,1}. We can represent an automoton as an integer between 0 and 256 where the boolean encoding of the integer specifies the outputs for each of the 8 possible inputs in the set {0,1}^3. For example, 4 = {0,0,1,0,0,0,0,0} so the automaton #4 maps {0,0,1} to 1 and every other input to 0. In the literature on cellular automata, #57 is considered to exhibit interesting behavior. Why?. *) ;[s] 3:0,0;84,1;95,0;669,-1; 2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,3,12,0,0,0; :[font = input; preserveAspect] (* Use the package released with Mathematica. Note that the Mathematica package has the same file name as this notebook, but a different path name. *) Needs["Examples`CellularAutomata`"] ;[s] 5:0,0;33,1;50,0;64,1;75,0;194,-1; 2:3,12,10,Courier,1,12,0,0,0;2,12,10,Courier,3,12,0,0,0; :[font = input; preserveAspect] (* NumberedRule[j] returns the rule for the automaton specified by j. EvolveCA[i,j,n] generates n boolean vectors, using the rule indicated by j, and starting from the state indicated by i. The length of the boolean vectors is determined by i. *) :[font = input; preserveAspect] Exercise0[m_,n_,i_] := ShowCA[EvolveCA[Table[Random[Integer],{j,0,m}], NumberedRule[i],n]] :[font = input; preserveAspect] Exercise1[min_,max_] := Block[{s = Table[Random[Integer],{i,0,50}]}, Do[ShowCA[EvolveCA[s,NumberedRule[i],50]], {i,min,max,2}]] :[font = input; preserveAspect] (* Here is a simple function for comparing rule sets. Think about what constitutes a "complex" rule set and try to design a comparator function to sort rules (automata) by their "complexity." *) :[font = input; preserveAspect] RuleCompare[x_,y_] := Less[Apply[Plus,Noop[x]], Apply[Plus,Noop[y]]] :[font = input; preserveAspect] RightHandSide[r_] := Last[r] :[font = input; preserveAspect] Noop[r_] := Abs[Plus[r[[1]][[2]],Minus[Last[r]]]] :[font = input; preserveAspect] SortedRules[min_,max_] := Sort[Table[NumberedRule[i],{i,min,max}], RuleCompare] :[font = input; preserveAspect] Exercise2[min_,max_] := Block[{s = Table[Random[Integer],{i,0,50}]}, If[And[Not[Less[min,0]], Not[Less[255,max]]], Map[Function[r,ShowCA[EvolveCA[s,r,50]]], SortedRules[min,max]]]] :[font = input; preserveAspect; startGroup] Exercise0[50,50,57] ; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% 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