(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 850423, 17812]*) (*NotebookOutlinePosition[ 851178, 17838]*) (* CellTagsIndexPosition[ 851134, 17834]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Algorithms on Finite Automata\ \>", "Title", TextAlignment->Center, TextJustification->0], Cell["Part I. Generating and Enumerating Words", "Subtitle", TextAlignment->Center, TextJustification->0], Cell[TextData[ "by Jaime Rangel Mondrag\[OAcute]n\n\nFacultad de Inform\[AAcute]tica\n\ Universidad Aut\[OAcute]noma de Quer\[EAcute]taro\nDivisi\[OAcute]n de \ Estudios de Posgrado\n\nCentro Universitario\nCerro de las Campanas s/n\nC.P. \ 76010 Quer\[EAcute]taro, Qro.\nMexico\n\njrangel@sunserver.uaq.mx\n\n\ Copyright \[Copyright]1999, UAQ, Facultad de Inform\[AAcute]tica"], "Subsubtitle", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox[ "Permission is hereby granted to reproduce part or all of this file for any \ purpose other than direct profit providing that the source is acknowledged.", FontColor->RGBColor[0, 0, 1]]], "Commentary", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[{ "This is the first of four parts on the implementation of algorithms on \ finite automata (FA). To activate the different functions comprising this \ folder, choose the ", StyleBox["Kernel-Evaluation-EvaluateInitialization", FontSlant->"Italic"], " menu path." }], "Subsubtitle"], Cell[CellGroupData[{ Cell["REFERENCES", "Section"], Cell["\<\ THE DESIGN AND ANALYSIS OF COMPUTER ALGORITHMS A. V. Aho, J. E. Hopcroft, J. D. Ullman Addison-Wesley INTRODUCTION TO COMPUTER THEORY D. I. Cohen John Wiley and Sons INTRODUCTION TO AUTOMATA THEORY, LANGUAGES AND COMPUTATION J. E. Hopcroft, J. D. Ullman Addison-Wesley REGULAR ALGEBRA AND FINITE MACHINES J. H. Conway Chapman and Hall DISCRETE AND COMBINATORIAL MATHEMATICS R. P. Grimaldi Addison-Wesley\ \>", "ItemizedTextNote"] }, Closed]], Cell[CellGroupData[{ Cell["1 INTRODUCTION", "Section"], Cell[TextData[{ "A finite automaton (FA) is described by a 5-tuple M=(Q, \[CapitalSigma], \ \[Delta],", Cell[BoxData[ \(TraditionalForm\`\(\ q\_0\)\)]], ", F), wherein Q is a finite set of ", StyleBox["states", FontSlant->"Italic"], ", \[Sum] is an ", StyleBox["alphabet", FontSlant->"Italic"], ", \[Delta] is the ", StyleBox["transition function", FontSlant->"Italic"], " \[Delta]: Q x \[CapitalSigma] \[Rule] Q, ", Cell[BoxData[ \(TraditionalForm\`q\_0\)]], "\[Epsilon]Q is the ", StyleBox["initial state", FontSlant->"Italic"], " and F \[SubsetEqual] Q is the set of", StyleBox[" final states", FontSlant->"Italic"], ". A word (string) made from elements in the alphabet is read one character \ at a time and forces the automaton to change states accordingly.\nIf a word \ ", Cell[BoxData[ \(TraditionalForm\`w = \(w\_1\) \(w\_2\) \[CenterEllipsis]w\_n\)]], "\[Element] ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " is such that \[Delta](\[CenterEllipsis]\[Delta](\[Delta](", Cell[BoxData[ \(TraditionalForm\`q\_0\)]], ",", Cell[BoxData[ \(TraditionalForm\`w\_1\)]], "),", Cell[BoxData[ \(TraditionalForm\`w\_2\)]], ")\[CenterEllipsis],", Cell[BoxData[ \(TraditionalForm\`w\_n\)]], ")\[Element]F, we say that M ", StyleBox["accepts", FontSlant->"Italic"], " or ", StyleBox["recognizes", FontSlant->"Italic"], " w. FAs are used to accept ", StyleBox["regular languages", FontSlant->"Italic"], ", that is, an automaton will accept ", StyleBox["only", FontVariations->{"Underline"->True}], " those words belonging to a particular regular language. \n", "Consider, for example, the FA accepting the language ", Cell[BoxData[ \(TraditionalForm\`\(\((a + b)\)\^*\)\)]], StyleBox["aba. ", FontSlant->"Italic"], "This automaton can be designed using four states.", StyleBox[" ", FontSlant->"Italic"], "In ", StyleBox["Mathematica", FontSlant->"Italic"], ", we can directly input ", Cell[BoxData[ \(TraditionalForm\`\(\(q\_0, \)\ \)\)]], "\[Delta] and F as:" }], "Text"], Cell[BoxData[ RowBox[{ \(q\_0 = 1\), ";", "\n", "\n", \(\[Delta][1, "\"] = 2\), ";", " ", \(\[Delta][1, "\"] = 1\), ";", " ", "\n", \(\[Delta][2, "\"] = 2\), ";", " ", \(\[Delta][2, "\"] = 3\), ";", " ", "\n", \(\[Delta][3, "\"] = 4\), ";", " ", \(\[Delta][3, "\"] = 1\), ";", " ", "\n", \(\[Delta][4, "\"] = 2\), ";", " ", \(\[Delta][4, "\"] = 3\), ";", " ", StyleBox["\n", FontSlant->"Italic"], "\n", \(F = {4}\), ";"}]], "Input", CellLabel->"In[1]:="], Cell[TextData[{ "We can test whether a given word is recognized by this automaton through \ function ", StyleBox["acceptedQ", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ \(acceptedQ[w_String] := MemberQ[F, Fold[\[Delta], q\_0, Characters[w]]]\)], "Input", CellLabel->"In[2]:="], Cell[CellGroupData[{ Cell[BoxData[ \(acceptedQ["\"]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[3]="] }, Closed]], Cell["\<\ We can also obtain the list of states traversed by the previous word in \ traversing the path to being accepted:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FoldList[\[Delta], q\_0, Characters["\"]]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({1, 2, 2, 3, 4, 3, 4, 3, 4}\)], "Output", CellLabel->"Out[4]="] }, Closed]], Cell[TextData[{ "However, this notation, although correct, makes the input of automata \ awkward and will not be followed in this work. \n", "Alternatively, we can describe FA by a list of lists. Consider, for \ example, the automaton ", StyleBox["evenEven", FontSlant->"Italic"], " accepting only those words in ", Cell[BoxData[ \(TraditionalForm\`\(\({0, 1}\^*\)\ \)\)]], " which have an even number of a\.b4s and an even number of b\.b4s defined \ by :" }], "Text"], Cell[BoxData[ RowBox[{ \(Q = {1, 2, 3, 4}\), ";", "\n", \(\[CapitalSigma] = {a, b}\), ";", "\n", \(\[Delta] = {{{1, a}, 4}, {{1, b}, 2}, {{2, a}, 3}, {{2, b}, 1}, {{3, a}, 2}, {{3, b}, 4}, {{4, a}, 1}, {{4, b}, 3}}\), ";", "\n", RowBox[{ FormBox[\(q\_0\), "TraditionalForm"], "=", "1"}], ";", "\n", \(F = {1}\), ";"}]], "Input", CellLabel->"In[5]:="], Cell["which we can encapsulate as:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({Q, \[CapitalSigma], \[Delta], q\_0, F}\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \({{1, 2, 3, 4}, {a, b}, {{{1, a}, 4}, {{1, b}, 2}, {{2, a}, 3}, {{2, b}, 1}, {{3, a}, 2}, {{3, b}, 4}, {{4, a}, 1}, {{4, b}, 3}}, 1, {1}}\)], "Output", CellLabel->"Out[6]="] }, Closed]], Cell[TextData[{ "This notation is still cumbersome. Let us adopt the following assumptions \ in order to expedite the description of FA: \n\n\[FilledSmallCircle] The set \ Q will always comprise consecutive numbers from 1 (assumed to be the initial \ state) to the number of states. This number can always be obtained from the \ entries in \[Delta] so there will be no need to explicitly provide Q. \n\ \[FilledSmallCircle] The previous assumption will also be applied to the \ alphabet \[CapitalSigma], which will enjoy the same fate as Q with the \ proviso that when we require to describe transitions (for instance in \ obtaining the regular expression corresponding to the accepted language) we \ will assume \[CapitalSigma] comprises all lower-case letters in lexicographic \ order from \"a\" up to the number of columns of vector \[Delta]. \n\ \[FilledSmallCircle] \[Delta] will be described by a matrix of dimension \ (number of states)\[Times](alphabet size) wherein \ \[Delta]\[LeftDoubleBracket] i, j \[RightDoubleBracket] will correspond to \ the state to which the automaton goes from state i under transition j.\n\n\ Hence the previous automaton ", StyleBox["evenEven", FontSlant->"Italic"], " is shortly described as:" }], "Text"], Cell[BoxData[ \(\(evenEven = {{{4, 2}, {3, 1}, {2, 4}, {1, 3}}, {1}}; \)\)], "Input", CellLabel->"In[7]:=", InitializationCell->True], Cell[TextData[ "Finally, as Length[First[evenEven]]=4 we know that |Q|=4 and, as \ Length[First[First[evenEven]]]=2, we know that \[Sum]={a,b}. In other words, \ for any automaton M, Dimensions[First[M]]={|Q|,|\[CapitalSigma]|}."], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["2 TEST AUTOMATA", "Section"], Cell[TextData[{ "Along with the automaton ", StyleBox["evenEven", FontSlant->"Italic"], " introduced before, the following automata will be used as test automata \ for the rest of our work. An alternative to the linear display of an \ automaton, obtained by just writing its name, is that provided by function ", StyleBox["showFA", FontSlant->"Italic"], " below." }], "Text"], Cell[BoxData[ \(showFA[{\[Delta]_, F_}] := MatrixForm\ /@\ {\[Delta], F}\)], "Input", CellLabel->"In[8]:=", InitializationCell->True], Cell["for instance:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(showFA[evenEven]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"(", GridBox[{ {"4", "2"}, {"3", "1"}, {"2", "4"}, {"1", "3"} }], ")"}], (MatrixForm[ #]&)], ",", InterpretationBox[\((1)\), MatrixForm[ {1}]]}], "}"}]], "Output", CellLabel->"Out[9]="] }, Closed]], Cell[CellGroupData[{ Cell[TextData["ON \[Sum]={a,b}:"], "Subsubsection"], Cell[TextData[{ "\[FilledSmallCircle]", StyleBox[" ", "Text"], "automaton ", StyleBox["aa", "Text", FontFamily->"Courier New", FontSlant->"Italic"], StyleBox[" recognizing the language ", "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\((a + b)\)\^*\) \(\(aa(a + b)\)\^*\)\ \)\)]], " of a", StyleBox["ll words having a double a somewhere is described as:", "Text"] }], "Text"], Cell[BoxData[ \(\(aa = {{{2, 1}, {3, 1}, {3, 3}}, {3}}; \)\)], "Input", CellLabel->"In[10]:=", InitializationCell->True], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["secondB", FontFamily->"Courier New", FontSlant->"Italic"], " accepting the language ", StyleBox["(ab+bb)", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\((a + b)\)\^*\)\)]], " of all words whose second character is b is:" }], "Text"], Cell[BoxData[ \(\(secondB = {{{2, 2}, {4, 3}, {3, 3}, {4, 4}}, {3}}; \)\)], "Input", CellLabel->"In[11]:=", InitializationCell->True], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["oddA", FontFamily->"Courier New", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Courier New"], "accepting those words having an odd number of a\.b4s:" }], "Text"], Cell[BoxData[ \(\(oddA = {{{2, 1}, {1, 2}}, {2}}; \)\)], "Input", CellLabel->"In[12]:=", InitializationCell->True], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["dif", FontFamily->"Courier New", FontSlant->"Italic"], " accepting the language", StyleBox[" ", FontSlant->"Italic"], "of those words which have no two same letters adjacent (Note this is not \ the language ", Cell[BoxData[ \(TraditionalForm\`\(\((ab)\)\^*\)\)], FontSlant->"Italic"], StyleBox["+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\((ba)\)\^*\)\)], FontSlant->"Italic"], "):" }], "Text"], Cell[BoxData[ \(\(dif = {{{2, 3}, {4, 3}, {2, 4}, {4, 4}}, {1, 2, 3}}; \)\)], "Input", CellLabel->"In[13]:=", InitializationCell->True], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["mod5", FontSlant->"Italic"], " accepting those words that, interpreted as the binary equivalent of a \ decimal number, are equal to 1 mod 5. For instance, as baabaa=", Cell[BoxData[ \(TraditionalForm\`100100\_2\)]], "=", Cell[BoxData[ \(TraditionalForm\`36\_10\)]], "\[Congruent]1 mod 5, the automaton recognises the word baabaa." }], "Text"], Cell[BoxData[ \(\(mod5 = {{{1, 2}, {3, 4}, {5, 1}, {2, 3}, {4, 5}}, {2}}; \)\)], "Input",\ CellLabel->"In[14]:=", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["ON \[Sum]={a,b,c}:"], "Subsubsection"], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["difABC", FontFamily->"Courier New", FontSlant->"Italic"], " accepting those words on \[CapitalSigma]={a,b,c} which have no two same \ letters adjacent:" }], "Text"], Cell[BoxData[ \(\(difABC = {{{2, 3, 4}, {5, 3, 4}, {2, 5, 4}, {2, 3, 5}, {5, 5, 5}}, {1, 2, 3, 4}}; \)\)], "Input", CellLabel->"In[15]:=", InitializationCell->True], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["evenEvenABC", FontSlant->"Italic"], " accepting all words on \[CapitalSigma]={a,b,c} having an even number of \ a\.b4s and an even number of b\.b4s:" }], "Text"], Cell[BoxData[ \(\(evenEvenABC = {{{4, 2, 1}, {3, 1, 2}, {2, 4, 3}, {1, 3, 4}}, {1}}; \)\)], "Input", CellLabel->"In[16]:=", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["ON \[Sum]={a,b,c, ...}:"], "Subsubsection"], Cell[TextData[{ "\[FilledSmallCircle] automaton ", StyleBox["modMxN", FontSlant->"Italic"], " accepting those words that, interpreted as the base M equivalent of a \ decimal number, are equal to x mod N. For instance, as cbad=", Cell[BoxData[ \(TraditionalForm\`2103\_4\)]], "=", Cell[BoxData[ \(TraditionalForm\`147\_10\)]], "\[Congruent]3 mod 6, automaton ", StyleBox["modMxN[4,3,6]", FontFamily->"Courier New"], " must accept word cbad. Automaton ", StyleBox["modMxN[2,1,5]", FontSlant->"Italic"], " is equal to automaton ", StyleBox["mod5.", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ \(modMxN[m_, x_, n_] := { Partition[Flatten[Table[Range[n], {m}]], m], {1 + x}}\)], "Input", CellLabel->"In[17]:=", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(showFA[modMxN[2, 1, 5]]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"}, {"5", "1"}, {"2", "3"}, {"4", "5"} }], ")"}], (MatrixForm[ #]&)], ",", InterpretationBox[\((2)\), MatrixForm[ {2}]]}], "}"}]], "Output", CellLabel->"Out[18]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(showFA[modMxN[3, 1, 4]]\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"(", GridBox[{ {"1", "2", "3"}, {"4", "1", "2"}, {"3", "4", "1"}, {"2", "3", "4"} }], ")"}], (MatrixForm[ #]&)], ",", InterpretationBox[\((2)\), MatrixForm[ {2}]]}], "}"}]], "Output", CellLabel->"Out[19]="] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["3 GENERATING WORDS OF LENGTH n ACCEPTED BY A GIVEN AUTOMATON", "Section"], Cell[TextData[{ "Given an FA, we want to obtain a list comprising all words of length n \ that it accepts. By polymorphism, we subordinate function ", StyleBox["acceptedWords", FontFamily->"Courier New", FontSlant->"Italic"], " below to one computing the words of length n accepted by M considering a \ given starting state q\[NotEqual]1. The result is generated in lexicographic \ order." }], "Text"], Cell[BoxData[{ \(acceptedWords[M_, n_] := acceptedWords[M, 1, n]\n\), \(acceptedWords[{_, F_}, q_, 0] := If[MemberQ[F, q], {"\<\>"}, {}]\n\), \(acceptedWords[M : {\[Delta]_, _}, q_, n_] := \(acceptedWords[M, q, n] = Module[{w, i, ans = {}}, \n\t\t\t\t\t w = Map[acceptedWords[M, #, n - 1]&, \[Delta]\[LeftDoubleBracket]q\[RightDoubleBracket]]; \n \t\t\t\t\t Do[\n\t\t\t\t\t\t\t\t If[Length[w\[LeftDoubleBracket]i\[RightDoubleBracket]] > 0, AppendTo[ans, Map[StringJoin[ FromCharacterCode[ToCharacterCode["\"] - 1 + i], #]& \ , w\[LeftDoubleBracket]i\[RightDoubleBracket]]]]\n \t\t\t\t\t\t, {i, Length[w]}]; \n\t\tFlatten[ans]]\)\)}], "Input",\ CellLabel->"In[20]:=", InitializationCell->True], Cell["Applying this function to our test automata:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(TableForm[Table[acceptedWords[evenEven, n], {n, 0, 4}]]\), \(Table[Length[acceptedWords[evenEven, n]], {n, 0, 10}]\)}], "Input", CellLabel->"In[23]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("aa"\), \("bb"\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("aaaa"\), \("aabb"\), \("abab"\), \("abba"\), \("baab"\), \("baba"\), \("bbaa"\), \("bbbb"\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{""}, {}, {"aa", "bb"}, {}, {"aaaa", "aabb", "abab", "abba", "baab", "baba", "bbaa", "bbbb"}}]]], "Output", CellLabel->"Out[23]//TableForm="], Cell[BoxData[ \({1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512}\)], "Output", CellLabel->"Out[24]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[{ \(TableForm[Table[acceptedWords[aa, n], {n, 0, 4}]]\), \(Table[Length[acceptedWords[aa, n]], {n, 0, 10}]\)}], "Input", CellLabel->"In[25]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("aa"\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("aaa"\), \("aab"\), \("baa"\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("aaaa"\), \("aaab"\), \("aaba"\), \("aabb"\), \("abaa"\), \("baaa"\), \("baab"\), \("bbaa"\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{}, {}, {"aa"}, {"aaa", "aab", "baa"}, {"aaaa", "aaab", "aaba", "aabb", "abaa", "baaa", "baab", "bbaa"}}]]], "Output", CellLabel->"Out[25]//TableForm="], Cell[BoxData[ \({0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880}\)], "Output", CellLabel->"Out[26]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[{ \(TableForm[Table[acceptedWords[secondB, n], {n, 0, 4}]]\), \(Table[Length[acceptedWords[secondB, n]], {n, 0, 10}]\)}], "Input", CellLabel->"In[27]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("ab"\), \("bb"\), \(""\), \(""\), \(""\), \(""\), \(""\), \(""\)}, {\("aba"\), \("abb"\), \("bba"\), \("bbb"\), \(""\), \(""\), \(""\), \(""\)}, {\("abaa"\), \("abab"\), \("abba"\), \("abbb"\), \("bbaa"\), \("bbab"\), \("bbba"\), \("bbbb"\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{}, {}, {"ab", "bb"}, {"aba", "abb", "bba", "bbb"}, {"abaa", "abab", "abba", "abbb", "bbaa", "bbab", "bbba", "bbbb"}}]]], "Output", CellLabel->"Out[27]//TableForm="], Cell[BoxData[ \({0, 0, 2, 4, 8, 16, 32, 64, 128, 256, 512}\)], "Output", CellLabel->"Out[28]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[{ \(TableForm[Table[acceptedWords[oddA, n], {n, 0, 4}]]\), \(Table[Length[acceptedWords[oddA, n]], {n, 0, 10}]\)}], "Input", CellLabel->"In[29]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {\(""\), \(""\), 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Possibly \ yes; transformation {a\[Rule]c, c\[Rule]a} is a close one."], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["4 PLOTTING ALL WORDS OF LENGTH n", "Section"], Cell["\<\ In order to display the complexity of the words of length n accepted by a \ given FA, we produce a density plot assigning different gray levels to each \ of the elements of the alphabet, starting with white for character \"a\" and \ ending with black for the last character used. 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WORDS OF LENGTH n ACCEPTED BY A GIVEN AUTOMATON\ \>", "Section"], Cell[TextData[{ "Let us find an expression for the number of words of length n accepted by \ an FA. To this end we design function ", StyleBox["arden", FontFamily->"Courier New", FontSlant->"Italic"], ". Note that a system of recursive equations is set up so that its solution \ is the answer. " }], "Text"], Cell[BoxData[ \(<< "\"\)], "Input", CellLabel->"In[51]:=", InitializationCell->True], Cell[BoxData[ \(arden[{\[Delta]_, F_}] := Module[{nStates, nSymbols, v = Flatten[ Range[ToCharacterCode["\"], ToCharacterCode["\"]]], w, eq, ans = {}, u, i, j}, \n \t\t\ \ {nStates, nSymbols} = Take[Dimensions[\[Delta]], 2]; \n \t\t\ \ v = FromCharacterCode/@\ Take[v, nStates]; \n\t\t\ \ v = \(("\"<>#&)\)/@v; \n\t\t\ \ Do[eq = v\[LeftDoubleBracket]i\[RightDoubleBracket]<>"\<[n]==\>"; \n \t\t\t\tDo[ eq = eq<> \((v\[LeftDoubleBracket]\[Delta]\[LeftDoubleBracket]i, j \[RightDoubleBracket]\[RightDoubleBracket]<> "\<[n-1]+\>")\), {j, nSymbols}]; \n\t\t\t\t If[StringTake[eq, \(-1\)] == "\<+\>", eq = StringDrop[eq, \(-1\)]]; \n\t\t\t\tAppendTo[ans, eq]; \n\t\t\t\t AppendTo[ans, v\[LeftDoubleBracket]i\[RightDoubleBracket]<>"\<[0]==\>"<> If[MemberQ[F, i], "\<1\>", "\<0\>"]], {i, nStates}]; \n\n\t\t w = ToExpression[\((#<>"\<[n]\>"&)\)\ /@\ v]; \n\t\t u = RSolve[ToExpression[ans], w, n]; \ \n\t\ \ FullSimplify[ Expand[\(Simplify[w /. \ u]\)\[LeftDoubleBracket]1, 1 \[RightDoubleBracket]]]]\)], "Input", CellLabel->"In[52]:=", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(arden[{{{1, 1}, {2, 2}}, {2}}]\)], "Input", CellLabel->"In[53]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[53]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[{{{1, 1}}, {}}]\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[54]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[{{{2, 2}, {2, 2}}, {1}}]\)], "Input", CellLabel->"In[55]:="], Cell[BoxData[ \(If[n == 0, 1, 0]\)], "Output", CellLabel->"Out[55]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[{{{1, 1}}, {1}}]\)], "Input", CellLabel->"In[56]:="], Cell[BoxData[ \(2\^n\)], "Output", CellLabel->"Out[56]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[{ \(arden[evenEven]\), \(Map[Table[#\ /. \ {n -> i}, \ {i, 0, 10}]&, %]\)}], "Input", CellLabel->"In[57]:="], Cell[BoxData[ \(1\/4\ \((\((\(-2\))\)\^n + 2\^n + 2\ If[n == 0, 1, 0])\)\)], "Output", CellLabel->"Out[57]="], Cell[BoxData[ \({1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512}\)], "Output", CellLabel->"Out[58]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[aa]\)], "Input", CellLabel->"In[59]:="], Cell[BoxData[ \(1\/5\ 2\^\(\(-1\) - n\)\ \((5\ 2\^\(1 + 2\ n\) - 5\ \((1 - \@5)\)\^n + 3\ \@5\ \((1 - \@5)\)\^n - \((1 + \@5)\)\^n\ \((5 + 3\ \@5)\))\)\)], "Output", CellLabel->"Out[59]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[secondB]\)], "Input", CellLabel->"In[60]:="], Cell[BoxData[ \(1\/2\ \((2\^n - If[n == 0, 1, 0] - 2\ If[n == 1, 1, 0])\)\)], "Output", CellLabel->"Out[60]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[oddA]\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ \(1\/2\ \((2\^n - If[n == 0, 1, 0])\)\)], "Output", CellLabel->"Out[61]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[dif]\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ \(2 - If[n == 0, 1, 0]\)], "Output", CellLabel->"Out[62]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[mod5]\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ \(1\/20\ \((5 - 5\ \((\(-1\))\)\^n + 2\^\(2 + n\) - 4\ Cos[\(n\ \[Pi]\)\/2] + 2\ Sin[\(n\ \[Pi]\)\/2])\)\)], "Output", CellLabel->"Out[63]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[difABC]\)], "Input", CellLabel->"In[64]:="], Cell[BoxData[ \(1\/2\ \((3\ 2\^n - If[n == 0, 1, 0])\)\)], "Output", CellLabel->"Out[64]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(arden[evenEvenABC]\)], "Input", CellLabel->"In[65]:="], Cell[BoxData[ \(1\/4\ \((2 + \((\(-1\))\)\^n + 3\^n)\)\)], "Output", CellLabel->"Out[65]="] }, Closed]], Cell["\<\ How many numbers of size n multiple of four are there in base 4?. Answer:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(arden[modMxN[4, 0, 4]]\)], "Input", CellLabel->"In[66]:="], Cell[BoxData[ \(1\/4\ \((4\^n + 3\ If[n == 0, 1, 0])\)\)], "Output", CellLabel->"Out[66]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["6 DRAWING AUTOMATA", "Section"], Cell[TextData[{ "There is no general canonical way of drawing the graph corresponding to \ FA. In this section we propose one that is useful for teaching purposes and \ which consists of the following features:\n\n\[FilledSmallCircle] the states \ are represented by circles labeled accordingly,\n\[FilledSmallCircle] final \ states are indicated in double circle,\n\[FilledSmallCircle] the initial \ state is colored in yellow and the other states in green,\n\ \[FilledSmallCircle] transitions are Bezier curves,\n\[FilledSmallCircle] \ transitions are ", StyleBox["not", FontVariations->{"Underline"->True}], " directed. We feel that arrowed transitions, although an essential \ feature, clutter the graph unnecessarily; our graph, while experimenting, \ gives a rough idea on the topology of the automata in hand, which after \ printing or exporting to standard software, can be provided of appropriate \ arrowheads by the user.\n\nFunction ", StyleBox["transitions", FontSlant->"Italic"], " below constructs the set of bezier curves joining two given points ", StyleBox["p0", FontSlant->"Italic"], " and ", StyleBox["p2", FontSlant->"Italic"], ", (possibly equal, in which case the curves will be circles of radii a \ multiple of a given parameter ", StyleBox["r", FontSlant->"Italic"], "). There will be ", StyleBox["numTrans", FontSlant->"Italic"], " curves drawn through ", StyleBox["numPoints", FontSlant->"Italic"], " segments. Two examples follow the definition." }], "Text"], Cell[BoxData[{ \(transitions[p0_, p0_, numTrans_, _, r_] := Module[{i}, \n\t\t Table[Circle[p0 \((1 + \(i\ r\)\/\(2\ \@\(p0.p0\)\))\), \(i\ r\)\/2], {i, numTrans + 1}]]\n\), \(transitions[p0_, p2_, numTrans_, numPoints_, _] := Module[{i, q, a, b, p1, t, c}, \n\t\t i = Range[1 - numTrans, \ numTrans - 1, 2]; \n\t\t{a, b} = p2 - p0; \n \t\t\tq = {\(-b\), a}; \n\t\t\t p1 = Map[\((\(p0 + p2\)\/2 + \(\ q\)\/\(3\ numTrans\)\ #)\)&, i]; \n \t\tc = Map[ \((\((1 - 2 t)\) \((1 - t)\) p0 + 4\ t\ \((1 - t)\)\ #\ + \ t \((2 t - 1)\)\ p2)\)&, p1]; \n\t\t Map[Line[Table[#, {t, 0, 1, 1/numPoints}]]&, c]]\)}], "Input", CellLabel->"In[67]:=", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[Graphics[transitions[{0, 0}, {5, 3}, 7, 20, 1]], AspectRatio \[Rule] Automatic]; \)\)], "Input", CellLabel->"In[69]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .72857 MathPictureStart /Mabs { Mgmatrix 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generates the graph of a given automata. Examples follow." }], "Text"], Cell[BoxData[ \(drawFA[{\[Delta]_, F_}] := Module[{n, alphSize, \[Theta], i, j, centers, p, r, connec, arcs = {}}, \n\t\t\t\t\n\t\t{n, alphSize} = Dimensions[\[Delta]]; \n\t\t connec = Table[0, {n}, {n}]; \n\t\t Do[\n\t\t\t \(connec\[LeftDoubleBracket]i, \[Delta]\[LeftDoubleBracket]i, j \[RightDoubleBracket]\[RightDoubleBracket]++\), {i, n}, {j, alphSize}]; \n\t\t Do[\n\t\t\t connec\[LeftDoubleBracket]i, j\[RightDoubleBracket] += connec\[LeftDoubleBracket]j, i\[RightDoubleBracket], {i, n - 1}, { j, i + 1, n}]; \n\t\t Do[\n\t\t\t If[connec\[LeftDoubleBracket]i, j\[RightDoubleBracket] > 0, AppendTo[ arcs, {i, j, connec\[LeftDoubleBracket]i, j\[RightDoubleBracket]}]], {i, n}, {j, i, n}]; \n\t\t\n\t\t\[Theta] = N[\(2 \[Pi]\)\/n]; \n\t\t r = 2\ Sin[\[Theta]\/8]; \n\t\t centers = Table[{Cos[i\ \[Theta]], Sin[i\ \[Theta]]}, {i, 0, n - 1}]; \n\t\t\t\t\n\t\t p = Map[transitions[ centers\[LeftDoubleBracket]#\[LeftDoubleBracket]1 \[RightDoubleBracket]\[RightDoubleBracket], centers\[LeftDoubleBracket]#\[LeftDoubleBracket]2 \[RightDoubleBracket]\[RightDoubleBracket], # \[LeftDoubleBracket]3\[RightDoubleBracket], 15, r]&, arcs]; \n\t\t\n\t\t Map[AppendTo[p, {RGBColor[0, 0.5, 0.5], Disk[#, r]}]&, Rest[centers]]; \n\t\t Map[AppendTo[p, Circle[#, r]]&, Rest[centers]]; \n\n\t\t AppendTo[ p, {RGBColor[1, 1, 0.5], Disk[centers\[LeftDoubleBracket]1\[RightDoubleBracket], r]}]; \n \t\tAppendTo[p, Circle[centers\[LeftDoubleBracket]1\[RightDoubleBracket], r]]; \n \t\t\n\t\t Map[\n\t\t\t AppendTo[p, Circle[centers\[LeftDoubleBracket]#\[RightDoubleBracket], \(2\ r\)\/3]]&, F]; \n\t\t\n\t\t Do[\nAppendTo[p, Text[FontForm[ToString[i], {"\", N[50\ r]}], centers\[LeftDoubleBracket]i\[RightDoubleBracket]]], {i, Length[centers]}]; \n\t\t\n\t\t\t Show[Graphics[p], AspectRatio \[Rule] Automatic, PlotRange \[Rule] All]; ]\)], "Input", CellLabel->"In[71]:=", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ 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