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Sc. and Ph. D. degrees in Applied \ Mathematics and Computation from the University College of North Wales in \ Bangor, Great Britain. He has held research positions in the School of \ Informatics at UCNW Bangor, U.K., the College of Mexico, the Center of \ Research and Advanced Studies, the Monterrey Institute of Technology and the \ University of Queretaro in Mexico, where he currently works at the Faculty of \ Informatics. Prolific contributor to ", StyleBox["MathSource", FontSlant->"Italic"], ", his current research include Combinatorics, Theory of Computing, \ Functional Languages, Computational Geometry and Recreational Mathematics." }], "Abstract"] }, Open ]], Cell[CellGroupData[{ Cell["Acknowledgements", "Section"], Cell["\<\ This work was finished at Wolfram Research Inc., in Champaign IL, USA, while \ the author was spending two months as a Visiting Scholar. Their assistance \ and enthusiastic support is gratefully acknowledged. My thanks are also given \ for the continuous support received from the University of Queretaro in \ Mexico.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["References", FontWeight->"Bold"]], "Section"], Cell[TextData[{ "Cellular Automata and Complexity ", StyleBox["\nS. Wolfram\nPerseus, 1994\n\nThe Algorithmic Beauty of Plants", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox["\nA. Lindenmeyer\t \nSpringer-Verlag, NY, 1990\n\nPower \ Programming with ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], ". The Kernel\nD. B. Wagner\nMcGraw-Hill, 1996\n\nTerm rewriting and \ Programming Paradigms\nR. E. Maeder\nin Mathematics with Vision\nProceedings \ of the First International Mathematica Symposium\nComputational Mechanics \ Publications 1995, pp. 7-19\n\nProgramming in Mathematica\nR. E. Maeder\n\ Addison-Wesley, second edition, 1991\n\nSelf-similar Mosaics\nN. Dolbilin\n\ Quantum, July/August 2000, pp.4-9\n" }], "Reference"] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "SectionFirst"], Cell[TextData[{ StyleBox["Rewriting systems (RS), also called ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Reduction Systems, String Rewriting Systems", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" or ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Term Rewriting Systems", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", have been widely used in contexts in which formulae \ manipulation plays a prominent role, such as symbolic and functional \ programming, formal grammars (useful in the design of compilers and \ interpreters), computer graphics, simulation, etc. This is due to their \ intrinsic ability to specify, through employing a set of ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["directed transformational equations", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", the basic mechanism of subterm substitution. Repeated \ applications of this mechanism on a given term are considered as leading to a \ \[OpenCurlyDoubleQuote]simpler form\[CloseCurlyDoubleQuote] of this term. \ However, it is possible that we might never arrive at this simple form - \ called ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["normal", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["form", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", due to either divergence or cycling of the terms generated.\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Courier New"], StyleBox["RS are determined by a series of productions meant to be applied \ to specific objects. This application consists in the substitution of a \ pattern occurring on the object also appearing on the left side of a \ production, by the corresponding pattern on the right side of this \ production. Consider, for example the following simple RS:\n", FormatType->StandardForm, FontFamily->"Times New Roman"], "\n\t\t21 \[Rule] 12\n\t\t31 \[Rule] 13\n\t\t32 \[Rule] 23\n\n", StyleBox["Throughout its repeated application on a given word w \[Element] \ {1, 2, 3}", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["*", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" (that is, the objects we are considering consist on all strings \ of symbols taken from the alphabet {1, 2, 3}), we obtain a sorted anagram of \ w. It does not matter in which order we apply the rules, the convergence to \ the normal form is inevitable and it leads to a word of the form 1*2*3*. Such \ systems are called ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["confluent", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" or ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Church-Rosser", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[". This paradigm is also present in the study of cellular \ automata, wherein the notion of transitions over discrete state spaces is \ characterized by this approach.\n", FormatType->StandardForm, FontFamily->"Times New Roman"], "\n", StyleBox["The aim of this work is to illustrate the application of RS on a \ variety of important settings throughout the following sections: \ One-dimensional Checkers, Group Theory, Equivalence Relations, Systems of \ Numeric Productions, Recursive Productions, L systems, ", FormatType->StandardForm, FontFamily->"Times New Roman"], "From Binary Numbers to Compositions and Huffman Coding,", StyleBox[" providing a smorgasbord of ideas on this fascinating topic. A \ series of animations supplementing several parts of the text is also \ provided. We are using ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Mathematica", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" version 5", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Abstract", TextAlignment->Left, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["1.- One-dimensional Checkers", "Section", FontColor->RGBColor[1, 0.889998, 0.00999466], Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell["\<\ Not all RS are confluent. Consider, for example, the following:\ \>", "Text"], Cell["\<\ 110 \[Rule] 001 011 \[Rule] 100\ \>", "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["which corresponds to the jumps made in the \ one-dimensional version of the game of checkers. If we consider as the normal \ form of a word that one containing only one occurrence of the symbol 1, \ disregarding the number of symbols 0 it contains, then not all words lead to \ normal forms, and even those that do might not reach it due to the \ non-deterministic character of the rewriting rules. Termination is, in \ general an undecidable property of RS. For example, the word 0110011110 \ possess a normal form but applying the first of the above rules to obtain \ 0110011001 leads it away from reaching it. \n\nLet us obtain all \"valid\" \ words, that is, those words of a given size n having normal form and hence \ leading to a successful solitaire game of one-dimensional checkers. Valid \ words are of the form 1x1 where x \[Element] {0, 1}* is a word of length n-2. \ We can readily obtain the set of all valid words from an initial list of \ candidates.", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(candidates[n_] := Map[Join[{1}, IntegerDigits[#, 2, n - 2], {1}] &, Range[0, 2\^\(n - 2\) - 1]]\)], "Input", CellLabel->"In[1]:="], Cell[CellGroupData[{ Cell[BoxData[ \(candidates[5]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \({{1, 0, 0, 0, 1}, {1, 0, 0, 1, 1}, {1, 0, 1, 0, 1}, {1, 0, 1, 1, 1}, {1, 1, 0, 0, 1}, {1, 1, 0, 1, 1}, {1, 1, 1, 0, 1}, {1, 1, 1, 1, 1}}\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell[TextData[StyleBox["And the rules can be coded as:", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(jump := {{X___, 1, 1, 0, Y__} \[Rule] {X, 0, 0, 1, Y}, {X__, 0, 1, 1, Y___} \[Rule] {X, 1, 0, 0, Y}}\)], "Input", CellLabel->"In[3]:=", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "The non-determinism in applying this rule is captured by ", StyleBox["Mathematica", FontSlant->"Italic"], "'s function ", StyleBox["ReplaceList", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ReplaceList[{1, 1, 1, 0, 1, 1, 1, 1}, jump]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({{1, 0, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 0, 0, 1, 1}}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ "In jumping at the end positions, we have to eliminate any leading and \ trailing zeroes that might appear. We accomplish this by making use of \ function ", StyleBox["compress", "Input"], " and changing the jumping transformations accordingly." }], "Text"], Cell[BoxData[{ \(compress[{0, x__}] := compress[{x}]\), "\[IndentingNewLine]", \(compress[{x__, 0}] := compress[{x}]\), "\[IndentingNewLine]", \(\(\(compress[x_] := x\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(jump := {{X___, 1, 1, 0, Y__} \[RuleDelayed] compress[{X, 0, 0, 1, Y}], {X__, 0, 1, 1, Y___} \[RuleDelayed] compress[{X, 1, 0, 0, Y}], \[IndentingNewLine]{1, 1} \[RuleDelayed] \ {1, 1}\ }\)}], "Input", CellLabel->"In[5]:="], Cell[TextData[{ "We are interested in succesively applying ", StyleBox["junp", "Input"], " to get to {1, 1} (or equivalently to 1; we choose this criteria so that \ we always have an ending pair of 1's). By using ", StyleBox["FixedPointList", "Input"], " we obtain all successive positions from a given starting one." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FixedPointList[ Flatten[\(ReplaceList[#, jump] &\)\ /@ \ #, 1] &, {{1, 1, 1, 0, 1, 1, 1, 1}}]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \({{{1, 1, 1, 0, 1, 1, 1, 1}}, {{1, 0, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 0, 0, 1, 1}}, {{1, 0, 1, 0, 0, 1, 1, 1}, {1, 1, 0, 0, 1, 0, 1, 1}, {1, 1, 1, 1, 0, 1}}, {{1, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 1}, {1, 1, 0, 0, 1, 1}, {1, 1, 0, 0, 1, 1}}, {{1, 0, 1, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}}, {{1, 1}, {1, 1}, {1, 1}, {1, 1}, {1, 1}}, {{1, 1}, {1, 1}, {1, 1}, {1, 1}, {1, 1}}}\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell["\<\ This means that there are five ways to succeed, and correspond to the \ building of the game tree\ \>", "Text"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgWoo0`002Goo0`00CWoo0`002Goo0`00=Goo002cOol30009Ool3 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1@001Woo10000Woo100017oo1@001goo1@001goo1@001goo1@00ogooXgoo 0016Ool40008Ool40008Ool40006Ool:0006Ool30009Ool30009Ool30009 Ool3003oOonSOol004Qoo`8000Yoo`8000Yoo`8000Moo`L000Uoo`8000Yo o`8000Yoo`8000Yoo`800?mooj=oo`00B7oo0P002Woo0P002Woo0P002Goo 10002Woo0P002Woo0P002Woo0P002Woo0P00ogooXgoo003oOoooOom3Ool0 0?mooomood=oo`00ogooogoo@goo003oOoooOom3Ool00?mooomood=oo`00 ogooogoo@goo003oOoooOom3Ool00?mooomood=oo`00ogooogoo@goo003o OoooOom3Ool00?mooomood=oo`00ogooogoo@goo003oOoooOom3Ool00?mo oomood=oo`00\ \>"], "PlacedGraphics", ImageSize->{332, 229}, ImageMargins->{{0.75, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Let us encapsulate all the above in predicate ", StyleBox["validQ", "Input"], " which decides whether a given word leads to a winning game." }], "Text"], Cell[BoxData[ \(validQ[ word_List] := \((FixedPoint[ Union[Flatten[\(ReplaceList[#, jump] &\)\ /@ \ #, 1]] &, {word}] == {{1, 1}})\)\)], "Input", CellLabel->"In[10]:="], Cell[TextData[{ "Note we take the ", StyleBox["union", "InlineInput"], " to expedite the decision, and in doing so we graft several branches of \ the game tree. So, for instance, the valid words of size 8 are obtained as:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FromDigits\ /@ \ Select[candidates[8], validQ]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \({10101011, 10111011, 10111111, 11001011, 11001111, 11010011, 11010101, 11011101, 11101111, 11110011, 11110111, 11111101}\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["The sequence of valid words begins as:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sort[ Flatten[Table[ FromDigits\ /@ \ Select[candidates[n], validQ], {n, 10}]]]\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \({11, 1011, 1101, 101011, 101111, 110011, 110101, 111101, 10101011, 10111011, 10111111, 11001011, 11001111, 11010011, 11010101, 11011101, 11101111, 11110011, 11110111, 11111101, 1010101011, 1011101011, 1011111011, 1011111111, 1100101011, 1100111011, 1100111111, 1101001011, 1101001111, 1101010011, 1101010101, 1101011101, 1101101111, 1101110011, 1101110111, 1101111101, 1110111011, 1111001011, 1111011011, 1111011111, 1111101111, 1111110011, 1111111101}\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell["(to which we have to include 1 of course)", "Text"], Cell["\<\ Note that there cannot be three consecutive 0's in any of the above words. \ Similarly, a valid word has to have even length. The reason for this is the \ following. By looking at the reverse process whereby we start at 11 and \ finish at a valid word we observe two kind of moves. Those that extend the \ word sideways - increasing its length by 2 adding a 0 and a 1, and those that \ only rearrange the internal configuration by eliminating a 0 and adding a 1 \ without changing the length of the word. Hence any valid word must have even \ length. Note also that, if 1X1 is a valid word, then words 110X1 and 1X011 are also \ valid, for any word X (valid or not). Apparently, this could be used to \ generate all valid words by the number of steps they require to reach word \ 11.\ \>", "Text"], Cell[BoxData[{ \(steps[1] := {{1, 1}}\), "\[IndentingNewLine]", \(steps[n_] := Flatten[Map[{Join[{1, 1, 0}, Rest[#]], Join[Delete[#, \(-1\)], {0, 1, 1}]} &, steps[n - 1]], 1]\)}], "Input", CellLabel->"In[13]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Table[ FromDigits\ /@ \ steps[n], {n, 5}]\ ;\)\), "\[IndentingNewLine]", \(Sort[Flatten[%]]\)}], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \({11, 1011, 1101, 101011, 110011, 110011, 110101, 10101011, 11001011, 11001011, 11001011, 11010011, 11010011, 11010011, 11010101, 1010101011, 1100101011, 1100101011, 1100101011, 1100101011, 1101001011, 1101001011, 1101001011, 1101001011, 1101001011, 1101001011, 1101010011, 1101010011, 1101010011, 1101010011, 1101010101}\)], "Output", CellLabel->"Out[16]="] }, Open ]], Cell[TextData[{ "However, this approach is flawed (can you see why? hint: how many steps \ needs the word 101111 to reach 1?). An interesting problem is: for what \ numbers n, m such that 0 < n \[LessEqual] m, the word ", Cell[BoxData[ \(TraditionalForm\`1\^n\)]], "0", Cell[BoxData[ \(TraditionalForm\`1\^m\)]], " is a valid one?" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["2.- Group Theory", "Section", FontColor->RGBColor[1, 0.889998, 0.00999466], Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell[TextData[{ StyleBox["Roughly speaking, a RS is a non-deterministic, pattern-oriented \ program transforming terms into terms. As an example, let us consider the \ algebraic structure ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["group", FormatType->StandardForm, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", by which a set S is classified in accordance to compliance to \ the following axioms (explicit bi-directional commutativity with the identity \ element and inverses is commonly assumed in general but it is not necessary, \ as we aim to show.)", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell["f(e, a)=a \t", "BulletedList", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "f(", Cell[BoxData[ \(TraditionalForm\`a\^\(-1\)\)]], ", a) = e\t" }], "BulletedList"], Cell["f(f(a, b), c) = f(a, f(b, c))\t", "BulletedList"], Cell[TextData[{ StyleBox["wherein f: S \[Times] S \[Rule] S is a closed", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["binary operation (not necessarily commutative) on S, e is a \ special element of S and a, b and c are arbitrary elements of S. The first \ axiom asserts the existence of a left identity, the second of a left inverse \ and the third the fact that f is associative. \n\nThese axioms enjoy a dual \ character in their interpretation; the first axiom establishes the way the \ left identity acts on any element of S and can be interpreted as that, \ wherever an f[e, a] appears it can be simplified to a. On the other hand, it \ also can be interpreted as that, wherever an element a appears, it can be \ expanded by f[e, a]. It is possible to derive valid general structural \ equations on S by regarding these axioms as rewriting rules. Let us \ illustrate this by solving the problem of deducing the existence of right \ identity and right inverses, being equal to their left counterparts.\n\nThe \ bi-directional character of the axioms above gives rise to the following six \ productions, wherein for the sake of clarity we have adopted the infix \ notation x \[Bullet] y to stand for f(x, y):", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[TextData[StyleBox["e \[Bullet] a \[Rule] a", FormatType->StandardForm, FontFamily->"Times New Roman"]], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["a \[Rule] e \[Bullet] a", FormatType->StandardForm, FontFamily->"Times New Roman"]], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[{ StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`a\^\(-1\)\)]], " ", StyleBox["\[Bullet] a \[Rule] e", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[{ StyleBox["e \[Rule]", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(a\^\(-1\)\)\)\)]], StyleBox[" \[Bullet] a", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["(a \[Bullet] b) \[Bullet] c \[Rule] a \[Bullet] (b \ \[Bullet] c)", FormatType->StandardForm, FontFamily->"Times New Roman"]], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["a \[Bullet] (b \[Bullet] c) \[Rule] (a \[Bullet] b) \ \[Bullet] c", FormatType->StandardForm, FontFamily->"Times New Roman"]], "NumberedEquation", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["We now proceed to show that a \[Bullet] e \[Rule] a \ through the following sequence of applications in which we indicate above the \ arrow which of the former six productions is being used at each step.", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[TextData[{ StyleBox["\t\t\t\ta", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["e \t", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \&2\)\)]], StyleBox[" (e", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["a)", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["e \n\t\t\t\t\t", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \&4\)\)]], StyleBox[" ((", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(a\^\(-1\)\), "TraditionalForm"], ")"}], \(-1\)], TraditionalForm]]], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`a\^\(-1\)\)]], StyleBox[")", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["a )", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["e \t\n\t\t\t\t\t", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \&5\)\)]], StyleBox[" (", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(a\^\(-1\)\), "TraditionalForm"], ")"}], \(-1\)], 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RowBox[{"(", FormBox[\(a\^\(-1\)\), "TraditionalForm"], ")"}], \(-1\)], TraditionalForm]]], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["e\n\t\t\t\t\t", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \&4\)\)]], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(a\^\(-1\)\), "TraditionalForm"], ")"}], \(-1\)], TraditionalForm]]], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["(", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`a\^\(-1\)\)]], StyleBox[" \[Bullet] ", "DisplayFormula", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["a )\n\t\t\t\t\t", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \&6\)\)]], 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Let us first find a closed commutative operation \ on this set. We make use of the following result.\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Theorem", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[": \t\nLet S be an arbitrary set and let (G, * ) be a group. \ Further, assume S is of the same cardinality than G and let g: S \[Rule] G be \ a bijection. Then (S, \[Bullet]) is a group isomorphic to (G, *) with the \ operation \[Bullet] defined on S as:", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[TextData[{ StyleBox["x \[Bullet] y = ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`g\^\(-1\)\)]], StyleBox["( g(x) * g(y) )", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Taking the set S to be the set of prime numbers and the set G to be the set \ of integer numbers we design function g as follows.\ \>", "Text"], Cell[BoxData[{ \(\(\(SetAttributes[g, Listable]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(g[p_] := Module[{n = 1}, \[IndentingNewLine]While[ Prime[n] \[NotEqual] p, \(n++\)]; \[IndentingNewLine]If[EvenQ[n], n/2, \((1 - n)\)/2]]\)}], "Input", CellLabel->"In[26]:="], Cell[CellGroupData[{ Cell[TextData[StyleBox["g[Prime[Range[20]]]", FormatType->StandardForm, FontFamily->"Courier New"]], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \({0, 1, \(-1\), 2, \(-2\), 3, \(-3\), 4, \(-4\), 5, \(-5\), 6, \(-6\), 7, \(-7\), 8, \(-8\), 9, \(-9\), 10}\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell["The inverse of which is:", "Text"], Cell[TextData[StyleBox["SetAttributes[gInv,Listable]\n\ngInv[0]:=2\n\ gInv[i_]:=If[Positive[i],Prime[2i],Prime[1-2i]]", FormatType->StandardForm, FontFamily->"Courier New"]], "Input", CellLabel->"In[29]:="], Cell[CellGroupData[{ Cell[TextData[StyleBox["gInv[g[Prime[Range[20]]]]", FormatType->StandardForm, FontFamily->"Courier New"]], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71}\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell[TextData[{ "So, the function ", StyleBox["f", "InlineInput"], " we seek is:" }], "Text"], Cell[TextData[StyleBox["f[p1_,p2_]:=gInv[g[p1]+g[p2]]", FormatType->StandardForm, FontFamily->"Courier New"]], "Input", CellLabel->"In[33]:="], Cell["The corresponding multiplication table starts as follows.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[Table[f[Prime[i], Prime[j]], {i, 10}, {j, 10}], TableHeadings \[Rule] {Prime[Range[10]], Prime[Range[10]]}]\)], "Input",\ CellLabel->"In[34]:="], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "2", "3", "5", "7", "11", "13", "17", "19", "23", "29"}, {"2", "2", "3", "5", "7", "11", "13", "17", "19", "23", "29"}, {"3", "3", "7", "2", "13", "5", "19", "11", "29", "17", "37"}, {"5", "5", "2", "11", "3", "17", "7", "23", "13", "31", "19"}, {"7", "7", "13", "3", "19", "2", "29", "5", "37", "11", "43"}, {"11", "11", "5", "17", "2", "23", "3", "31", "7", "41", "13"}, {"13", "13", "19", "7", "29", "3", "37", "2", "43", "5", "53"}, {"17", "17", "11", "23", "5", "31", "2", "41", "3", "47", "7"}, {"19", "19", "29", "13", "37", "7", "43", "3", "53", "2", "61"}, {"23", "23", "17", "31", "11", "41", "5", "47", "2", "59", "3"}, {"29", "29", "37", "19", "43", "13", "53", "7", "61", "3", "71"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{2, 3, 5, 7, 11, 13, 17, 19, 23, 29}, { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29}}]]]], "Output", CellLabel->"Out[34]//TableForm="] }, Open ]], Cell[TextData[StyleBox["We leave this section with two questions for the \ reader. How can we design a non-commutative group operation on the set of \ prime numbers? How do we turn Z into a non-abelian group through finding a \ suitable operation?", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["3.- Equivalence Relations", "Section", FontColor->RGBColor[1, 0.889998, 0.00999466], Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell["\<\ Let us consider the decomposition of a set into disjoint classes imposed by \ a given equivalence relation defined on it. We can concisely describe the \ procedure to obtain these classes by building them up through successively \ inserting elements as an RS process. Specifically, consider the following set \ of derivations, which defines a function t that, given a pair {u, v} in the \ relation, selects the appropriate transformation to perform on a partially \ constructed set of classes.\ \>", "Text"], Cell[BoxData[ \(t[{u_, v_}] := {{a___, {b___, \((u | v)\), c___}, d___, {e___, \((u | v)\), f___}, g___} \[Rule] {a, {b, u, v, c, e, f}, d, g}, \[IndentingNewLine]\ {a___, {b___, \((u | v)\), c___}, d___} \[Rule] {a, {b, u, v, c}, d}, \[IndentingNewLine]{a___} \[Rule] {a, {u, v}}}\)], "Input", CellLabel->"In[36]:="], Cell[TextData[{ StyleBox["Accordingly, we can define function ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["classes", "InlineInput", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" which, when given a relation, through mapping each of its \ elements to t, produces the required set. \:f35f", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[ \(classes[rel_] := Module[{clas = {}}, \[IndentingNewLine]Map[\((clas = \((clas\ /. \ t[#])\))\) &, \ rel]; \[IndentingNewLine]clas]\)], "Input",\ CellLabel->"In[38]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(\(relations = {{1, 5}, {6, 8}, {7, 2}, {9, 8}, {3, 7}, {4, 2}, {9, 3}};\)\), "\[IndentingNewLine]", \(classes[relations]\)}], "Input", CellLabel->"In[39]:="], Cell[BoxData[ \({{1, 5}, {6, 9, 3, 8, 7, 4, 2}}\)], "Output", CellLabel->"Out[40]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4.- Systems of Numeric Productions", "Section", FontColor->RGBColor[1, 0.889998, 0.00999466], Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell[TextData[StyleBox["Let n be a positive integer, and x and y be arbitrary \ numbers. Consider the RS:", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[TextData[StyleBox["< n > \[Rule] < n - 1 > x - \ \[LeftDoubleBracketingBar] n - 1\[RightDoubleBracketingBar] y\n\ \[LeftDoubleBracketingBar] n \[LeftDoubleBracketingBar] \[Rule] \ \[LeftDoubleBracketingBar] n - 1 \[RightDoubleBracketingBar] x + < n - 1 > y\n\ < 0 > \[Rule] 1\n\[LeftDoubleBracketingBar] 0 \[RightDoubleBracketingBar] \ \[Rule] 0", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["where the brackets surrounding n are meant to provide \ two different kinds of contexts and the operations on the right are the usual \ arithmetic ones. The last two derivations make this system terminating. For \ instance, ", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[TextData[{ StyleBox["< 2 > \[Rule] < 1 > x - \[LeftDoubleBracketingBar] 1\ \[RightDoubleBracketingBar] y ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \&*\)\)]], StyleBox[" x", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" - y", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}] }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["Where the star above the arrow means that we have \ omitted internal productions involved in this derivation. Let us denote by A \ \[DoubleRightArrow] N the fact that N is a normal form of A.", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[BoxData[ \(<< Utilities`Notation`\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[{ RowBox[{"Notation", "[", RowBox[{ TagBox[\(\(<\)\(n_\)\(>\)\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[\(b1[n_]\), NotationBoxTag, TagStyle->"NotationTemplateStyle"]}], "]"}], "\n", RowBox[{"Notation", "[", RowBox[{ TagBox[\(\[LeftDoubleBracketingBar]n_\[RightDoubleBracketingBar]\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[\(b2[n_]\), NotationBoxTag, TagStyle->"NotationTemplateStyle"]}], "]"}]}], "Input", CellLabel->"In[2]:="], Cell[BoxData[{ \(b1[0] := 1\), "\n", \(b2[0] := 0\), "\n", \(b1[n_] := \(b1[n] = Expand[b1[n - 1] x - b2[n - 1] y]\)\), "\n", \(b2[n_] := \(b2[n] = Expand[b2[n - 1] x + b1[n - 1] y]\)\)}], "Input", CellLabel->"In[4]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\(\(<\)\(j\)\(>\)\), {j, 6}]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \({x, x\^2 - 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455\ x\^12\ y\^3 + 3003\ x\^10\ y\^5 - 6435\ x\^8\ y\^7 + 5005\ x\^6\ y\^9 - 1365\ x\^4\ y\^11 + 105\ x\^2\ y\^13 - y\^15\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["\<\ The following animation shows the complexity of function .\ \>", "Text"], Cell["ANIMATION", "Text", FontSize->18, FontColor->RGBColor[0.0299992, 0.179995, 0.329992], Background->RGBColor[0.737301, 0.5608, 0.5608]], Cell[BoxData[ \(\(Map[\(Plot3D[\(\(<\)\(#\)\(>\)\), {x, \(-1\), 1}, {y, \(-1\), 1}, PlotPoints \[Rule] 100, Mesh \[Rule] False, Boxed \[Rule] False, Axes \[Rule] False];\) &, Join[Range[15], Range[15, 1, \(-1\)]]];\)\)], "Input", CellLabel->"In[31]:="], Cell[TextData[{ StyleBox["Theorem", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[" \tIf \[DoubleRightArrow] A and [n] \[DoubleRightArrow] B \ then A", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2 ", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["+ B", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" = (x", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" + y", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[")", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\t", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[\((\(<\)\(10\)\(>\))\)\^2 + \ \((\[LeftDoubleBracketingBar]10\ \[RightDoubleBracketingBar])\)\^2]\)], "Input", CellLabel->"In[52]:="], Cell[BoxData[ \(\((x\^2 + y\^2)\)\^10\)], "Output", CellLabel->"Out[52]="] }, Open ]], Cell[TextData[{ StyleBox["Theorem", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[" \tIf x=Cos[1], and y=Sin[1] then", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[TextData[StyleBox[" \[DoubleRightArrow] Cos[n] \n[n] \ \[DoubleRightArrow] Sin[n]", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[BoxData[{ \(N[\(\(<\)\(10\)\(>\)\) /. \ {x \[Rule] Cos[1], \ y \[Rule] Sin[1]}]\), "\[IndentingNewLine]", \(N[Cos[10]]\)}], "Input", CellLabel->"In[53]:="], Cell[BoxData[ \(\(-0.8390715290764517`\)\)], "Output", CellLabel->"Out[53]="], Cell[BoxData[ \(\(-0.8390715290764524`\)\)], "Output", CellLabel->"Out[54]="] }, Open ]], Cell[TextData[StyleBox["The truth of the previous theorems comes from the \ identity: ", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(\ \ \)\(Cos[2 \[Theta]]\)\)}, {\(\(\ \)\(Sin[2 \[Theta]]\)\)} }], "\[NegativeThinSpace] ", ")"}], TraditionalForm]]], StyleBox[" = ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ RowBox[{"(", " ", GridBox[{ {\(Cos[\[Theta]]\), \(Sin[\[Theta]]\)}, {\(\(\ \)\(Sin[\[Theta]]\)\), \(Cos[\[Theta]]\)} }], ")"}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{"(", " \[NegativeThinSpace]", GridBox[{ {\(\(\ \)\(Sin[\[Theta]]\)\)}, {\(\(\ \)\(Cos[\[Theta]]\)\)} }], "\[NegativeThinSpace] ", ")"}], TraditionalForm]]] }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["describing a counterclockwise rotation trough an \ angle \[Theta] around the origin. Using this it can be shown that", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(<\)\(n\)\(>\)\)}, {\([n]\)} }], "\[NegativeThinSpace]", ")"}], " ", "\[DoubleRightArrow]", " ", RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ {"x", \(-y\)}, {"y", "x"} }], ")"}], \(n - 1\)], RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"x"}, {"y"} }], "\[NegativeThinSpace]", ")"}]}]}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["5.- Recursive Productions", "Section", FontColor->RGBColor[1, 0.889998, 0.00999466], Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell[TextData[StyleBox["Consider the following rules, invented by the \ logician Raymond Smullyan:", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[TextData[StyleBox["2X \[Rule] X\nIf X \[Rule] Y then 3X \[Rule] Y2Y", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text", TextAlignment->Center, TextJustification->0], Cell[BoxData[{ \(smullyan[{"\<2\>", X__}] := {X}\), "\[IndentingNewLine]", \(smullyan[{"\<3\>", X__}] := Module[{Y = smullyan[{X}]}, Join[Y, {"\<2\>"}, Y]]\)}], "Input", CellLabel->"In[55]:="], Cell[TextData[{ StyleBox["There are novel features in this RS. Those rules are applicable \ to integers regarded as words built from concatenating digits, but not all \ integers are \"acceptable\" words. These rules apply only to words of the \ form 3", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["2X for some n\[GreaterEqual]0 and arbitrary word X. The second \ rule is posed in a conditional manner in a way that does not resemble any \ context dependence. It is this rule that allows us to conclude, through a \ simple application of mathematical induction that:\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Theorem ", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}] }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`3\^n\ 2\ X\)]], " \[Rule] ", Cell[BoxData[ \(TraditionalForm\`\(\((X\ 2)\)\^\(2\^n - 1\)\) X\)]] }], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[BoxData[ \(StringJoin\ @@ \ smullyan[Characters["\<3332456\>"]]\)], "Input", CellLabel->"In[57]:="], Cell[BoxData[ \("4562456245624562456245624562456"\)], "Output", CellLabel->"Out[57]="] }, Open ]], Cell[TextData[{ StyleBox["Also, it is easy to find out that there is a unique word N such \ that N \[Rule] N, that is, the RS admits only one", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" fixed point", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox[". More generally, we can assert the following.\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Theorem", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[" (Fixed Point Theorem) Given any word A, there exists a word Y \ such that Y \[Rule] AY\n\nFor example, if A=332332, then Y=32A3 is such word. \ This Y might not be unique though, for example, Y=33233 also complies with \ the theorem for this particular A. A limited converse of the former theorem \ applies.\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Theorem", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[" \tLet N= ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SuperscriptBox[ FormBox[\(\(3\^n\ 2\)\()\)\), "TraditionalForm"], "m"], " ", \(3\^n\)}]}], TraditionalForm]]], StyleBox[". Then N \[Rule] NA with A =", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"(", RowBox[{"2", " ", SuperscriptBox[ FormBox[\(\(3\^n\)\()\)\), "TraditionalForm"], \(m\ 2\^m\ - m - 1\)]}]}]}], TraditionalForm]]], StyleBox["\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Proof", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[":\tWe have that ", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[TextData[{ StyleBox["N= ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SuperscriptBox[ FormBox[\(\(\(3\^n\) 2\)\()\)\), "TraditionalForm"], "m"], " ", \(3\^n\)}]}], TraditionalForm]]], StyleBox[" = ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\(3\^n\) 2\)]], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SuperscriptBox[ FormBox[\(\(\(3\^n\) 2\)\()\)\), "TraditionalForm"], \(m - 1\)], " ", \(3\^n\)}]}], TraditionalForm]]], StyleBox[" \[Rule] ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", " ", RowBox[{ FormBox[ RowBox[{"(", SuperscriptBox[ FormBox[\(\(\(3\^n\) 2\)\()\)\), "TraditionalForm"], \(m - 1\)]}], "TraditionalForm"], " ", FormBox[\(3\^n\), "TraditionalForm"]}], ")"}], \(2\^n - 1\)], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"(", SuperscriptBox[ FormBox[\(\(\(3\^n\) 2\)\()\)\), "TraditionalForm"], \(m - 1\)]}], "TraditionalForm"], " ", FormBox[\(3\^n\), "TraditionalForm"]}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"(", SuperscriptBox[ FormBox[\(\(\(3\^n\) 2\)\()\)\), "TraditionalForm"], \(m(2\^n - 1) + m - 1\)]}], "TraditionalForm"], " ", FormBox[\(3\^n\), "TraditionalForm"]}], TraditionalForm]]] }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ StyleBox["which coincides with the value of the word NA. \[FilledSquare]\n\ \nNote that not all words have normal forms; among those we can mention words \ of the form 32", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["3 for n>1, which have periodic behavior, or of the form 32", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["32, which have exponential-growth behavior. Moreover we claim \ that:\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Theorem", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox["\t\nA number of the form ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\((32)\)\^a\_1\)]], " 2 ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^a\_2\)]], StyleBox[" 2 ... 2 ", FormatType->StandardForm, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\((32)\)\^a\_n\)]], StyleBox[" 3, where the exponents are non-negative integers and n>0, does \ not possess a normal form\n\n", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["Proof", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"Underline"->True}], StyleBox[":\tWriting the above form as [a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["1", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis], a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["] we can deduce that", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[TextData[{ StyleBox["[a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["1", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis], a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["] \[Rule] [a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["1", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["-1, a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis], a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n-1,", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["1", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["+a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", \[Ellipsis], a", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["]", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ StyleBox["which proves our claim. ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["\[FilledSquare]", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["\n\nWe can code this notation as:", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[{ \(\(\(Clear[a, b, c, t]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(t[{a_, b___, c_}] := {a - 1, b, a + c, b, c}\), "\n", \(t[{0, b__}] := {b}\), "\n", \(t[{a_}] := {2 a - 1}\)}], "Input", CellLabel->"In[58]:="], Cell[TextData[StyleBox["So that, for example:", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NestList[t, {1}, 6]\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ \({{1}, {1}, {1}, {1}, {1}, {1}, {1}}\)], "Output", CellLabel->"Out[62]="] }, Open ]], Cell["corresponding to the behavior of the fixed point 323.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NestList[t, {2}, 6]\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ \({{2}, {3}, {5}, {9}, {17}, {33}, {65}}\)], "Output", CellLabel->"Out[63]="] }, Open ]], Cell[TextData[{ "corresponding to the productions ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^2\)]], "3 \[Rule] ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^3\)]], "3 \[Rule] ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^5\)]], "3 \[Rule] ..." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NestList[t, {1, 1}, 6]\)], "Input", CellLabel->"In[64]:="], Cell[BoxData[ \({{1, 1}, {0, 2, 1}, {2, 1}, {1, 3, 1}, {0, 3, 2, 3, 1}, {3, 2, 3, 1}, {2, 2, 3, 4, 2, 3, 1}}\)], "Output", CellLabel->"Out[64]="] }, Open ]], Cell[TextData[{ "corresponding to the productions ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^1\)]], "2 ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^1\)]], "3 = 322323 \[Rule] 232322323= ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^0\)]], "2 ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^2\)]], "2 ", Cell[BoxData[ \(TraditionalForm\`\((32)\)\^1\)]], "3 \[Rule] ..." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["6.- L Systems", "Section", FontColor->RGBColor[1, 0.889998, 0.00999466], Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell[TextData[{ "Sometimes a useful purpose can be found for RS having no normal form. \ Consider the L systems (also called ", StyleBox["Lindenmayer systems", FontSlant->"Italic"], " or ", StyleBox["parallel string-rewrite systems", FontSlant->"Italic"], ") to generate words that can be interpreted by the LOGO turtle (an \ imaginary graphical entity) in suitable ways as to render eye-catching \ objects. These systems have not normal forms although they often converge to \ fractal curves. Although those forms are not reachable in a finite number of \ steps successive derivations are often intrinsically interesting. An L system \ is created by starting with an axiom and replacing it through some prescribed \ productions so that a more complex expression is formed. This expression is \ assumed to act as LOGO instructions on a starting figure, often a single line \ segment. \n\n", StyleBox["Assume the symbol F means drawing a unit-length segment forward \ in the current direction. Assume also that the symbol + conveys a clockwise \ turn by a fixed angle from the direction we are heading and symbol - a \ counterclockwise turn by the same amount. Function ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["logo", "Input", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" below comprises a very small logo interpreter powerful enough to \ render striking graphics.", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[ \(\(SetOptions[Graphics, \ AspectRatio \[Rule] \ Automatic, PlotRange \[Rule] All];\)\)], "Input", CellLabel->"In[133]:="], Cell[BoxData[ RowBox[{\(logo[s_String, \[Theta]_]\), ":=", RowBox[{"Module", "[", RowBox[{\({a = {0, 0}, b = {1, 0}, h = Cos[\[Theta]], k = Sin[\[Theta]], t}\), ",", "\[IndentingNewLine]", RowBox[{\(t = {a}\), ";", "\[IndentingNewLine]", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Switch", "[", RowBox[{ "#", ",", "\"\\"", ",", \(AppendTo[t, a = a + b]\), ",", "\"\<+\>\"", ",", RowBox[{"b", "=", RowBox[{ RowBox[{"(", GridBox[{ {"h", \(-k\)}, {"k", "h"} }], ")"}], 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instructive ways of generating \ Kock curves, one of which is provided by function ", StyleBox["sF", "Input"], " below which sets up the parameters needed for the recursive function ", StyleBox["snowFlake", "Input"], ". Its advantage is that it offers a finer control on the parameters \ defining this family. Function ", StyleBox["snowflake", "InlineInput"], "'s first two parameters correspond to the ends of the generated curve and \ the last parameter to its order. The reader might find interesting altering \ the values of the other parameters to find their purpose in generating \ mutated members of this family. 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Sagan \ in The Mathematical Intelligencer , Vol. 15 num. 4, 1993 pp. 37-43). The \ code for generating this attractive curve is the following. 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subsets of {1, 2, \[Ellipsis], n}, 2) the first ", Cell[BoxData[ \(TraditionalForm\`2\^n\)]], " binary words starting from ", Cell[BoxData[ \(TraditionalForm\`0\^n\)]], " and ending at ", Cell[BoxData[ \(TraditionalForm\`1\^n\)]], " , 3) the integer numbers from 0 to ", Cell[BoxData[ \(TraditionalForm\`2\^n\)]], "-1 and 4) the compositions of n+1. The table below shows these \ isomorphisms for the case n=4. We illustrate the use of RS in the translation \ from a binary word to a composition, that is, given a binary word we will set \ up an RS in such a way that the normal form to which it converges is the \ required composition. This is accomplished by the rules " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`1\^n\)]], "0 \[Rule] (n+1)+\n\"+ \" \[Rule] \"\" " }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ (note the space after the plus sign in the last production) or \ equivalently:\ \>", "Text"], Cell["\<\ 10 \[Rule] 2+ 1x+ \[Rule] (x+1)+ 0 \[Rule] 1+ \"+ \" \[Rule] \"\"\ \>", "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ We will append a 0 at a given binary string before applying these \ productions. For instance, to find out which composition corresponds to the binary \ sequence 111011 the productions act as follows:\ \>", "Text"], Cell["\<\ 1110110 \[Rule] 112+110 \[Rule] 112+12+ \[Rule] 13+12+ \[Rule] 4+12+ \ \[RightArrow] 4+3+ \[Rule] 4+3\ \>", "Text", TextAlignment->Center, TextJustification->0], Cell["So, the associated composition is 4+3.", "Text"], Cell[BoxData[{ \(subSets[0] := {{}}\), "\n", \(\(\(subSets[n_] := Module[{s = subSets[n - 1]}, Join[s, Map[Append[#, n] &, s]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(binary[n_, s_] := Module[{bn = Table[0, {n}]}, bn\_\(\(\[LeftDoubleBracket]\)\(s\)\(\[RightDoubleBracket]\)\) = 1; bn]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(dec[{}] := 0\), "\n", \(\(\(dec[{x___, t_}] := 2\ dec[{x}] + t\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(compositions[u_] := \ Module[{v = Append[u, 0]}, \[IndentingNewLine]v = \((v\ //. \ \ {{a___, 1, 0, c___} \[Rule] {a, 2, "\<+\>", c}, \[IndentingNewLine]{a___, 1, b_, "\<+\>", c___} \[Rule] {a, b + 1, "\<+\>", c}, \[IndentingNewLine]{a___, 0, c___} \[Rule] {a, 1, "\<+\>", c}})\); \[IndentingNewLine]StringJoin[ ToString\ /@ \ Drop[v, \(-1\)]]]\)}], "Input", CellLabel->"In[105]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(\(n = 4;\)\), "\n", \(m = Map[{#, u = binary[n, #]; \ StringJoin[ToString\ /@ \ u], dec[u], compositions[u]} &, subSets[n]]\ // \ MatrixForm\)}], "Input", CellLabel->"In[111]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({}\), "\<\"0000\"\>", "0", "\<\"1+1+1+1+1\"\>"}, {\({1}\), "\<\"1000\"\>", "8", "\<\"2+1+1+1\"\>"}, {\({2}\), "\<\"0100\"\>", "4", "\<\"1+2+1+1\"\>"}, {\({1, 2}\), "\<\"1100\"\>", "12", "\<\"3+1+1\"\>"}, {\({3}\), "\<\"0010\"\>", "2", "\<\"1+1+2+1\"\>"}, {\({1, 3}\), "\<\"1010\"\>", "10", "\<\"2+2+1\"\>"}, {\({2, 3}\), "\<\"0110\"\>", "6", "\<\"1+3+1\"\>"}, {\({1, 2, 3}\), "\<\"1110\"\>", "14", "\<\"4+1\"\>"}, {\({4}\), "\<\"0001\"\>", "1", "\<\"1+1+1+2\"\>"}, {\({1, 4}\), "\<\"1001\"\>", "9", "\<\"2+1+2\"\>"}, {\({2, 4}\), "\<\"0101\"\>", "5", "\<\"1+2+2\"\>"}, {\({1, 2, 4}\), "\<\"1101\"\>", "13", "\<\"3+2\"\>"}, {\({3, 4}\), "\<\"0011\"\>", "3", "\<\"1+1+3\"\>"}, {\({1, 3, 4}\), "\<\"1011\"\>", "11", "\<\"2+3\"\>"}, {\({2, 3, 4}\), "\<\"0111\"\>", "7", "\<\"1+4\"\>"}, {\({1, 2, 3, 4}\), "\<\"1111\"\>", "15", "\<\"5\"\>"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[112]//MatrixForm="] }, Open ]], Cell["\<\ By interpreting the numbers appearing in the last column as distances to be \ traversed by the LOGO interpreter, we can get a visualization of the set of \ compositions of a given number. 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Background->RGBColor[0.4392, 0.502007, 0.564706]], Cell[TextData[{ "In dealing with the storage of a great amount of textual information we \ face the task of encoding it so that any redundancy in the text can be \ exploited to our advantage. Dictionaries like the Mexican Spanish Dictionary, \ a major on-going project that started about twenty years ago at El Colegio de \ Mexico, a prestigious research institution in Mexico City, faced this \ important challenge. The author was then a member of the team that suggested, \ implemented and tested a variety of approaches to alleviate the computational \ burden of dealing with massive amounts of textual information when we came \ across the fact that the optimal way to accomplish the compression that \ exploits the inherent redundancy in any discrete interchange of information \ was already discovered by D.Huffman and reported on his paper \"A Method for \ the construction of minimum-redundancy codes\" at the Proceedings of the IRE, \ vol.40, 1952.\n\nHuffman coding involves the building of a tree of \ frequencies of each of the symbols forming the original message. Once this \ tree is constructed, the leave nodes can be coded to a binary specification \ of the path needed to reach them from the root.\n\nThe details are now part \ of the standard content of any undergraduate course on algorithm design and \ here we want to illustrate the application of transformational grammars, as \ RS are called in this context, to this type of problems (for an alternative \ way of implementing run-length encoding consult Version 3.0 Programming \ Features by Roman Maeder, the Mathematica Journal vol.6 issue 4, Fall 1996, \ p. 40-43). First we design function ", StyleBox["frequencies", "Input"], " to count the frequencies of the symbols appearing in a given sentence." }], "Text"], Cell[BoxData[ \(frequencies[mesg_String] := With[{m = Characters[mesg]}, \[IndentingNewLine]\(({#, Count[m, #]} &)\) /@ Union[m]]\)], "Input", CellLabel->"In[41]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(\(message = "\";\)\), "\ \[IndentingNewLine]", \(frequencies[message]\)}], "Input", CellLabel->"In[44]:="], Cell[BoxData[ \({{",", 1}, {" ", 6}, {"a", 2}, {"c", 1}, {"d", 2}, {"e", 3}, {"g", 1}, {"h", 1}, {"H", 1}, {"i", 2}, {"k", 1}, {"l", 2}, {"m", 1}, {"o", 2}, {"r", 1}, {"t", 2}, {"u", 2}, {"y", 1}}\)], "Output", CellLabel->"Out[45]="] }, Open ]], Cell[TextData[StyleBox["or, alternatively", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[BoxData[ \(runEncode[mesg_String] := Map[{#, \ 1}\ &, \ \n\ \ \ \ \ \ Characters[ mesg]]\ //. {x___, {a_, \ i_}, \ y___, {a_, \ j_}, z___}\ \[Rule] \ {x, \ {a, \ i\ + \ j}, \ y, \ z}\)], "Input", CellLabel->"In[46]:="], Cell[CellGroupData[{ Cell[BoxData[ \(runEncode[message]\)], "Input", CellLabel->"In[47]:="], Cell[BoxData[ \({{"H", 1}, {"i", 2}, {" ", 6}, {"t", 2}, {"h", 1}, {"e", 3}, {"r", 1}, {",", 1}, {"g", 1}, {"l", 2}, {"a", 2}, {"d", 2}, {"y", 1}, {"o", 2}, {"u", 2}, {"c", 1}, {"m", 1}, {"k", 1}}\)], "Output", CellLabel->"Out[47]="] }, Open ]], Cell[TextData[StyleBox["We use nested lists for the encoding of trees; for \ instance, the tree", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgLcg_?Nl00`000Sg_00=>LgooOol0NWoo1P00 17oo00=>Lcg_00000`0000Ool00cg_0000000;00000cg_Oomo o`1gOol600000goo?Nl0000;00000cg_Oomoo`09Ool00cg_0000000<0000 0dicOomoo`0;Ool00cg_0000000800000cg_Oomoo`05Ool00cg_0000000: 00000ekg?Nl0000400000cg_Oomoo`3oOoo9Ool7002POol7000DOol00LgooOol01Goo00=>L`000000400000=NmgooOol027oo 00Lekg1Goo00LgooOomNmdic00P007Ioo`T00003CW=oogoo00=o 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o`13Ool7003oOooNOol00?mooomoo`9oo`L000aoo`03G_Lmk`0000L00003 G_Moogoo04=oo`L00?moomioo`00ogooogoo0Woo7@00AWoo1`00ogoogWoo 003oOoooOol2OolL00000ekgOomoo`14Ool00ekg00000004003oOooNOol0 0?mooomoo`9ooa/00003?Nmoogoo04Moo`03CW<0000000800?moomioo`00 ogooogoo0Woo6P0000L`000000ogoogWoo003oOooo Ool2OolI00000ekgOomoo`1"], "PlacedGraphics", ImageSize->{281, 136.25}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["is represented by the list: ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["\:f35f", FormatType->StandardForm, FontFamily->"Courier New", FontSize->10] }], "Text"], Cell[BoxData[ \(\(t = {Root, {12, {5, {6, "\"}}, "\", 28, {11, {4, 7}}}};\)\)], "Input", CellLabel->"In[48]:="], Cell[TextData[{ StyleBox["We also need function ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["sons", "InlineInput", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[" to obtain all the sons of a given tree:", FormatType->StandardForm, FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[{ \(sons[s_] := s\), "\n", \(sons[{a___, {b_, c__}, d___}] := sons[{a, b, d}]\)}], "Input", CellLabel->"In[49]:="], Cell[CellGroupData[{ Cell[BoxData[ \(sons[Last[t]]\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \({12, 5, "alpha", 28, 11}\)], "Output", CellLabel->"Out[51]="] }, Open ]], Cell["\<\ We will consider only numeric trees representing the frequencies of their \ respective words. We now set up the patterns comprising the Huffman \ algorithm. All of them implement one step of the iterative process which will \ build the Huffman tree. This process can be described as follows. Given the \ two smallest elements x, y of a list, we encapsulate them to form a subtree \ labeled x+y.\ \>", "Text"], Cell[BoxData[ \(patterns[x_, y_] := {{a___, x, b___, y, c___} | {a___, y, b___, x, c___} \[Rule] {a, b, {x + y, {x, y}}, c}, {a___, {x, t__}, b___, {y, g__}, c___} \[Rule] {a, b, {x + y, {{x, t}, {y, g}}}, c}, {a___, {y, t__}, b___, {x, g__}, c___} \[Rule] {a, b, {x + y, {{y, t}, {x, g}}}, c}, {a___, {x, t__}, b___, y, c___} \[Rule] {a, b, {x + y, {{x, t}, y}}, c}, {a___, {y, t__}, b___, x, c___} \[Rule] {a, b, {x + y, {{y, t}, x}}, c}, {a___, x, b___, {y, g__}, c___} \[Rule] {a, b, {x + y, {x, {y, g}}}, c}, {a___, y, b___, {x, g__}, c___} \[Rule] {a, b, {x + y, {y, {x, g}}}, c}}\)], "Input", CellLabel->"In[52]:="], Cell[TextData[{ StyleBox["using those transformations in processing the given vector of \ frequencies we arrive at function ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["huffman", "InlineInput", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[".", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["\:f35f", FormatType->StandardForm, FontFamily->"Courier New", FontSize->10] }], "Text"], Cell[BoxData[ \(huffman[frec_] := Module[{x, y}, \[IndentingNewLine]FixedPoint[\(({x, y} = Take[Sort[sons[#]], 2]; # /. patterns[x, y])\) &, frec, Length[frec] - 1]]\)], "Input", CellLabel->"In[53]:="], Cell[CellGroupData[{ Cell[BoxData[ \(huffman[{20, 28, 4, 17, 12, 7}]\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \({{88, {{37, {17, 20}}, {51, {28, {23, {12, {11, {4, 7}}}}}}}}}\)], "Output", CellLabel->"Out[54]="] }, Open ]], Cell[TextData[StyleBox["The successive steps of the process to arrive at this \ result are the following:", FormatType->StandardForm, FontFamily->"Times New Roman"]], "Text"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({20, 28, 4, 17, 12, 7}\)}, {\({20, 28, 17, 12, {11, {4, 7}}}\)}, {\({20, 28, 17, {23, {12, {11, {4, 7}}}}}\)}, {\({28, {37, {17, 20}}, {23, {12, {11, {4, 7}}}}}\)}, {\({{37, {17, 20}}, {51, {28, {23, {12, {11, {4, 7}}}}}}}\)}, {\({{88, {{37, {17, 20}}, {51, {28, {23, {12, {11, {4, 7}}}}}}}}}\)} }], "\[NoBreak]", ")"}], MatrixForm[ {{20, 28, 4, 17, 12, 7}, {20, 28, 17, 12, {11, {4, 7}}}, { 20, 28, 17, {23, {12, {11, {4, 7}}}}}, {28, {37, {17, 20}}, {23, { 12, {11, {4, 7}}}}}, {{37, {17, 20}}, {51, {28, {23, {12, {11, {4, 7}}}}}}}, {{88, {{37, {17, 20}}, {51, {28, {23, {12, {11, {4, 7}}}}}}}}}}]]], "Output"], Cell[TextData[{ "To recover the original labels, i.e., substitute the numerical labels for \ their original characters at the leaves nodes, we need to apply rewriting on \ the Huffman tree. Function ", StyleBox["lettersTree", "Input"], " below receives a sentence, obtains its frequencies, obtains its Huffman \ tree and performs this rewriting." }], "Text"], Cell[BoxData[ \(lettersTree[mesg_] := Module[{f = frequencies[mesg], t, i, j, k, p, q, x, y, z}, \[IndentingNewLine]t = First[huffman[Last /@ \ f]]; \[IndentingNewLine]t = t\ //. \ {i_Integer, {x___, j_Integer, y___}} \[Rule] {i, {x, j + 0.1, y}}; \[IndentingNewLine]While[ f \[NotEqual] {}, \[IndentingNewLine]{i, j} = First[f]; \[IndentingNewLine]p = First[Position[t, j + 0.1]]; \[IndentingNewLine]t = ReplacePart[t, i, p]; \[IndentingNewLine]f = Rest[f]]; \[IndentingNewLine]t]\)], "Input", CellLabel->"In[55]:="], Cell[CellGroupData[{ Cell[BoxData[ \(t1 = lettersTree[message]\)], "Input", CellLabel->"In[56]:="], Cell[BoxData[ \({32, {{14, {" ", {8, {{4, {{2, {",", "c"}}, {2, {"g", "h"}}}}, {4, {"a", "d"}}}}}}, {18, {{8, {{4, {"i", "l"}}, {4, {{2, {"H", "k"}}, {2, {"m", "r"}}}}}}, {10, {{4, {"o", "t"}}, {6, {"e", {3, 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"\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], "dt"->RowBox[ { SubscriptBox[ "\[PartialD]", "\[Placeholder]"], " ", "\[SelectionPlaceholder]"}], "notation"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation>"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation<"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "symb"->RowBox[ {"Symbolize", "[", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], "]"}], "infixnotation"->RowBox[ {"InfixNotation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "addia"->RowBox[ {"AddInputAlias", "[", RowBox[ {"\"\[Placeholder]\"", "\[Rule]", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "pattwraper"->TagBox[ "\[Placeholder]", NotationPatternTag, TagStyle -> "NotationPatternWrapperStyle"], "madeboxeswraper"->TagBox[ "\[Placeholder]", NotationMadeBoxesTag, TagStyle -> "NotationMadeBoxesWrapperStyle"]}, Magnification->1, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. Make modifications to any definition using commands in the \ Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], ScriptMinSize->9], Cell[StyleData[All, "Presentation"], ScriptMinSize->9], Cell[StyleData[All, "SlideShow"], PageWidth->WindowWidth, ScrollingOptions->{"PagewiseDisplay"->True, "VerticalScrollRange"->Fit}, ShowCellBracket->False, ScriptMinSize->9], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ShowCellLabel->False, ImageSize->{200, 200}, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the Notebook level.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{True, True}, PrintingOptions->{"FirstPageHeader"->False, "FacingPages"->True}, CellLabelAutoDelete->False, CellFrameLabelMargins->6, StyleMenuListing->None], Cell[StyleData["Notebook", "Presentation"], FontSize->12] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{0, 0}, {0, 0.25}}, CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->9, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{0.95, 11}, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->36], Cell[StyleData["Title", "Presentation"], CellMargins->{{25, 30}, {40, 30}}, CellFrameMargins->{{6, 10}, {10, 14}}, FontSize->54], Cell[StyleData["Title", "Printout"], CellMargins->{{18, 30}, {4, 0}}, CellFrameMargins->4, FontSize->30] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{18, 30}, {0, 10}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 0}, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->24, FontSlant->"Italic"], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{25, 40}, {55, 10}}, FontSize->36], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{18, 30}, {0, 10}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SectionFirst"], CellFrame->{{0, 0}, {0, 3}}, CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameMargins->3, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["SectionFirst", "Presentation"], CellFrame->{{0, 0}, {0, 8}}, CellMargins->{{25, 30}, {65, 10}}, FontSize->27], Cell[StyleData["SectionFirst", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{25, 30}, {30, 15}}, FontSize->27], Cell[StyleData["Section", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{25, 30}, {30, 20}}, FontSize->21], Cell[StyleData["Subsection", "Printout"], FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{18, 30}, {4, 12}}, CellGroupingRules->{"SectionGrouping", 60}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Subsubsection", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{25, 30}, {10, 10}}, FontSize->18], Cell[StyleData["Subsubsection", "Printout"], FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{18, 10}, {Inherited, 6}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> 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Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{55, 10}, {5, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellMargins->{{80, 10}, {8, 15}}, FontSize->18], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 55}, {0, 10}}, ShowCellLabel->False, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{55, 10}, {8, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelPositioning->Left, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{80, 10}, {30, 12}}, FontSize->16], Cell[StyleData["Output", "Printout"], CellMargins->{{55, 55}, {10, 10}}, ShowCellLabel->False, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellDingbat->"\[LongDash]", CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, AutoStyleOptions->{"UnmatchedBracketStyle"->None}, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Message", "Presentation"], CellMargins->{{80, 10}, {15, 8}}, FontSize->15], Cell[StyleData["Message", "Printout"], CellMargins->{{55, 55}, {0, 3}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{80, 10}, {12, 8}}, FontSize->16], Cell[StyleData["Print", "Printout"], CellMargins->{{54, 72}, {2, 10}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{80, 10}, {10, 10}}, FontSize->16], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, 55}, {0, 15}}, ImageSize->{0.0625, 0.0625}, ImageMargins->{{35, Inherited}, {Inherited, 0}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], CellMargins->{{9, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontSlant->"Oblique"], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{45, Inherited}, {2, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Author", "Presentation"], CellMargins->{{65, 30}, {4, 30}}, FontSize->21], Cell[StyleData["Author", "Printout"], CellMargins->{{36, Inherited}, {2, 30}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Address"], CellMargins->{{45, Inherited}, {2, 2}}, CellGroupingRules->{"TitleGrouping", 30}, PageBreakBelow->False, LineSpacing->{1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->12, FontSlant->"Italic"], Cell[StyleData["Address", "Presentation"], CellMargins->{{65, 30}, {40, 2}}, FontSize->18], Cell[StyleData["Address", "Printout"], CellMargins->{{36, Inherited}, {2, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellMargins->{{45, 75}, {Inherited, 30}}, LineSpacing->{1, 0}], Cell[StyleData["Abstract", "Presentation"], CellMargins->{{65, 30}, {8, 25}}, FontSize->18], Cell[StyleData["Abstract", "Printout"], CellMargins->{{36, 67}, {Inherited, 50}}, Hyphenation->True, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{18, 40}, {2, 2}}, TextJustification->1, LineSpacing->{1, 0}], Cell[StyleData["Reference", "Presentation"], CellMargins->{{25, 40}, {2, 2}}, FontSize->18], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 40}, {Inherited, 0}}, Hyphenation->True, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", 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