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Indeed, these structures have turned out to be very rare and hard \ to find. Basically, only two examples, and slight modifications based on \ them, have been found so far. We have also encountered highly nonlinear \ phenomena which considerably affect our everyday calculations and usually \ make their accomplishment hard. However, quite recently, we have gained new \ insight of why these structures are so very rare. Consequently, the present \ work has a potential to make the future explorations easier. T", StyleBox["he rareness of long words avoiding abelian squares ", FontFamily->"Times New Roman"], "can be explained, at least partly, ", StyleBox["by using the concept of an unfavourable factor. The purpose of \ this paper is to ", FontFamily->"Times New Roman"], "describe", StyleBox[" the utilisation of ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" in searching and suppressing these factors. In principle, the \ same code can be used with slight modifications for other kinds of patterns \ as well.", FontFamily->"Times New Roman"] }], "Abstract", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "In the year 1961, Paul Erd\[ODoubleDot]s [E] raised the question whether \ abelian squares can be avoided (as factors) in infinitely long words (also \ called strings). An abelian square is a non-empty word ", StyleBox["uv", FontSlant->"Italic"], ", where ", StyleBox["u", FontSlant->"Italic"], " and ", StyleBox["v", FontSlant->"Italic"], " are permutations (anagrams) of each other. In 1969, Pleasants [Pl] solved \ positively the question by Erd\[ODoubleDot]s in the case of a five letter \ alphabet, but the four letter case remained open till 1992 when we presented \ an abelian square-free (a-2-free) endomorphism ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], " over the four letter alphabet ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], " = {", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], ", ", StyleBox["d", FontSlant->"Italic"], "}. This endomorphism ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], " was found after long computer experiments, and, until quite recently, \ all known methods for constructing arbitrarily long a-2-free words on ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], " have been based on the structure of this ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], ". " }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "In 2002, after over 11 years of exhaustive searches, we found a completely \ new endomorphism ", Cell[BoxData[ \(TraditionalForm\`g\_98\)]], " of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], "*, the iteration of which produces an infinite abelian square-free word. \ The endomorphism ", Cell[BoxData[ \(TraditionalForm\`g\_98\)]], " is not an a-2-free endomorphism itself, since it does not preserve the \ a-2-freeness of all words of length 7. However, ", Cell[BoxData[ \(TraditionalForm\`g\_98\)]], " can be used together with ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], " to produce a-2-free DT0L-languages of unlimited size. Nowadays, abelian \ square-free words have also found their way to applications, for instance, in \ number theory, algorithmic music, and only recently in cryptography (Rivest \ [R] in 2005)." }], "Text", CellAutoOverwrite->False], Cell[TextData[{ StyleBox["In spite of these findings and ", Background->GrayLevel[1]], "applications", StyleBox[", our understanding of a-2-free words has not advanced very much \ at all contrasted to the situation of ordinary repetition-free words. ", Background->GrayLevel[1]], "Excluding ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], ", Carpi's [C2] modification of it, and ", Cell[BoxData[ \(TraditionalForm\`g\_98\)]], ", every systematic attempt for constructing long abelian square-free \ words over 4 letters has failed. Moreover, all the succesful constructions \ have been found only by exhaustive computerised searches through extremely \ large number of candidates. There is also an important analogous avoidability \ problem for the 3 letter case in which one allows short abelian repetitions \ of the form ", StyleBox["xx", FontSlant->"Italic"], " or ", StyleBox["xxx", FontSlant->"Italic"], " for a letter ", StyleBox["x ", FontSlant->"Italic"], "\[Element]", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_3\)]], " = {", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], "}. This open problem was posed by Sami M\[ADoubleDot]kel\[ADoubleDot] [M\ \[ADoubleDot]] in 2002. " }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "Quite recently, however, we have gained new insight of why these \ structures are so very rare. We are now able to explain, at least partly, t", StyleBox["his rareness of long words avoiding abelian squares by using the \ concept of an unfavourable factor. The purpose of the paper is to ", FontFamily->"Times New Roman"], "describe", StyleBox[" the utilisation of ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" in searching and suppressing these factors. In principle, the \ same code can be used with slight modifications for other kinds of (possibly \ non-abelian) patterns as well.", FontFamily->"Times New Roman"] }], "Text", CellAutoOverwrite->False], Cell["\<\ We have carefully studied the options to make the code efficient with regards \ to running time and the required memory space. At the same time, the \ structures need to be flexible for further development. Indeed, it is \ expected that this code will be run interactively for a quite long time. The \ code is also likely to be transformed into various computational environments \ (such as GRID, C++, FPGA). This work has already been started. In the process \ of these computations, efficiency will still be increasing due to the natural \ reduction process. Indeed, the long lists of words that we now have to use, \ can most likely be considerably reduced as we move on to study longer \ words.\ \>", "Text", CellAutoOverwrite->False], Cell[TextData[{ "The code and some of the structures involved are quite complicated and \ would have been really hard to develope without using ", StyleBox["Mathematica", FontSlant->"Italic"], ". We have been using integer coding and cumulative integer lists for words \ incorporated into quite extensive precomputations. Moreover, certain symbolic \ representations for words and ", StyleBox["Mathematica", FontSlant->"Italic"], "'s pattern matching properties (for lists and strings) have been extremely \ useful for us." }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "At the same time, we feel that it would be really great for the \ programmability and computational efficiency, if in ", StyleBox["Mathematica", FontSlant->"Italic"], " there was an efficient way of saying, for example, that when matching the \ pattern ", StyleBox["{___,u__,v__}", FontFamily->"Courier New", FontWeight->"Bold"], ", we are actually interested only in those cases of ", StyleBox["u", FontFamily->"Courier New", FontWeight->"Bold"], " and ", StyleBox["v", FontFamily->"Courier New", FontWeight->"Bold"], " in which their lengths are equal. Indeed, our words can contain \ thousands of letters, and the absence of restricted pattern matching led us \ to write part of the code in a quite tedious C\[Dash]like fashion. \ Fortunately, to our big relief, debuging turned out to be a comfortable \ process." }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "We define an ", StyleBox["unfavourable", FontSlant->"Italic"], " (or ", StyleBox["forbidden", FontSlant->"Italic"], ") word or factor to be an abelian square-free (a-2-free) word over a fixed \ alphabet \[CapitalSigma] (in our case \[CapitalSigma] = ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], " = {", StyleBox["a", FontSlant->"Italic"], ",", StyleBox["b", FontSlant->"Italic"], ",", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["d", FontSlant->"Italic"], "}, or \[CapitalSigma] = ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_3\)]], " = {", StyleBox["a", FontSlant->"Italic"], ",", StyleBox["b", FontSlant->"Italic"], ",", StyleBox["c", FontSlant->"Italic"], "} in which case we allow ", StyleBox["xx", FontSlant->"Italic"], " or ", StyleBox["xxx", FontSlant->"Italic"], " for a letter ", StyleBox["x", FontSlant->"Italic"], ") to be an a-2-free word over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " which cannot anyhow occur as a proper factor inside any infinite \ a-2-free word. That is to say, over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", an unfavourable a-2-free word cannot be continued infinitely long to \ the left and to the right without necessarily creating an abelian square at \ some point. However, it might well be possible to extend such a word \ boundlessly to one direction, say to the right, without producing any abelian \ squares. Experiments support this conjecture but the existence of such \ unfavourable factors remains an open question." }], "Text", CellAutoOverwrite->False, TextAlignment->Left, TextJustification->1], Cell[TextData[{ "In what follows, we explain shortly our search for unfavourable factors. \ Let the alphabet \[CapitalSigma] be fixed and consider words over it. We \ take a word (finally we actually need to consider all the a-2-free words of a \ given length) and try to extend it in a-2-free fashion to the right and to \ the left with all possible ways up to a given upper bound for the total \ length. At a time, the length of the word increases only by a given fixed \ length. We extend alternately to right and left, and backtrack if need be. If \ the upper bounds are reached then the original word is a ", StyleBox["so-far-favourable", FontSlant->"Italic"], " one (it may still turn out to be unfavourable on later experiments). If \ there is no way to reach the upper bounds, then the original word is \ classified, without any doupt, to be unfavourable. Thus for a given length we \ obtain three kinds of words: ", StyleBox["unfavourable ", FontSlant->"Italic"], "(", StyleBox["bad", FontSlant->"Italic"], "), ", StyleBox["so-far-favourable ", FontSlant->"Italic"], "(", StyleBox["so-far-so-good", FontSlant->"Italic"], "), and ", StyleBox["favourable ", FontSlant->"Italic"], "(", StyleBox["good", FontSlant->"Italic"], "). At present, the latter type consists of words occurring as factor in \ a-2-free words obtained by using ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], ", ", Cell[BoxData[ \(TraditionalForm\`g\_98\)]], ", and Carpi's modification of ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], "." }], "Text", CellAutoOverwrite->False, TextAlignment->Left, TextJustification->1], Cell[TextData[{ "It is a remarkable phenomenon that already relatively short \ so-far-favourable words turn out to be unfavourable factors after being \ 'safely' extendable (to right and left) for quite a long distance (and with a \ huge number of branches). One might have expected the quite long buffers to \ be sufficient for further grow. Due to Carpi [C2], we know that the number \ of a-2-free words over four letters grows exponentially (\[GreaterEqual] ", Cell[BoxData[ \(TraditionalForm\`1.000021\^n\)]], ") with respect to word length ", StyleBox["n. ", FontSlant->"Italic"], "But how do these words grow then? We conjecture that in spite of the \ exponential growth, the ratio between the number of properly extendable, \ i.e., favourable, words of length ", StyleBox["n", FontSlant->"Italic"], ", and between the number of unfavourable factors of the same length, \ actually tends to zero when the length ", StyleBox["n ", FontSlant->"Italic"], " tends to infinity! Thus we suspect that the vast majority of a-2-free \ words over four (and, arguably, three) letters cannot occur as proper factors \ in the middle of very long (infinite) words. In a way, this would also \ explain why it is so extremely difficult to find a-2-free endomorphisms over \ four letters. At present we know, for example, that just a little bit over \ half (50.5737 %) of the a-2-free words over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], " of length 20 are indeed unfavourable. In the future, this and other \ similar observations could lead to discovery of new (now so hard to find) \ abelian square-free endomorphisms over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], ". This, in turn, could lead to a better understanding of the structures \ involved. For example, the new a-2-f endomorphisms could allow to establish \ sharper bounds for the growth of the number of a-2-f words over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_4\)]], ". " }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "As an example, in the case of 4 letters, the following words together with \ their permutations and mirror images are unfavourable (forbidden) factors:\n\n\ ", StyleBox["unfavourableFactorsOfLength8 = \ {\"abacdaba\",\"abacdbab\",\"abcabdab\",\"abcabdcb\",\"abcadbad\"};", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox["\n\n", FontSize->9], StyleBox["someUnfavourableFactorsOfLength20 = ", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox["\n\ {\"abacabadbacbcdbacdbd\",\"abacabdcadcdbcbacbdc\",\"abacadcabcbdcabdcdad\",\"\ abacdabcbdbcadabdacd\", \n\ \"abcabadbcadbdcbcabcd\",\"abcabadcbadbdcbcabcd\",\"abcacdabcdadbabcabda\",\"\ abcacdabdcadbabcabda\",\n\ \"abcacdbacdadbabcabda\",\"abcacdcbabdcdacdcbad\",\"abcbabcdcacdbdabacbc\",\"\ abcbabdabcacdabcdadb\",\n\ \"abcbadbcacdcabdbcdba\",\"abcbadcbdcdacabcadcb\",\"abcbdbcbacbdcdacbdac\",\"\ abcdbdadcbcadabdadcb\"};", FontFamily->"Courier New", FontSize->10, FontWeight->"Bold"] }], "Text", CellAutoOverwrite->False, TextAlignment->Left, TextJustification->0], Cell[TextData[{ "The latter words form a very interesting case, leading to a million-fold \ fluctuational phenomenon (with respect to needed running time, and memory \ space, if we wish to see the tree of all possibilities). This, actually quite \ frequently appearing, case is illustrated (in a bit different setting than \ the one we use elsewhere in this paper) online at [", ButtonBox["K5", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/research/ForbiddenFactors.html"], None}, ButtonStyle->"Hyperlink"], "]. For example, there is the following illustration for ", StyleBox["\"abcbdbcbacbdcdacbdac\"", FontFamily->"Courier New", FontWeight->"Bold"], ". 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The length of the words increases only by one letter at a time \ (alternately to right and left). All possible words are gathered to a list \ and the number of words are plotted according to the length of the words. For \ each word there are four possibilities: either it can be continued with 3, \ 2, 1, or 0 letters. The case of 0 letters means the word cannot be \ continued (in an a-2-f way) at all. If no words can be continued, then we get \ an empty list and the computation ends. The resulting list consists merely of \ unfavourable (or bad, or forbidden) factors that start from the originally \ given word, and this starting word is 'in the middle' of every non-empty word \ in the list. Indeed, all the computations for the words in the list of ", StyleBox["someUnfavourableFactorsOfLength20", FontFamily->"Courier New", FontWeight->"Bold"], ", together with a great many other similar words, do end with an empty \ list (i.e. with a dead end) even though the width of the bidirectional-tree, \ and the computational time, can be quite huge. All this can make finding of \ unfavourable factors really challenging. 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even be cut \ shorther from the right or left ends to (shortened) factors which, ", StyleBox["after a ", FontFamily->"Times New Roman"], StyleBox["transitional phase", CellAutoOverwrite->False, FontFamily->"Times New Roman"], StyleBox[",", FontFamily->"Times New Roman"], " have the same final trajectory." }], "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Preliminaries", FontWeight->"Bold"], " " }], "Section"], Cell["\<\ In this section we present some mathematical notations and terminology. Our \ terminology is more or less standard in the field of combinatorics on words. \ Consequently, the reader might consult this section later, if need arises. \ \>", "Text", CellAutoOverwrite->False], Cell[TextData[{ "An ", StyleBox["alphabet", FontSlant->"Italic"], " \[CapitalSigma] is a finite non-empty set of abstract symbols called \ ", StyleBox["letters", FontSlant->"Italic"], ". A ", StyleBox["word", FontSlant->"Italic"], " (", StyleBox["string", FontSlant->"Italic"], ") over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " is a finite (unless otherwise indicated) string, or sequence, of letters \ belonging to ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". The set of all words [non-empty words] over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " is denoted by ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " [", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^+\)\)]], "]. On the ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], ", the associative binary operation of ", StyleBox["catenation", FontSlant->"Italic"], " is defined. For words ", StyleBox["u ", FontSlant->"Italic"], "and ", StyleBox["v", FontSlant->"Italic"], ", it is the juxtaposition ", StyleBox["uv", FontSlant->"Italic"], ". The ", StyleBox["empty word", FontSlant->"Italic"], ", which is the neutral element of catenation, is denoted by ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)]], ". The algebraic structures ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^+\)\)]], " are called, respectively, the free monoid and the free semigroup \ generated by ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". " }], "Text"], Cell[TextData[{ "Let ", StyleBox["w", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[CenterEllipsis]\)]], " ", Cell[BoxData[ \(TraditionalForm\`x\_m\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\_i\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Element] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". The ", StyleBox["length", FontSlant->"Italic"], " of the word ", StyleBox["w", FontSlant->"Italic"], ", denoted by |", StyleBox[" ", FontSize->7], StyleBox["w", FontSlant->"Italic"], StyleBox[" ", FontSize->7, FontSlant->"Italic"], "|, is the number of occurrences of letters in ", StyleBox["w", FontSlant->"Italic"], ", i.e., |", StyleBox[" ", FontSize->7], StyleBox["w", FontSlant->"Italic"], StyleBox[" ", FontSize->7, FontSlant->"Italic"], "| = ", StyleBox["m", FontSlant->"Italic"], ". Let ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " = { ", Cell[BoxData[ \(TraditionalForm\`a\_1\)]], ", \[Ellipsis] , ", Cell[BoxData[ \(TraditionalForm\`a\_n\)]], "}. The number of occurrences of one letter ", StyleBox["x", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\` \[Element] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " in ", StyleBox["w", FontSlant->"Italic"], " is denoted by ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(w\)\( | \_x\)\)\)]], ", or simply by ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(w\)\( | \_i\)\)\)]], " if ", StyleBox["x", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`a\_i\)]], ". The notation ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_\[CapitalSigma]\)]], "(", StyleBox["w", FontSlant->"Italic"], ") stands for the ", StyleBox["Parikh vector", FontSlant->"Italic"], " of ", StyleBox["w", FontSlant->"Italic"], ", i.e., ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_\[CapitalSigma]\)]], "(", StyleBox["w", FontSlant->"Italic"], ") = (", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(w\)\( | \_1\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Ellipsis]\)]], " , ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(w\)\( | \_n\)\)\)]], "). Usually we will omit the subscript ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " and write simply ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\)]], " instead of ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_\[CapitalSigma]\)]], ". Quite interchangeably with the Parikh vector notation also formal sums \ ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_s\)]], "(", StyleBox["w", FontSlant->"Italic"], ") = ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["\[Sum]", RowBox[{"x", "\[Element]", RowBox[{"{", RowBox[{ FormBox[\(a\_1\), "TraditionalForm"], ",", "\[Ellipsis]", ",", " ", FormBox[\(a\_n\), "TraditionalForm"]}], "}"}]}]], " ", RowBox[{ FormBox[\(k\_x\), "TraditionalForm"], " ", StyleBox["x", FontSlant->"Italic"]}]}], TraditionalForm]]], ", with ", Cell[BoxData[ \(TraditionalForm\`k\_x\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Element] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalN]\)]], ", are used. For example ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_s\)]], "(", StyleBox["abacaba", FontSlant->"Italic"], ") = 4", StyleBox["a", FontSlant->"Italic"], " + 2", StyleBox["b", FontSlant->"Italic"], " + ", StyleBox["c", FontSlant->"Italic"], ". Thus ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_s\)]], "(", StyleBox["w", FontSlant->"Italic"], "), with ", StyleBox["w ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\` \[Element] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", is an element of the abelian free monoid ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalN]\^\[CapitalSigma]\)]], " generated by ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". We will also consider differences of Parikh vectors and differences of \ formal sums. Consequently, these vectors and sums are extended into elements \ of the abelian free group ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\^\[CapitalSigma]\)]], " generated by ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". The neutral element of ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\^\[CapitalSigma]\)]], " is denoted by ", Cell[BoxData[ FormBox[ StyleBox["0", FontWeight->"Bold"], TraditionalForm]]], ".\t" }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "A word ", StyleBox["u", FontSlant->"Italic"], " is called a ", StyleBox["factor", FontSlant->"Italic"], " (some authors call it ", StyleBox["subword", FontSlant->"Italic"], ")", " of a word ", StyleBox["w", FontSlant->"Italic"], ", if ", StyleBox["w", FontSlant->"Italic"], " = ", StyleBox["p", FontSlant->"Italic"], StyleBox[" ", FontSize->7, FontSlant->"Italic"], StyleBox["u", FontSlant->"Italic"], StyleBox[" ", FontSize->7, FontSlant->"Italic"], StyleBox["s ", FontSlant->"Italic"], "for some words ", StyleBox["p", FontSlant->"Italic"], " and ", StyleBox["s", FontSlant->"Italic"], ". If ", StyleBox["p", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)]], " [", StyleBox["s", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)]], "], then ", StyleBox["u", FontSlant->"Italic"], " is called a ", StyleBox["prefix", FontSlant->"Italic"], " [a ", StyleBox["suffix", FontSlant->"Italic"], "] of ", StyleBox["w", FontSlant->"Italic"], ". " }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "Let ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\` \[GreaterEqual] \)]], " 2 be a given integer. A ", StyleBox["k-repetition", FontSlant->"Italic"], " [an ", StyleBox["abelian k-repetition", FontSlant->"Italic"], "] is a non-empty word of the form ", Cell[BoxData[ \(TraditionalForm\`R\^k\)]], " [", Cell[BoxData[ \(TraditionalForm\`P\_1\)]], Cell[BoxData[ \(TraditionalForm\`\[CenterEllipsis]\)]], Cell[BoxData[ \(TraditionalForm\`P\_k\)]], ", where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Psi]", "(", FormBox[\(P\_\[Mu]\), "TraditionalForm"], ")"}], " ", "=", " ", RowBox[{"\[Psi]", "(", FormBox[\(P\_\[Nu]\), "TraditionalForm"], ")"}]}], TraditionalForm]]], " for all ", Cell[BoxData[ \(TraditionalForm\`1\ \[LessEqual] \ \[Mu]\ < \ \[Nu]\ \[LessEqual] \ \ k\)]], ", i.e., ", Cell[BoxData[ \(TraditionalForm\`P\_i\)]], ":s are permutations of each other]. Instead of [abelian] 2- and \ 3-repetitions, terms [", StyleBox["abelian", FontSlant->"Italic"], "] ", StyleBox["squares", FontSlant->"Italic"], " and [", StyleBox["abelian", FontSlant->"Italic"], "] ", StyleBox["cubes", FontSlant->"Italic"], " are often used. A word or an ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], "-word (explained below) is called ", StyleBox["k-repetition free", FontSlant->"Italic"], " [", StyleBox["abelian k-repetition free", FontSlant->"Italic"], ", or ", StyleBox["k-free in the abelian sense", FontSlant->"Italic"], "], or in short ", StyleBox["k-free", FontSlant->"Italic"], " [", StyleBox["a-k-free", FontSlant->"Italic"], "], if it does not contain any ", StyleBox["k", FontSlant->"Italic"], "-repetition [abelian ", StyleBox["k", FontSlant->"Italic"], "-repetition] as a factor. A word sequence or a word set is ", StyleBox["k-free", FontSlant->"Italic"], " [", StyleBox["a-k-free", FontSlant->"Italic"], "], if all words in it are ", StyleBox["k", FontSlant->"Italic"], "-free [a-", StyleBox["k", FontSlant->"Italic"], "-free]. If, for a fixed ", StyleBox["k", FontSlant->"Italic"], ", it is possible to construct arbitrarily long (infinite) a-", StyleBox["k", FontSlant->"Italic"], "-free (or other pattern-free) words over a given alphabet ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", then we say that abelian ", StyleBox["k", FontSlant->"Italic"], "-repetitions (or those patterns) are ", StyleBox["avoidable", FontSlant->"Italic"], " over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". " }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "A ", StyleBox["morphism", FontSlant->"Italic"], " ", StyleBox["h", FontSlant->"Italic"], " is a mapping between free monoids ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalDelta]\^*\)\)]], " with ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["uv", FontSlant->"Italic"], ") = ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["u", FontSlant->"Italic"], ")", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["v", FontSlant->"Italic"], ") for every ", StyleBox["u", FontSlant->"Italic"], " and ", StyleBox["v", FontSlant->"Italic"], " in ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], ". Especially, ", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)]], ". A morphism ", StyleBox["h", FontSlant->"Italic"], ": ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Rule] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalDelta]\^*\)\)]], ", being compatible with the catenation of words, is uniquely defined, if \ the word ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") ", Cell[BoxData[ \(TraditionalForm\` \[Element] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalDelta]\^*\)\)]], " is (effectively) given for each ", Cell[BoxData[ \(TraditionalForm\`x\ \[Element] \ \[CapitalSigma]\)]], ". If ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\ = \ \[CapitalSigma]\)]], ", we call ", StyleBox["h ", FontSlant->"Italic"], " an ", StyleBox["endomorphism", FontSlant->"Italic"], " (and usually write ", StyleBox["g ", FontSlant->"Italic"], " instead of ", StyleBox["h", FontSlant->"Italic"], "). For a morphism ", StyleBox["h", FontSlant->"Italic"], " and a language ", StyleBox["L", FontSlant->"Italic"], " we define ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["L", FontSlant->"Italic"], ") = {", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["w", FontSlant->"Italic"], ") | ", Cell[BoxData[ \(TraditionalForm\`w\ \[Element] \ L\)]], "}. A morphism ", StyleBox["h", FontSlant->"Italic"], " is termed ", StyleBox["uniformly growing", FontSlant->"Italic"], ", if |", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")| = |", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")| ", Cell[BoxData[ \(TraditionalForm\` \[GreaterEqual] \)]], " 2 for every ", StyleBox["x", FontSlant->"Italic"], " and ", Cell[BoxData[ \(TraditionalForm\`y\ \[Element] \ \[CapitalSigma]\)]], "." }], "Text"], Cell[TextData[{ "For a given integer ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\` \[GreaterEqual] \)]], " 2, a morphism ", StyleBox["h", FontSlant->"Italic"], ": ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Rule] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalDelta]\^*\)\)]], " is called ", StyleBox["k-free", FontSlant->"Italic"], " [", StyleBox["a-k-free", FontSlant->"Italic"], "], if ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["w", FontSlant->"Italic"], ") is ", StyleBox["k", FontSlant->"Italic"], "-free [a-", StyleBox["k", FontSlant->"Italic"], "-free] for every ", StyleBox["k", FontSlant->"Italic"], "-free [a-", StyleBox["k", FontSlant->"Italic"], "-free] word ", Cell[BoxData[ FormBox[ RowBox[{"w", " ", "\[Element]", " ", FormBox[\(\[CapitalSigma]\^*\), "TraditionalForm"]}], TraditionalForm]]], ".\t" }], "Text"], Cell[TextData[{ "With regards to ", StyleBox["L-systems", FontSlant->"Italic"], " (Aristid Lindenmayer 1925", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "1989), we specify the following concepts. A ", StyleBox["D0L-system", FontSlant->"Italic"], " is a triple ", StyleBox["G", FontSlant->"Italic"], " = (", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", ", StyleBox["g", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), where ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], " is an alphabet, ", StyleBox["g", FontSlant->"Italic"], ": ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Rule] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " is an endomorphism, and ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ", called the ", StyleBox["axiom", FontSlant->"Italic"], ", is a word over ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". The (", StyleBox["word", FontSlant->"Italic"], ") ", StyleBox["sequence", FontSlant->"Italic"], " ", StyleBox["S", FontSlant->"Italic"], "(", StyleBox["G", FontSlant->"Italic"], ") generated by ", StyleBox["G", FontSlant->"Italic"], " consists of the words\n\t\n\t\t", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], " = ", Cell[BoxData[ \(TraditionalForm\`g\^0\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), ", Cell[BoxData[ \(TraditionalForm\`g\^1\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), ", Cell[BoxData[ \(TraditionalForm\`g\^2\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), ", Cell[BoxData[ \(TraditionalForm\`g\^3\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), ... , \n\nwhere ", Cell[BoxData[ \(TraditionalForm\`g\^i\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ") = ", StyleBox["g", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`g\^\(i - 1\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ")) for ", Cell[BoxData[ \(TraditionalForm\`i\ \[GreaterEqual] \ 1\)]], ". The ", StyleBox["language", FontSlant->"Italic"], " of ", StyleBox["G", FontSlant->"Italic"], " is defined by ", StyleBox["L", FontSlant->"Italic"], "(", StyleBox["G", FontSlant->"Italic"], ") = {", Cell[BoxData[ \(TraditionalForm\`g\^i\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ") | ", Cell[BoxData[ \(TraditionalForm\`i\ \[GreaterEqual] \ 0\)]], "}. Languages [sequences] defined by a D0L-system are referred to as ", StyleBox["D0L-languages", FontSlant->"Italic"], " [", StyleBox["D0L-sequences", FontSlant->"Italic"], "]. D0L-systems provide a very convenient way for defining languages and \ infinite words. Furthermore, if ", StyleBox["g", FontSlant->"Italic"], " and ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], " are ", StyleBox["k", FontSlant->"Italic"], "-free [a-", StyleBox["k", FontSlant->"Italic"], "-free], then the iteration of ", StyleBox["g", FontSlant->"Italic"], " will yield a ", StyleBox["k", FontSlant->"Italic"], "-free [a-", StyleBox["k", FontSlant->"Italic"], "-free] D0L-sequence. An ", StyleBox["HD0L-system", FontSlant->"Italic"], " is a 5-tuple ", Cell[BoxData[ \(TraditionalForm\`G\_1\)]], " = (", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], ", ", StyleBox["g", FontSlant->"Italic"], ", ", StyleBox["h", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), where (", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", ", StyleBox["g", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ") is a D0L-system, called the underlying D0L-system of ", Cell[BoxData[ \(TraditionalForm\`G\_1\)]], ", \[CapitalDelta] is an alphabet, and ", StyleBox["h", FontSlant->"Italic"], ": ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Rule] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalDelta]\^*\)\)]], " is a morphism. The ", StyleBox["HD0L-sequence", FontSlant->"Italic"], " ", StyleBox["S", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`G\_1\)]], ") generated by ", Cell[BoxData[ \(TraditionalForm\`G\_1\)]], " consists of the words\n\n\t\t", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ") = ", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`g\^0\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ")), ", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`g\^1\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ")), ", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`g\^2\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ")), ", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`g\^3\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ")), ... , \n\nand the ", StyleBox["HD0L-language ", FontSlant->"Italic"], "of ", Cell[BoxData[ \(TraditionalForm\`G\_1\)]], " is the set ", StyleBox["L", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`G\_1\)]], ") = {", StyleBox["h", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`g\^i\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ")) | ", Cell[BoxData[ \(TraditionalForm\`i\ \[GreaterEqual] \ 0\)]], "}. A ", StyleBox["DT0L", FontSlant->"Italic"], StyleBox["-system", FontSlant->"Italic"], " is a triple ", Cell[BoxData[ \(TraditionalForm\`G\_2\)]], " = (", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", ", StyleBox["H", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], "), where ", StyleBox["H", FontSlant->"Italic"], " is a finite non-empty set of morphisms (called tables) and (", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", ", StyleBox["h", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], ") is a D0L-system for every ", Cell[BoxData[ \(TraditionalForm\`h\ \[Element] \ H\)]], ". The ", StyleBox["DT0L-language ", FontSlant->"Italic"], "of ", Cell[BoxData[ \(TraditionalForm\`G\_2\)]], " is the set ", StyleBox["L", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`G\_2\)]], ") = {", StyleBox["w ", FontSlant->"Italic"], "| ", StyleBox["w", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], " or ", StyleBox["w", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(h\_k\), "TraditionalForm"], "\[CenterEllipsis]", RowBox[{ FormBox[\(h\_1\), "TraditionalForm"], "(", FormBox[\(\[Alpha]\_0\), "TraditionalForm"], ")"}]}], TraditionalForm]]], ", where the compositions ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(h\_k\), "TraditionalForm"], "\[CenterEllipsis]", FormBox[\(h\_1\), "TraditionalForm"]}], TraditionalForm]]], " of morphims are constructed from ", Cell[BoxData[ \(TraditionalForm\`h\_1\)]], ", ..., ", Cell[BoxData[ \(TraditionalForm\`h\_k\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Element] \)]], " ", StyleBox["H", FontSlant->"Italic"], "}. Obviously, a DT0L-system can be regarded as a D0L-system, when ", StyleBox["H", FontSlant->"Italic"], " contains only one (endo)morphism. For a thorough discussion of various \ L-systems we refer the reader to Rozenberg and Salomaa [RS]." }], "Text"], Cell[TextData[{ "An ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], "-", StyleBox["word", FontSlant->"Italic"], " is an infinite sequence, from left to right, of letters of an alphabet \ ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". Thus an ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], "-word can be identified with a mapping of ", Cell[BoxData[ \(TraditionalForm\`\(\[DoubleStruckCapitalN]\_+\)\)]], " into ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ". One can construct an ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], "-word, for example, by iterating an endomorphism ", StyleBox["g", FontSlant->"Italic"], ": ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[Rule] \)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^*\)\)]], " such that ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\ \[NotElement] \ g(\[CapitalSigma])\)]], " and ", StyleBox["g", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = ", StyleBox["xw ", FontSlant->"Italic"], " for some ", Cell[BoxData[ \(TraditionalForm\`x\ \[Element] \ \[CapitalSigma]\)]], ", ", Cell[BoxData[ FormBox[ RowBox[{"w", " ", "\[Element]", " ", FormBox[\(\[CapitalSigma]\^+\), "TraditionalForm"]}], TraditionalForm]]], ". Such a morphism ", StyleBox["g", FontSlant->"Italic"], " is called ", StyleBox["prefix preserving", FontSlant->"Italic"], " for the reason that ", Cell[BoxData[ \(TraditionalForm\`g\^i\)]], "(", StyleBox["x", FontSlant->"Italic"], ") is a proper prefix of ", Cell[BoxData[ \(TraditionalForm\`g\^\(i + 1\)\)]], "(", StyleBox["x", FontSlant->"Italic"], ") whenever ", StyleBox["i", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\` \[GreaterEqual] \)]], " 0. An ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], "-word is obtained as the \"limit\" of the sequence ", Cell[BoxData[ \(TraditionalForm\`g\^i\)]], "(", StyleBox["a", FontSlant->"Italic"], "); ", StyleBox["i", FontSlant->"Italic"], " = 0, 1, 2, \[Ellipsis] . " }], "Text", TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Suppression", StyleBox[" ", FontSize->14], "of", StyleBox[" ", FontSize->14], "Unfavourable", StyleBox[" ", FontSize->14], "Factors", StyleBox[" ", FontSlant->"Italic"], "with", StyleBox[" Mathematica", FontSlant->"Italic"], " " }], "Section", CellAutoOverwrite->False], Cell[TextData[{ "In this section we give examples of the developed ", StyleBox["Mathematica", FontSlant->"Italic"], " program. The correctness of the code and the related packages has been \ tested but exhaustive computer runs are still likely to take a long time in \ the future. The full code and further versions of it can be downloaded from \ ", ButtonBox["[K6]", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/research/\ UnfavourableFactorsInPatternAvoidance/Programs&Notebooks&Packages.html"], None}, ButtonStyle->"Hyperlink"], ", or from ", ButtonBox["[K7], ", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/StructuresGraphicsMusic.html"], None}, ButtonStyle->"Hyperlink"], "the latter of which is a general link page for the topic. In addition, the \ code (in a partly preliminary form) is attached at the end of this notebook, \ and the reader is invited to make experiments with it. For this purpose, one \ may copy and edit the Input Cells of the examples presented below. The code \ consists of Initialization Cells and will be activated automatically. \ Nevertheless, it should be noted that, intially, only the 4 letter case can \ be tried out. The 3 letter case requires the user to activate the very last \ Input Cell of this notebook which lies inside the last section." }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "In the following examples, many of the function definitions are omitted \ (they can be found at the end). It should be noted that all the structures, \ variables and constants starting with ", StyleBox["\[Euro]", FontFamily->"Courier New", FontWeight->"Bold"], " are global. For example, the global variable ", StyleBox["\[Euro]state", FontFamily->"Courier New", FontWeight->"Bold"], " represents the state of the construction. Its values are strings, \ including, as an example, the following ones:", StyleBox[" ", FontFamily->"Courier New"], StyleBox["\"extendOrChangeRight\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"extendOrChangeLeft\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"testRight\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"testLeft\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"failBoth\"", FontFamily->"Courier New", FontWeight->"Bold"], ", and ", StyleBox["\"succeeded\"", FontFamily->"Courier New", FontWeight->"Bold"], "." }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "As mentioned in the Introduction, we firstly fix the alphabet \ \[CapitalSigma] and consider words over it. We take a word (in the final \ investigation, we will actually take all the a-2-free words of a given \ length) and try to extend it in a-2-free fashion to the right and to the left \ with all possible ways up to a given upper bound for the total length. At a \ time, the length of the word increases only by a given fixed length. We \ extend alternately to right and left, and backtrack when necessary. If the \ upper bounds are reached then the original word is a ", StyleBox["so-far-favourable", FontSlant->"Italic"], " one. If there is no way to reach the upper bounds, then the original word \ is classified, without any doupt, to be ", StyleBox["unfavourable", FontSlant->"Italic"], ". At present, the ", StyleBox["favourable ", FontSlant->"Italic"], "words (4 letter case) consist of only those occurring as factors in \ a-2-free words obtained by using the endomorphims ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], ", ", Cell[BoxData[ \(TraditionalForm\`g\_98\)]], ", and Carpi's modification of ", Cell[BoxData[ \(TraditionalForm\`g\_85\)]], "." }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "In the final program we use integer coding for letters of the alphabet \ \[CapitalSigma] and cumulative integer lists for words. This makes the \ detection of abelian squares fast. We use two different cumulative integer \ lists, ", StyleBox["\[Euro]cumulIntListRight ", FontFamily->"Courier New", FontWeight->"Bold"], " for the right hand, and ", StyleBox["\[Euro]cumulIntListLeft", FontFamily->"Courier New", FontWeight->"Bold"], " for the left hand extensions. This makes addition of integers (addition \ of cumulative integer lists, in fact) fast but forces us to write the testing \ function ", StyleBox["testA2RLpairCumulIntList", FontFamily->"Courier New", FontWeight->"Bold"], " in a more complicated fashion (nevertheless, still maintaining the \ necessary high speed). In building all the structures, quite extensive \ precomputations are needed." }], "Text", CellAutoOverwrite->False], Cell["\<\ Fistly, let us define the alphabets by using letters (strings of length one) \ and integers:\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[{ \(\(\[Euro]alphTwoLet = {"\", "\"};\)\), "\[IndentingNewLine]", \(\(\[Euro]alphTwoInt = {0, 1};\)\), "\[IndentingNewLine]", \(\(\[Euro]alphThreeLet = {"\", "\", "\"};\)\), "\ \[IndentingNewLine]", \(\[Euro]alphThreeInt = {0, 1, 2\^16}; \ \ \ (*\ Words\ of\ length\ \[LessEqual] \ \((2\^16 - 1)\)*4/3\ = \ 87380\ \[IndentingNewLine]are\ safe\ to\ use\ - \ provided\ that\ they\ do\ not\ contain\ xxxx\ for\ a\ letter\ x\ in\ \ \[Euro]alphThreeLet\ *) \[IndentingNewLine]\[Euro]alphFourLet = {"\", "\ \", "\", "\"};\), "\[IndentingNewLine]", \(\[Euro]alphFourInt = {0, 1, 2\^10, 2\^20}; \ \ \ (*\ Words\ of\ length\ \[LessEqual] \ \((2\^10 - 1)\)*2\ = \ 2046\ \[IndentingNewLine]are\ safe\ to\ use\ - \ provided\ that\ they\ do\ not\ contain\ xx\ for\ a\ letter\ x\ in\ \ \[Euro]alphFourLet\ *) \[IndentingNewLine]\[Euro]alphThreeLetInt = \ {\[Euro]alphTwoLet, \[Euro]alphTwoInt};\), "\[IndentingNewLine]", \(\(\[Euro]alphThreeLetInt = {\[Euro]alphThreeLet, \ \[Euro]alphThreeInt};\)\), "\[IndentingNewLine]", \(\(\[Euro]alphFourLetInt = {\[Euro]alphFourLet, \ \[Euro]alphFourInt};\)\)}], "Input", CellLabel->"", CellAutoOverwrite->False], Cell["\<\ The first example of the use of cumulative Parikh vectors is a symbolic one \ (Parikh vector is explained in the Preliminaries section): \ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(convertStrToCumulIntList["\", \ {{"\", "\", \ "\"}, {"\", "\", "\"}}]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({"b", "a" + "b", "a" + "b" + "c", 2\ "a" + "b" + "c", 2\ "a" + 2\ "b" + "c", 2\ "a" + 2\ "b" + 2\ "c", 3\ "a" + 2\ "b" + 2\ "c", 3\ "a" + 2\ "b" + 3\ "c", 4\ "a" + 2\ "b" + 3\ "c", 4\ "a" + 3\ "b" + 3\ "c"}\)], "Output"] }, Open ]], Cell["\<\ In actual computations, we use integer representation for these cumulative \ Parikh vectors: \ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(convertStrToCumulIntList["\", \ \ \[Euro]alphThreeLetInt]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({1, 1, 65537, 65537, 65538, 131074, 131074, 196610, 196610, 196611}\)], "Output"] }, Open ]], Cell["\<\ Transforming back to the string representation over three letters:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(convertCumulIntListToStr[%, \[Euro]alphThreeLetInt]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \("bacabcacab"\)], "Output"] }, Open ]], Cell["The case of four letters can be handled in a similar way:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(convertStrToCumulIntList["\", \[Euro]alphFourLetInt]\)], \ "Input", CellAutoOverwrite->False], Cell[BoxData[ \({1, 1, 1025, 1025, 1026, 1049602, 1049602, 1050626, 1050626, 2099202}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(convertCumulIntListToStr[%, \[Euro]alphFourLetInt]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \("bacabdacad"\)], "Output"] }, Open ]], Cell[TextData[{ "When extending the words (to right or left) the program looks at the \ suffix ", StyleBox["s", FontSlant->"Italic"], " (of fixed length) of the constructed word and then catenates a new word \ ", StyleBox["w", FontSlant->"Italic"], " of a fixed length to it. This new word ", StyleBox["w", FontSlant->"Italic"], " is selected, in an orderly fashion, from a precomputed list for ", StyleBox["s", FontSlant->"Italic"], " and it is always good, firstly, in the sense that no abelian squares are \ produced in the new suffix ", StyleBox["sw", FontSlant->"Italic"], ", and, secondly, in the sense that ", StyleBox["sw", FontSlant->"Italic"], " itself is never an ", "unfavourable", " (i.e. bad) factor. However, new longer abelian squares can appear in the \ whole structure in which case the catenation is not successful and the next \ candidate ", StyleBox["w", FontSlant->"Italic"], ", if it exists, will be tried out. The process is depicted in Figure 1. \ We have also taken care that the selection of the next proper candidate is \ done in an efficient way. If all the possible candidates for ", StyleBox["w", FontSlant->"Italic"], " have already been tried out, then the program backtracks to change the \ other end's suffix. Indeed, the process is the same for for the right and \ left hand extensions (both of their cumulative list representations grow from \ left to right), so we can really speak of suffixes only. The lengths for the \ suffix ", StyleBox["s", FontSlant->"Italic"], " and the extension ", StyleBox["w", FontSlant->"Italic"], " need to be fixed at the beginning of the computation, and changing \ (re-fixing) the lengths usually requires new quite extensive precomputations. \ In the code at the end of this notebook, the lengths have been selected to be \ | ", StyleBox["s ", FontSlant->"Italic"], "| = 4 and | ", StyleBox["w", FontSlant->"Italic"], " | = 4, but they can be, and usually are, selected differentely as well. \ Up to now, we have been using e.g. the lengths | ", StyleBox["s ", FontSlant->"Italic"], "| = 8, | ", StyleBox["w", FontSlant->"Italic"], " | = 4, and | ", StyleBox["s ", FontSlant->"Italic"], "| = 12, | ", StyleBox["w", FontSlant->"Italic"], " | = 4. Especially, longer suffixes ", StyleBox["s", FontSlant->"Italic"], " would increase the computaional time efficiency considerably but, on the \ other hand, the structures might need too much memory in our present \ environment for distributed computing. Moreover, selecting | ", StyleBox["w", FontSlant->"Italic"], " | = 1 would allow the maximal avoidance of ", "unfavourable", " factors in the extensions. However, the setting | ", StyleBox["w", FontSlant->"Italic"], " | > 1 probably allows to detect the abelian squares more quickly, albeit \ the final decision of this still requires quite extensive experiments. " }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "\nFigure 1:\n\t\t\[LongLeftArrow] \[Ellipsis] ", Cell[BoxData[ FormBox[ RowBox[{"|", FormBox[\(\[LongDash]\[LongDash]\&\(w\_6\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"|", FormBox[\(\[LongDash]\[LongDash]\&\(w\_4\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"|", FormBox[\(\[LongDash]\[LongDash]\&\(w\_2\)\), "TraditionalForm"], "|"}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[LongDash]\[LongDash]\[LongDash]\[LongDash]\ \[LongDash]\[LongDash]\[LongDash]\[LongDash]\[LongDash]\[LongDash]\&\(given\ \ word\ for\ the\ test\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(\[LongDash]\[LongDash]\&\(w\_1\)\)\)\)]], Cell[BoxData[ FormBox[ UnderscriptBox[ RowBox[{"|", FormBox[\(\[LongDash]\[LongDash]\&\(w\_3\)\), "TraditionalForm"], "|", FormBox[\(\[LongDash]\[LongDash]\&\(w\_5\)\), "TraditionalForm"], "|"}], \(\(\(\ \ \)\(\(|\)\(\[Dash]\[LongDash]\[LongDash]\[LongDash]\[LongDash]\[LongDash]\ \[Dash]\[Dash]\)\(|\)\)\)\+\(\(\ \)\(s\)\)\)], TraditionalForm]]], Cell[BoxData[ FormBox[ UnderscriptBox[" ", AdjustmentBox[ UnderscriptBox[\(\(|\)\(\[LongDash]\[LongDash]\[LongDash]\)\(|\)\)\ , AdjustmentBox["w", BoxMargins->{{0, 0}, {0.5, -0.5}}, BoxBaselineShift->-0.5]], BoxMargins->{{0, 0}, {-1.07143, 1.07143}}, BoxBaselineShift->1.07143]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " \[Ellipsis] \[LongRightArrow]\n\t\t" }], "Text", CellAutoOverwrite->False], Cell[TextData[{ "In the example below, we add all possible words ", StyleBox["w", FontSlant->"Italic"], " of length 4, represented by lists that follow all possible ", StyleBox["reduced", FontSlant->"Italic"], " suffixes ", StyleBox["s", FontSlant->"Italic"], " of length 4. Here the term ", StyleBox["reduced", FontSlant->"Italic"], " means that we pay attention only to the structure of the word \[Dash] the \ other possibilites can straightforwardly be obtained by permutations, i.e., \ by renaming letters. The example originates from the 3 letter case in which \ we allow short abelian repetitions of the form ", StyleBox["xx", FontSlant->"Italic"], " or ", StyleBox["xxx", FontSlant->"Italic"], " for a letter ", StyleBox["x ", FontSlant->"Italic"], "\[Element]", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_3\)]], " = {", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], "}. For simplicity, we show the words only in the string form: " }], "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(\[Euro]reducedWordListSuff4Add4 = \ \(selectWordsWithProperPrefixes[#, addWordList8StartWitha] &\) /@ \ reducedWordList4\)], "Input", Active->True, CellAutoOverwrite->False], Cell[BoxData[ \({{"aaab", {"aaac", "aaca", "aacb", "aacc", "acaa", "acab", "acba", "acbb", "acbc", "accb", "accc", "bbaa", "bbac", "bbca", "bbcb", "bbcc", "bcaa", "bcab", "bcac", "bcba", "bcbb", "bcca", "bccc", "caaa", "caab", "caba", "cabb", "cacc", "cbaa", "cbab", "cbba", "cbbb", "ccaa", "ccac", "ccca", "cccb"}}, {"aaba", {"aaca", "aacb", "aacc", "acaa", "acab", "acba", "acbb", "acbc", "accb", "accc", "caaa", "caab", "caba", "cabb", "cbaa", "cbab", "cbba", "cbbb", "cbcc", "ccbb", "ccbc", "ccca", "cccb"}}, {"aabb", {"baaa", "baac", "baca", "bacb", "bacc", "bcaa", "bcab", "bcac", "bcba", "bcbb", "bcca", "bccc", "caaa", "caab", "caba", "cabb", "cabc", "cacb", "cacc", "cbaa", "cbab", "cbac", "cbba", "cbbb", "ccaa", "ccab", "ccac", "ccca", "cccb"}}, {"aabc", {"aaab", "aaac", "aaba", "aabb", "abaa", "abbb", "abbc", "accb", "accc", "baaa", "babb", "babc", "bbaa", "bbab", "bbac", "bbba", "caaa", "cacb", "cacc", "ccaa", "ccab", "ccac", "ccba", "ccbb"}}, {"abaa", {"acaa", "acab", "acba", "acbb", "acbc", "accb", "accc", "caaa", "caab", "caba", "cabb", "cabc", "cbaa", "cbab", "cbac", "cbba", "cbbb", "cbca", "cbcc", "ccba", "ccbb", "ccbc", "ccca", "cccb"}}, {"abac", {"aaab", "aaba", "aabb", "abaa", "abbb", "abbc", "baaa", "babb", "bbaa", "bbab", "bbba", "bbbc", "bcca", "bccc", "cbba", "cbbb", "cbca", "cbcc", "ccaa", "ccab", "ccba", "ccbb", "ccbc"}}, {"abbb", {"aaab", "aaac", "aaca", "aacb", "aacc", "acaa", "acab", "acba", "acbb", "acbc", "accb", "accc", "caaa", "caab", "caba", "cabb", "cabc", "cacb", "cacc", "cbaa", "cbab", "cbac", "cbba", "cbbb", "ccaa", "ccab", "ccac", "ccca", "cccb"}}, {"abbc", {"aaab", "aaac", "aaba", "aabb", "abaa", "abbb", "abcc", "acbc", "accb", "accc", "baaa", "baac", "babb", "baca", "bacc", "bbaa", "bbab", "bbba", "caaa", "caab", "caba", "cabc", "cacb", "cacc", "ccaa", "ccab", "ccac", "ccba", "ccbb"}}, {"abca", {"aaba", "aabb", "aacc", "abaa", "abbb", "abbc", "baaa", "bbba", "bbbc", "bbcb", "bbcc", "ccba", "ccbb", "ccbc", "cccb"}}, {"abcb", {"aaab", "aaac", "aaca", "aacc", "abbb", "abcc", "baaa", "baac", "babb", "baca", "bacc", "bbaa", "bbab"}}, {"abcc", {"aaab", "aaac", "aaba", "aabb", "aabc", "acba", "acbb", "accc", "caaa", "caab", "caba", "cabb", "cacc", "cbaa", "cbab", "cbba", "cbbb"}}}\)], "Output", FontSize->9] }, Open ]], Cell["\<\ To save memory and to make the computations as fast as possible, the final \ calculations in fact use symbols instead of strings. The cumulative integer \ lists are associated to symbols by using rules. Moreover, to save memory, the \ permutations are added only when needed. In the following example we show a \ rule from symbols to cumulative integer lists (in an abbreviated form):\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(\[Euro]ruleSymbolsToCumulIntegerLists = {aaab \[Rule] {0, 0, 0, 1}, aaba \[Rule] {0, 0, 1, 1}, aabb \[Rule] {0, 0, 1, 2}, aabc \[Rule] {0, 0, 1, 65537}, abaa \[Rule] {0, 1, 1, 1}, abac \[Rule] {0, 1, 1, 65537}, abbb \[Rule] {0, 1, 2, 3}, abbc \[Rule] {0, 1, 2, 65538}, abca \[Rule] {0, 1, 65537, 65537}, abcb \[Rule] {0, 1, 65537, 65538}, abcc \[Rule] {0, 1, 65537, 131073}, ... , \ acbb \[Rule] {0, 65536, 65537, 65538}, bbba \[Rule] {1, 2, 3, 3}, ... , bcaa \[Rule] {1, 65537, 65537, 65537}, ccca \[Rule] {65536, 131072, 196608, 196608}, ... , \ cbaa \[Rule] {65536, 65537, 65537, 65537}};\)\)], "Input", Active->True, CellAutoOverwrite->False], Cell[TextData[{ "In our present setting, we cut the suffix ", StyleBox["s", FontSlant->"Italic"], " from the cumulative integer list(s) for the so-far constructed word and \ then select the new extension ", StyleBox["w", FontSlant->"Italic"], " from the precomputed symbolic list for ", StyleBox["s", FontSlant->"Italic"], ". After this, the corresponding cumulative integer list for ", StyleBox["w", FontSlant->"Italic"], " will be added to (the cumulative integer list of) the previously \ constructed word. This is done in order to detect the possible new abelian \ squares in a quick manner. One function we use for selecting the new \ extensions is ", StyleBox["tryProperExtensionVisList", FontFamily->"Courier New", FontWeight->"Bold"], ". Below we present, as an example, one rule (of a great many) connected \ to it. This time the example is from the 4 letter, and | ", StyleBox["s ", FontSlant->"Italic"], "| = 12, | ", StyleBox["w", FontSlant->"Italic"], " | = 4, case: " }], "Text", CellAutoOverwrite->False], Cell[BoxData[ \(tryProperExtensionVisList[{0, 1, 1, 1025, 1025, 1026, 1026, 1049602, 1049603, 1050627, 1050627, 1051651}] = {bcda, bcdb, bcdc, daca, dacb, dadb, dcab, dcba, dcbc, dcbd}\)], "Input"], Cell["\<\ These kinds of precomputed rules are quite fast to use, but, of course, a \ considerable amount of memory is needed to store all the structures.\ \>", "Text", CellAutoOverwrite->False], Cell[TextData[{ "The following function ", StyleBox["generateRLthree", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Courier New"], "(or, equivalently, ", StyleBox["generateRL", FontFamily->"Courier New", FontWeight->"Bold"], ") constructs the extensions for a given word. Its arguments are states \ (strings that are also values of the global variable ", StyleBox["\[Euro]state", FontFamily->"Courier New", FontWeight->"Bold"], "). As explained also earlier, some examples of these states include ", StyleBox["\"extendOrChangeRight\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"extendOrChangeLeft\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"testRight\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"testLeft\"", FontFamily->"Courier New", FontWeight->"Bold"], ", ", StyleBox["\"failBoth\"", FontFamily->"Courier New", FontWeight->"Bold"], ", and ", StyleBox["\"succeeded\"", FontFamily->"Courier New", FontWeight->"Bold"], ". By opening the cell brackets, the code for ", StyleBox["generateRLthree", FontFamily->"Courier New", FontWeight->"Bold"], " can be viewed from below. Note that even the code was originally \ developed for the 3 letter case (allowing short abelian repetitions of the \ form ", StyleBox["xx", FontSlant->"Italic"], " or ", StyleBox["xxx", FontSlant->"Italic"], " for a letter ", StyleBox["x ", FontSlant->"Italic"], "\[Element]", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\_3\)]], " = {", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], "}), it works perfectly for the 4 letter case as well in all our settings. \ Therefore, the more generic name ", StyleBox["generateRL", FontFamily->"Courier New", FontWeight->"Bold"], " can be, and will be, also used for it. " }], "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(?generateRLthree\)\)], "Input", CellLabel->"", Active->True, CellAutoOverwrite->False], Cell["Global`generateRLthree", "Print", CellTags->"Info3352010319-3690529"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(generateRLthree[ "changeLeft"] := \((\[Euro]state = "failBoth")\) /; Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft && Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight\)}, {" "}, {\(generateRLthree[ "changeLeft"] := \((\[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \ \[Euro]addWordLength]; \[Euro]howFarExtendedLeft = 0; \[Euro]state = "extendOrChangeLeft")\) /; Length[\[Euro]cumulIntListLeft] > \ \[Euro]bottomBoundaryLengthLeft + \[Euro]addWordLength\)}, {" "}, {\(generateRLthree[ "changeLeft"] := \((\[Euro]state = "changeRight")\) /; Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft && Length[\[Euro]cumulIntListRight] > \ \[Euro]bottomBoundaryLengthRight\)}, {" "}, {\(generateRLthree[ "changeLeft"] := \((\[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \ \[Euro]addWordLength]; \[Euro]howFarExtendedLeft = 0; \[Euro]state = "extendLeftBottomEnd")\) /; Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft + \[Euro]addWordLength\)}, {" "}, {\(generateRLthree[ "changeRight"] := \((\[Euro]state = "failBoth")\) /; Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight && Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft\)}, {" "}, {\(generateRLthree[ "changeRight"] := \((\[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \ \[Euro]addWordLength]; \[Euro]howFarExtendedRight = 0; \[Euro]state = "extendOrChangeRight")\) /; Length[\[Euro]cumulIntListRight] > \ \[Euro]bottomBoundaryLengthRight + \[Euro]addWordLength\)}, {" "}, {\(generateRLthree[ "changeRight"] := \((\[Euro]state = "changeLeft")\) /; Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight && Length[\[Euro]cumulIntListLeft] > \ \[Euro]bottomBoundaryLengthLeft\)}, {" "}, {\(generateRLthree[ "changeRight"] := \((\[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \ \[Euro]addWordLength]; \[Euro]howFarExtendedRight = 0; \[Euro]state = "extendRightBottomEnd")\) /; Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight + \[Euro]addWordLength\)}, {" "}, {\(generateRLthree["extendLeftBottomEnd"] = If[\(({\[Euro]successTF, \[Euro]whatToCatenateLeft, \ \[Euro]pointerExtendLeft, \[Euro]howFarExtendedLeft} = nextProperTrialForNewExtensionFin[\[Euro]\ cumulIntListReverseOfGivenWordForTest, Last[\[Euro]indexListLeft], \ \[Euro]howFarExtendedLeft])\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[Euro]cumulIntListLeft = catenateWithKnownAddRL[\[Euro]cumulIntListLeft, \ \[Euro]whatToCatenateLeft]; \[Euro]indexListLeft = ReplacePart[\[Euro]indexListLeft, \ \[Euro]pointerExtendLeft, Length[\[Euro]indexListLeft]]; \[Euro]state = "testLeftBottomEnd", \[Euro]indexListLeft = Take[\[Euro]indexListLeft, Length[\[Euro]indexListLeft] - 1]; \[Euro]state = "changeRight"]\)}, {" "}, {\(generateRLthree["extendOrChangeLeft"] := If[Length[\[Euro]cumulIntListRight] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthRight + \[Euro]extensionBoundaryLengthRight, \ \[Euro]howFarExtendedLeft = 0; \[Euro]state = "succeeded", \[Euro]howFarExtendedLeft = 0; \[Euro]indexListRight = Append[\[Euro]indexListRight, 0]; \[Euro]state = "extendOrChangeRight"] /; Length[\[Euro]cumulIntListLeft] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthLeft + \[Euro]extensionBoundaryLengthLeft\)}, {" "}, {\(generateRLthree["extendOrChangeLeft"] := If[\(({\[Euro]successTF, \[Euro]whatToCatenateLeft, \ \[Euro]pointerExtendLeft, \[Euro]howFarExtendedLeft} = nextProperTrialForNewExtensionFin[\[Euro]\ cumulIntListLeft, Last[\[Euro]indexListLeft], \ \[Euro]howFarExtendedLeft])\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[Euro]cumulIntListLeft = catenateWithKnownAddRL[\[Euro]cumulIntListLeft, \ \[Euro]whatToCatenateLeft]; \[Euro]indexListLeft = ReplacePart[\[Euro]indexListLeft, \ \[Euro]pointerExtendLeft, Length[\[Euro]indexListLeft]]; \[Euro]state = "testLeft", \[Euro]indexListLeft = Take[\[Euro]indexListLeft, Length[\[Euro]indexListLeft] - 1]; \[Euro]state = "changeRight"]\)}, {" "}, {\(generateRLthree["extendOrChangeRight"] := If[Length[\[Euro]cumulIntListLeft] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthLeft + \[Euro]extensionBoundaryLengthLeft, \ \[Euro]howFarExtendedRight = 0; \[Euro]state = "succeeded", \[Euro]howFarExtendedRight = 0; \[Euro]indexListLeft = Append[\[Euro]indexListLeft, 0]; \[Euro]state = "extendOrChangeLeft"] /; Length[\[Euro]cumulIntListRight] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthRight + \[Euro]extensionBoundaryLengthRight\)}, {" "}, {\(generateRLthree["extendOrChangeRight"] := If[\(({\[Euro]successTF, \[Euro]whatToCatenateRight, \ \[Euro]pointerExtendRight, \[Euro]howFarExtendedRight} = nextProperTrialForNewExtensionFin[\[Euro]\ cumulIntListRight, Last[\[Euro]indexListRight], \ \[Euro]howFarExtendedRight])\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[Euro]cumulIntListRight = catenateWithKnownAddRL[\[Euro]cumulIntListRight, \ \[Euro]whatToCatenateRight]; \[Euro]indexListRight = ReplacePart[\[Euro]indexListRight, \ \[Euro]pointerExtendRight, Length[\[Euro]indexListRight]]; \[Euro]state = "testRight", \[Euro]indexListRight = Take[\[Euro]indexListRight, Length[\[Euro]indexListRight] - 1]; \[Euro]state = "changeLeft"]\)}, {" "}, {\(generateRLthree["extendRightBottomEnd"] = If[\(({\[Euro]successTF, \[Euro]whatToCatenateRight, \ \[Euro]pointerExtendRight, \[Euro]howFarExtendedRight} = nextProperTrialForNewExtensionFin[\[Euro]\ cumulIntListOfGivenWordForTest, Last[\[Euro]indexListRight], \ \[Euro]howFarExtendedRight])\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[Euro]cumulIntListRight = catenateWithKnownAddRL[\[Euro]cumulIntListRight, \ \[Euro]whatToCatenateRight]; \[Euro]indexListRight = ReplacePart[\[Euro]indexListRight, \ \[Euro]pointerExtendRight, Length[\[Euro]indexListRight]]; \[Euro]state = "testRightBottomEnd", \[Euro]indexListRight = Take[\[Euro]indexListRight, Length[\[Euro]indexListRight] - 1]; \[Euro]state = "changeLeft"]\)}, {" "}, {\(generateRLthree["failBoth"] = Print["For ", \[Euro]givenWordForTest, " the extension failed!"]\)}, {" "}, {\(generateRLthree[ "startConstruction"] = \((\[Euro]wordLeft = StringTake[\[Euro]givenWordForTest, Ceiling[ StringLength[\[Euro]givenWordForTest]\/2]]; \ \[Euro]wordRight = StringTake[\[Euro]givenWordForTest, \(-Floor[ StringLength[\[Euro]givenWordForTest]\/2]\)]; \ \[Euro]cumulIntListLeft = convertStrToCumulIntList[ StringReverse[\[Euro]wordLeft], \[Euro]alphLetInt]; \ \[Euro]cumulIntListRight = convertStrToCumulIntList[\[Euro]wordRight, \ \[Euro]alphLetInt]; \[Euro]bottomBoundaryLengthRight = Length[\[Euro]cumulIntListRight]; \ \[Euro]bottomBoundaryLengthLeft = Length[\[Euro]cumulIntListLeft]; \[Euro]state = "extendRightBottomEnd"; )\)\)}, {" "}, {\(generateRLthree["succeeded"] = Print["For ", \[Euro]givenWordForTest, " the extension was successful!"]\)}, {" "}, {\(generateRLthree["testLeft"] = Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListLeft, \ \[Euro]cumulIntListRight, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedLeft]}, If[testResult\[LeftDoubleBracket]1\[RightDoubleBracket],\ \[Euro]howFarExtendedLeft = 0; \[Euro]indexListRight = Append[\[Euro]indexListRight, 0]; \[Euro]state = "extendOrChangeRight", \[Euro]howFarExtendedLeft = testResult\[LeftDoubleBracket]2\[RightDoubleBracket]\ - 1; \[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \ \[Euro]addWordLength]; \[Euro]state = "extendOrChangeLeft"]]\)}, {" "}, {\(generateRLthree["testLeftBottomEnd"] = Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListLeft, \ \[Euro]cumulIntListRight, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedLeft]}, If[testResult\[LeftDoubleBracket]1\[RightDoubleBracket],\ \[Euro]howFarExtendedLeft = 0; \[Euro]indexListRight = Append[\[Euro]indexListRight, 0]; \[Euro]state = "extendOrChangeRight", \[Euro]howFarExtendedLeft = testResult\[LeftDoubleBracket]2\[RightDoubleBracket]\ - 1; \[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \ \[Euro]addWordLength]; \[Euro]state = "extendLeftBottomEnd"]]\)}, {" "}, {\(generateRLthree["testRight"] = Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListRight, \ \[Euro]cumulIntListLeft, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedRight]}, If[testResult\[LeftDoubleBracket]1\[RightDoubleBracket],\ \[Euro]howFarExtendedRight = 0; \[Euro]indexListLeft = Append[\[Euro]indexListLeft, 0]; \[Euro]state = "extendOrChangeLeft", \[Euro]howFarExtendedRight = testResult\[LeftDoubleBracket]2\[RightDoubleBracket]\ - 1; \[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \ \[Euro]addWordLength]; \[Euro]state = "extendOrChangeRight"]]\)}, {" "}, {\(generateRLthree["testRightBottomEnd"] = Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListRight, \ \[Euro]cumulIntListLeft, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedRight]}, If[testResult\[LeftDoubleBracket]1\[RightDoubleBracket],\ \[Euro]howFarExtendedRight = 0; \[Euro]indexListLeft = Append[\[Euro]indexListLeft, 0]; \[Euro]state = "extendLeftBottomEnd", \[Euro]howFarExtendedRight = testResult\[LeftDoubleBracket]2\[RightDoubleBracket]\ - 1; \[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \ \[Euro]addWordLength]; \[Euro]state = "extendRightBottomEnd"]]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ "generateRLthree"], Editable->False]], "Print", CellTags->"Info3352010319-3690529"] }, Closed]], Cell[TextData[{ "Before starting the construction, we need to intialise the global (at this \ stage) variables and download the proper package (", StyleBox["alphThreeLetIntCase.m", FontFamily->"Courier New", FontWeight->"Bold"], " (18 KB), or ", StyleBox["alphFourLetIntCase.m", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "(35 KB)) depending on whether we use 3 or 4 letters. This is accomplished \ by using the ", StyleBox["intialise", FontFamily->"Courier New", FontWeight->"Bold"], " function that has the following main structure: " }], "Text", CellAutoOverwrite->False], Cell[BoxData[ \(initialise[howManyStepsLeftAndRight_, givenWordForTest_, alphLetInt_: \[Euro]alphThreeLetInt]\)], "Input", Active->True], Cell[TextData[{ "The variables", StyleBox[" \[Euro]extensionBoundaryLengthRight = howManyStepsLeftAndRight*\ \[Euro]addWordLength ", FontFamily->"Courier New", FontWeight->"Bold"], "and", StyleBox["\n\[Euro]extensionBoundaryLengthLeft = howManyStepsLeftAndRight*\ \[Euro]addWordLength ", FontFamily->"Courier New", FontWeight->"Bold"], "stand for extension lengths. Both of these lengths should be equal and of \ the form ", StyleBox["k", FontSlant->"Italic"], " ", StyleBox["* ", FontWeight->"Bold"], StyleBox["\[Euro]addWordLength", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Courier New"], "for a positive integer ", StyleBox["k", FontSlant->"Italic"], ". In our examples, and in our present code, the value for ", StyleBox["\[Euro]addWordLength ", FontFamily->"Courier New", FontWeight->"Bold"], "(length of ", StyleBox["w", FontSlant->"Italic"], " above) is equal to 4. The full definition of the ", StyleBox["intialise", FontFamily->"Courier New", FontWeight->"Bold"], " function is as follows:" }], "Text", CellAutoOverwrite->False], Cell[BoxData[{ \(Clear[initialise]; \), "\[IndentingNewLine]", \(initialise[howManyStepsLeftAndRight_, givenWordForTest_, alphLetInt_: \[Euro]alphThreeLetInt] \ := \[IndentingNewLine]\((\[Euro]alphLetInt = alphLetInt; \ \ (*\ \[Euro]alphLetInt\ needs\ to\ be\ set\ to\ \ \[Euro]alphThreeLetInt\ or\ to\ \[Euro]alphFourLetInt\ \ *) \[IndentingNewLine]If[ alphLetInt == \[Euro]alphThreeLetInt, << \ "\", << "\"]; \ \[IndentingNewLine]\[Euro]addWordLength\ = \ 4; \ \[IndentingNewLine]\[Euro]preCheckedSuffLength = \ 8; \ \ \[IndentingNewLine]\[Euro]\ suffLengthForWhichTryProperExtension = \[Euro]preCheckedSuffLength - \ \[Euro]addWordLength; \ \[IndentingNewLine]\[Euro]reducedWordListSuffNAddNToExpressions = \ \[Euro]reducedWordListSuff4Add4ToExpressions; \ \[IndentingNewLine]ordinalIndexForCumulIntegerListN = ordinalIndexForCumulIntegerList4; \ (*\ function\ names\ *) \[IndentingNewLine]\[Euro]pointerExtendRight = 0; \n\[Euro]pointerExtendLeft = 0; \n\[Euro]extensionBoundaryLengthRight = howManyStepsLeftAndRight*\[Euro]addWordLength; \[IndentingNewLine]\ \[Euro]extensionBoundaryLengthLeft = howManyStepsLeftAndRight*\[Euro]addWordLength; \ (*\ Indeed, \ both\ of\ these\ should\ be\ equal\ and\ of\ the\ form\ \ k\ *\ \ \[Euro]addWordLength\ for\ a\ positive\ integer\ \ k\ *) \[IndentingNewLine]\ \[Euro]howFarExtendedRight = 0; \[IndentingNewLine]\[Euro]howFarExtendedLeft = 0; \[IndentingNewLine]\[Euro]indexListRight = {0}; \ \[IndentingNewLine]\[Euro]indexListLeft = {}; \ \[IndentingNewLine]\[Euro]givenWordForTest = givenWordForTest; \ \[IndentingNewLine]\[Euro]cumulIntListOfGivenWordForTest = convertStrToCumulIntList[\[Euro]givenWordForTest, \ \ \[Euro]alphLetInt]; \ \[IndentingNewLine]\[Euro]cumulIntListReverseOfGivenWordForTest = convertStrToCumulIntList[ StringReverse[\[Euro]givenWordForTest], \ \[Euro]alphLetInt])\); \ \)}], "Input", CellLabel->"In[10]:=", InitializationCell->True, CellAutoOverwrite->False], Cell["\<\ For the purpose of this notebook, however, the user does not need to download \ any packages. The code will be activated automatically for the 4 letter case \ (the 3 letter case requires the evaluation of the very last Input Cell of \ this notebook).\ \>", "Text", CellAutoOverwrite->False], Cell["\<\ An example of the three letter case follows (note that this Input Cell is not \ activated):\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(initialise[ 3, "\"];\)\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(Print["\<\[Euro]givenWordForTest = \>", \ \[Euro]givenWordForTest];\)\), "\[IndentingNewLine]", \(\(generateRLthree["\"];\)\), "\[IndentingNewLine]", \ \(While[\ \[Euro]state =!= "\"\ \[And] \ \[Euro]state =!= \ "\", generateRLthree[\[Euro]state]\ ] // Timing\), "\[IndentingNewLine]", \(generateRLthree[\[Euro]state]\)}], "Input", Active->True, CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]givenWordForTest = \ "\[InvisibleSpace]"aaabbbaaacccaaabbb"\), SequenceForm[ "\[Euro]givenWordForTest = ", "aaabbbaaacccaaabbb"], Editable->False]], "Print", CellLabel->"From In[302]:="], Cell[BoxData[ \({0.010000000000001563`\ Second, Null}\)], "Output", CellLabel->""], Cell[BoxData[ InterpretationBox[\("For "\[InvisibleSpace]"aaabbbaaacccaaabbb"\ \[InvisibleSpace]" the extension failed!"\), SequenceForm[ "For ", "aaabbbaaacccaaabbb", " the extension failed!"], Editable->False]], "Print", CellLabel->"From In[302]:="] }, Open ]], Cell[TextData[{ "Thus, the given word is an ", StyleBox["unfavourable", FontSlant->"Italic"], " one", StyleBox[".", FontSlant->"Italic"] }], "Text", CellAutoOverwrite->False], Cell["\<\ The examples below deal with the four letter case. Fistly, we try to extend a \ word of length 20:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \("\" // StringLength\)], "Input", CellLabel->"", CellAutoOverwrite->False], Cell[BoxData[ \(20\)], "Output", CellLabel->""] }, Open ]], Cell[TextData[{ "The extra condition in the ", StyleBox["While", FontFamily->"Courier New", FontWeight->"Bold"], " loop guarantees that computation will not last too long:" }], "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(initialise[ 8, "\", \[Euro]alphFourLetInt];\)\(\ \[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(generateRL["\"]; \ iii = 1; Print["\<\[Euro]givenWordForTest = \>", \[Euro]givenWordForTest];\), "\ \[IndentingNewLine]", \(While[\ iii \[LessEqual] 50000 \[And] \[Euro]state =!= "\"\ \[And] \ \ \[Euro]state =!= "\", generateRL[\[Euro]state]; \(iii++\)\ ] // Timing\), "\[IndentingNewLine]", \(generateRL[\[Euro]state]\)}], "Input", CellLabel->"", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]givenWordForTest = \ "\[InvisibleSpace]"abacabadbacbcdbacdbd"\), SequenceForm[ "\[Euro]givenWordForTest = ", "abacabadbacbcdbacdbd"], Editable->False]], "Print", CellLabel->"From In[195]:="], Cell[BoxData[ \({7.501`\ Second, Null}\)], "Output", CellLabel->""], Cell[BoxData[ InterpretationBox[\("For "\[InvisibleSpace]"abacabadbacbcdbacdbd"\ \[InvisibleSpace]" the extension was successful!"\), SequenceForm[ "For ", "abacabadbacbcdbacdbd", " the extension was successful!"], Editable->False]], "Print", CellLabel->"From In[195]:="] }, Open ]], Cell[TextData[{ "Well, after all, the computation was not too long. At this point, we know \ only that the given word is a ", StyleBox["so-far-favourable", FontSlant->"Italic"], " one", "." }], "Text", CellAutoOverwrite->False], Cell["\<\ We may look at some of the inner values and structures produced by the \ previous computation:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(myPrint["\<4 letters\>"]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]cumulIntListLeft as reverse string | \ \[Euro]cumulIntListRight as string = \ "\[InvisibleSpace]"cbcacbcdcabdbabcdabcacdadbdadcadabacabadba"\ \[InvisibleSpace]"|"\[InvisibleSpace]\ "cbcdbacdbdadbcacdacabdabcbdadcdbcbadbcbdbc"\), SequenceForm[ "\[Euro]cumulIntListLeft as reverse string | \[Euro]cumulIntListRight \ as str = ", "cbcacbcdcabdbabcdabcacdadbdadcadabacabadba", "|", "cbcdbacdbdadbcacdacabdabcbdadcdbcbadbcbdbc"], Editable->False]], "Print", CellLabel->"From In[310]:="], Cell[BoxData[ InterpretationBox[\("\[Euro]howFarExtendedLeft = "\[InvisibleSpace]0\), SequenceForm[ "\[Euro]howFarExtendedLeft = ", 0], Editable->False]], "Print", CellLabel->"From In[310]:="], Cell[BoxData[ InterpretationBox[\("\[Euro]indexListLeft = "\[InvisibleSpace]{23, 1, 2, 12, 4, 24, 12, 25}\), SequenceForm[ "\[Euro]indexListLeft = ", {23, 1, 2, 12, 4, 24, 12, 25}], Editable->False]], "Print", CellLabel->"From In[310]:="], Cell[BoxData[ InterpretationBox[\("\[Euro]howFarExtendedRight = "\[InvisibleSpace]0\), SequenceForm[ "\[Euro]howFarExtendedRight = ", 0], Editable->False]], "Print", CellLabel->"From In[310]:="], Cell[BoxData[ InterpretationBox[\("\[Euro]indexListRight = "\[InvisibleSpace]{24, 11, 3, 21, 20, 12, 14, 34, 0}\), SequenceForm[ "\[Euro]indexListRight = ", {24, 11, 3, 21, 20, 12, 14, 34, 0}], Editable->False]], "Print", CellLabel->"From In[310]:="], Cell[BoxData[ InterpretationBox[\("\[Euro]state = "\[InvisibleSpace]"succeeded"\), SequenceForm[ "\[Euro]state = ", "succeeded"], Editable->False]], "Print", CellLabel->"From In[310]:="] }, Open ]], Cell[TextData[{ "Above the lists ", StyleBox["\[Euro]indexListLeft", FontFamily->"Courier New", FontWeight->"Bold"], " and ", StyleBox["\[Euro]indexListRight ", FontFamily->"Courier New", FontWeight->"Bold"], "contain the pointer values that indicate which words ", StyleBox["w", FontSlant->"Italic"], " from the precomputed lists should be next catenated for testing to \ corresponding suffixes ", StyleBox["s. ", FontSlant->"Italic"], "Variables ", StyleBox["\[Euro]howFarExtendedLeft", FontFamily->"Courier New", FontWeight->"Bold"], " and ", StyleBox["\[Euro]howFarExtendedRight ", FontFamily->"Courier New", FontWeight->"Bold"], "are used for efficient a-2-freeness testing and for finding the next \ proper word ", StyleBox["w", FontSlant->"Italic"], " for catenation as efficiently as possible." }], "Text", CellAutoOverwrite->False], Cell["\<\ Below we test that the reached extension is indeed an abelian square-free \ word:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(testA2["\<\ cbcacbcdcabdbabcdabcacdadbdadcadabacabadbacbcdbacdbdadbcacdacabdabcbdadcdbcbad\ bcbdbc\>"]\)], "Input", CellLabel->"", CellAutoOverwrite->False], Cell[BoxData[ \(True\)], "Output", CellLabel->""] }, Open ]], Cell[TextData[{ "We will extend the same word ", StyleBox["\"abacabadbacbcdbacdbd\"", FontFamily->"Courier New", FontWeight->"Bold"], " of length 20 one more step (of length 4) to the right and to the left:" }], "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(initialise[ 9, "\", \[Euro]alphFourLetInt];\)\(\ \[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(generateRL["\"]; \ iii = 1; Print["\<\[Euro]givenWordForTest = \>", \[Euro]givenWordForTest];\), "\ \[IndentingNewLine]", \(While[\ iii \[LessEqual] 100000 \[And] \[Euro]state =!= "\"\ \[And] \ \ \[Euro]state =!= "\", generateRL[\[Euro]state]; \(iii++\)\ ] // Timing\), "\[IndentingNewLine]", \(generateRL[\[Euro]state]\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]givenWordForTest = \ "\[InvisibleSpace]"abacabadbacbcdbacdbd"\), SequenceForm[ "\[Euro]givenWordForTest = ", "abacabadbacbcdbacdbd"], Editable->False]], "Print"], Cell[BoxData[ \({37.894999999999996`\ Second, Null}\)], "Output"], Cell[BoxData[ \("testRight"\)], "Output"] }, Open ]], Cell[TextData[{ "Once again, we know only that the given word is a", StyleBox[" so-far-favourable ", FontSlant->"Italic"], "one." }], "Text", CellAutoOverwrite->False], Cell["Let us consider the following word of length 84:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \("\" // StringLength\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(84\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(initialise[ 3, "\", \[Euro]alphFourLetInt];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(generateRL["\"]; \ iii = 1; Print["\<\[Euro]givenWordForTest = \>", \[Euro]givenWordForTest];\), "\ \[IndentingNewLine]", \(While[\ iii \[LessEqual] 100000 \[And] \[Euro]state =!= "\"\ \[And] \ \ \[Euro]state =!= "\", generateRL[\[Euro]state]; \(iii++\)\ ] // Timing\), "\[IndentingNewLine]", \(generateRL[\[Euro]state]\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]givenWordForTest = \ "\[InvisibleSpace]\ "cbcacbcdcabdbabcdabcacdadbdadcadabacabadbacbcdbacdbdadbcacdacabdabcbdadcdbcba\ dbcbdbc"\), SequenceForm[ "\[Euro]givenWordForTest = ", "cbcacbcdcabdbabcdabcacdadbdadcadabacabadbacbcdbacdbdadbcacdacabdabcbd\ adcdbcbadbcbdbc"], Editable->False]], "Print"], Cell[BoxData[ \({0.029999999998835847`\ Second, Null}\)], "Output"], Cell[BoxData[ InterpretationBox[\("For \ "\[InvisibleSpace]\ "cbcacbcdcabdbabcdabcacdadbdadcadabacabadbacbcdbacdbdadbcacdacabdabcbdadcdbcba\ dbcbdbc"\[InvisibleSpace]" the extension failed!"\), SequenceForm[ "For ", "cbcacbcdcabdbabcdabcacdadbdadcadabacabadbacbcdbacdbdadbcacdaca\ bdabcbdadcdbcbadbcbdbc", " the extension failed!"], Editable->False]], "Print"] }, Open ]], Cell[TextData[{ "In this case we know definitely that the given word of length 84 is an", StyleBox[" unfavourable ", FontSlant->"Italic"], "one. Actually, it turns out that the given word is already the longest \ possible extension of ", StyleBox["w", FontSlant->"Italic"], " = \"", StyleBox["abacabadbacbcdbacdbd", FontFamily->"Courier New", FontWeight->"Bold"], "\" ! Note that above the ", StyleBox["\[Euro]givenWordForTest", FontFamily->"Courier New", FontWeight->"Bold"], " has the form of ", StyleBox["\"cbcacbcdcabdbabcdabcacdadbdadcad\" <> ", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox["w", FontSlant->"Italic"], StyleBox[" <> \"adbcacdacabdabcbdadcdbcbadbcbdbc\"", FontFamily->"Courier New", FontWeight->"Bold"], ". However, we will not elaborate on this here." }], "Text", CellAutoOverwrite->False], Cell["Let us consider another given word:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(initialise[ 8, "\", \[Euro]alphFourLetInt];\)\(\ \[IndentingNewLine]\) \)\), "\n", \(generateRL["\"]; Print["\<\[Euro]givenWordForTest = \>", \[Euro]givenWordForTest];\), "\n\ ", \(While[\[Euro]state =!= "\"\ \[And] \ \[Euro]state =!= \ "\", \(generateRL[\[Euro]state];\)\ ] // Timing\), "\n", \(generateRL[\[Euro]state]\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]givenWordForTest = \ "\[InvisibleSpace]"abcbdbcbacbdcdacbdac"\), SequenceForm[ "\[Euro]givenWordForTest = ", "abcbdbcbacbdcdacbdac"], Editable->False]], "Print"], Cell[BoxData[ \({10.95300000000043`\ Second, Null}\)], "Output"], Cell[BoxData[ InterpretationBox[\("For "\[InvisibleSpace]"abcbdbcbacbdcdacbdac"\ \[InvisibleSpace]" the extension was successful!"\), SequenceForm[ "For ", "abcbdbcbacbdcdacbdac", " the extension was successful!"], Editable->False]], "Print"] }, Open ]], Cell[TextData[{ "Thus, ", "for the time being", ", our case is a", StyleBox[" so-far-favourable", FontSlant->"Italic"], " one. The computation did not take a long time even the extension was \ quite long (8\[CenterDot]4 = 32 to both sides)." }], "Text", CellAutoOverwrite->False], Cell["\<\ We may view some of the inner values and structures produced by the previous \ computation:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(myPrint["\<4 letters\>"]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]cumulIntListLeft as reverse string | \ \[Euro]cumulIntListRight as string = \ "\[InvisibleSpace]"abdabacbdbadbdcdadbabcacdbcabadbabcbdbcbac"\ \[InvisibleSpace]"|"\[InvisibleSpace]\ "bdcdacbdacabacdbdabdbcabcdcabadacdcbacdcac"\), SequenceForm[ "\[Euro]cumulIntListLeft as reverse string | \[Euro]cumulIntListRight \ as str = ", "abdabacbdbadbdcdadbabcacdbcabadbabcbdbcbac", "|", "bdcdacbdacabacdbdabdbcabcdcabadacdcbacdcac"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\[Euro]howFarExtendedLeft = "\[InvisibleSpace]0\), SequenceForm[ "\[Euro]howFarExtendedLeft = ", 0], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\[Euro]indexListLeft = "\[InvisibleSpace]{10, 12, 14, 18, 19, 10, 16, 13}\), SequenceForm[ "\[Euro]indexListLeft = ", {10, 12, 14, 18, 19, 10, 16, 13}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\[Euro]howFarExtendedRight = "\[InvisibleSpace]0\), SequenceForm[ "\[Euro]howFarExtendedRight = ", 0], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\[Euro]indexListRight = "\[InvisibleSpace]{28, 40, 13, 32, 25, 23, 16, 23, 0}\), SequenceForm[ "\[Euro]indexListRight = ", {28, 40, 13, 32, 25, 23, 16, 23, 0}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\[Euro]state = "\[InvisibleSpace]"succeeded"\), SequenceForm[ "\[Euro]state = ", "succeeded"], Editable->False]], "Print"] }, Open ]], Cell["\<\ Let us test that the reached extension really is an abelian square-free word:\ \ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(testA2["\<\ abdabacbdbadbdcdadbabcacdbcabadbabcbdbcbacbdcdacbdacabacdbdabdbcabcdcabadacdcb\ acdcac\>"]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["This is correct!", "Text", CellAutoOverwrite->False], Cell[TextData[{ "The final example shows that our word ", StyleBox["\"abcbdbcbacbdcdacbdac\" ", FontFamily->"Courier New", FontWeight->"Bold"], "above turns out to be an ", StyleBox["unfavourable ", FontSlant->"Italic"], "one. However, this time the computation takes nearly two hours (in a 1.6 \ GHz machine) instead of 11 seconds above:" }], "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(initialise[ 9, "\", \[Euro]alphFourLetInt];\)\(\n\) \)\), "\[IndentingNewLine]", \(generateRL["\"]; Print["\<\[Euro]givenWordForTest = \>", \[Euro]givenWordForTest];\), "\n\ ", \(While[\[Euro]state =!= "\"\ \[And] \ \[Euro]state =!= \ "\", \(generateRL[\[Euro]state];\)\ ] // Timing\), "\n", \(generateRL[\[Euro]state]\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[\("\[Euro]givenWordForTest = \ "\[InvisibleSpace]"abcbdbcbacbdcdacbdac"\), SequenceForm[ "\[Euro]givenWordForTest = ", "abcbdbcbacbdcdacbdac"], Editable->False]], "Print"], Cell[BoxData[ \({6659.5`\ Second, Null}\)], "Output"], Cell[BoxData[ InterpretationBox[\("For "\[InvisibleSpace]"abcbdbcbacbdcdacbdac"\ \[InvisibleSpace]" the extension failed!"\), SequenceForm[ "For ", "abcbdbcbacbdcdacbdac", " the extension failed!"], Editable->False]], "Print"] }, Open ]], Cell[TextData[{ "One might have expected that the long buffers (found before this final \ trial) of length 8\[CenterDot]4 = 32 to the both directions of ", StyleBox["\"abcbdbcbacbdcdacbdac\" ", FontFamily->"Courier New", FontWeight->"Bold"], "would already quarantee the given word to be ", StyleBox["favourable", FontSlant->"Italic"], ". Surprisingly enough, this turns out not to be the case." }], "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Conclusions", FontWeight->"Bold"]], "Section", CellAutoOverwrite->False], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has versatile structures that firmly support the development of concepts. \ This has been extremely useful for constructing prototypes and full \ stand-alone code. The use of ", StyleBox["Mathematica", FontSlant->"Italic"], " has enabled us to discover phenomena which previously were either \ unbelievable or really hard to experiment on. Even so, it would be useful, if \ in the future, in ", StyleBox["Mathematica", FontSlant->"Italic"], ",", " one could also use restricted pattern matching in a fast way that was \ explained in the Introduction section. Lastly, we expect that the presented \ code, and its future updates, will be used for a quite long time in our \ research." }], "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Acknowledgements", FontWeight->"Bold"]], "Section"], Cell["\<\ We gratefully acknowledge the participation of the following individuals most \ of who have been students at the Rovaniemi Polytechnic / University of \ Applied Sciences over the course of the research from 1990 to 2006. These \ people have made a number of computer programs for searching strings with \ desirable properties, helped in a crucial way to set up the computing \ environments, or made interactive graphical and musical representations of \ the structures. Starting year is given in parenthesis: Kari Tuovinen (1990); \ Minna Iivonen, Anja Keskinarkaus, Marko Manninen (1993); Abdeljalil Chabani, \ Tomi Laakso (1994); Mika Moilanen, Juha S\[ADoubleDot]rest\[ODoubleDot]niemi \ (1996); Juho Alfthan (1999); Olli-Pentti Saira (2000); Marja Kentt\ \[ADoubleDot], Ville Mattila (2001); Lauri Autio, and Marianna \ M\[ODoubleDot]ll\[ADoubleDot]ri (2002); Antti Eskola (2003); Antti Karhu, \ Veli-Matti Lahtela, Olli-Pekka Siivola (2004); Esa Nyrhinen, Sami Vuolli \ (2005); Esa Taskila, and Mikhail Kalkov (2006).\ \>", "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "[AS1]", StyleBox[" ", FontWeight->"Bold"], "J.-P. Allouche and J. Shallit. The ubiquitous Prouhet-Thue-Morse \ sequenece. In C. Ding, T. Helleseth, and H. Niederreiter, editors, Sequences \ and Their Applications, Proc. SETA '98, 1-16. Springer-Verlag, 1999.", StyleBox["\n", FontWeight->"Bold"], "[AS2] ", StyleBox[" ", FontWeight->"Bold"], "J.-P. Allouche and J. Shallit. Automatic Sequences - Theory, Applications \ Generalizations. Cambridge University Press, 2003.", StyleBox["\n", FontWeight->"Bold"], "[B1]", StyleBox[" ", FontWeight->"Bold"], "J. Berstel. Some recent results on square-free words. In M. Fontet and K. \ Melhorn, editors, Proc. STACS '84, Lecture Notes in Comp. Sci., 166:14-25. \ Springer-Verlag, Berlin, 1984.", StyleBox["\n", FontWeight->"Bold"], "[B2] ", StyleBox[" ", FontWeight->"Bold"], "J. Berstel. Axel Thue's work on repetitions in words: a translation, \ Publications du LCIM 20, 16 pages. Universit\[EAcute] du Quebec \[AGrave] \ Montr\[EAcute]al, 1994.\n[C1] A. Carpi. On abelian power-free morphisms. \ Int. J. Algebra Comput., 3:151", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "167. World Sci. Publ. Company, 1993.\n[C2] A. Carpi. On the number of \ abelian square-free words on four letters. Discrete Appl. Math., 81:155", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "167. Elsevier, 1998.\n[C3] A. Carpi. On abelian squares and \ substitutions. Theor. Comp. Sci., 218:61", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "81. Elsevier, 1999.\n[CC] J. Cassaigne and J.D. Currie. Words strongly \ avoiding fractional powers. Europ. J. Combinatorics, 20:725", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "737. Academic Press, 1999.\n[E]", StyleBox[" ", FontWeight->"Bold"], "P. Erd\[ODoubleDot]s. Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\ \[OAcute] Int. K\[ODoubleDot]zl., 6:221", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "254, 1961.\n[K1] V. Ker\[ADoubleDot]nen. Abelian squares are avoidable on \ 4 letters. In W. Kuich, editor, Proc. ICALP '92, Lecture Notes in Comp. Sci., \ 623:41", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "52. Springer-Verlag, Berlin, 1992.\n[K2] V. Ker\[ADoubleDot]nen. \ Mathematica", StyleBox[" ", FontSlant->"Italic"], "in research of avoidable patterns in strings. In V. Ker\[ADoubleDot]nen \ and P. Mitic, editors, Mathematics with Vision, Proc. First International \ Mathematica Symposium (IMS '95, Southampton, England), 259", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "266. Computational Mechanics Publications, 1995.\n[K3] V. \ Ker\[ADoubleDot]nen, New abelian square-free DT0L-languages over 4 letters. \ Proc. Fifth International Arctic Seminar (", StyleBox["IAS", FontSlant->"Italic"], " 2002, May 15 - 17, 2002, Murmansk, Russia), Murmansk State Pedagogical \ Institute, 2002, 14 pages. Available online at\n", ButtonBox["http://south.rotol.ramk.fi/keranen/ias2002/ias2002papers.html.", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/ias2002/ias2002papers.html"], None}, ButtonStyle->"Hyperlink"], "\n[K4] V. Ker\[ADoubleDot]nen. On abelian square-free \ DT0L\[Dash]languages over 4 letters. In T. Harju and J. \ Karhum\[ADoubleDot]ki, editors, Proc. 4th International Conference on \ Combinatorics on Words 27:95\[Dash]109. Turku Centre for Computer Science, \ Turku, 2003.\n[K5] V. Ker\[ADoubleDot]nen. Forbidden abelian square-free \ factors over 4 Letters, 2004. Available online at\n", ButtonBox["http://south.rotol.ramk.fi/keranen/research/ForbiddenFactors.\ html", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/research/ForbiddenFactors.html"], None}, ButtonStyle->"Hyperlink"], ".\n[K6] V. Ker\[ADoubleDot]nen. Programs & notebooks & packages. ", StyleBox["Mathematica", FontSlant->"Italic"], " code for suppression of unfavourable factors in pattern avoidance, 2006. \ Available online at ", ButtonBox["http://south.rotol.ramk.fi/keranen/research/\ UnfavourableFactorsInPatternAvoidance/Programs&Notebooks&Packages.html.\n", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/research/\ UnfavourableFactorsInPatternAvoidance/Programs&Notebooks&Packages.html"], None}, ButtonStyle->"Hyperlink"], "[K7] V. Ker\[ADoubleDot]nen. 2006. Avoidable regularities in strings. A \ collection of links to web pages on structures, graphics, and music of \ abelian square-free strings, 1996\[Dash]2006. Available online at", ButtonBox[" ", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/research/\ UnfavourableFactorsInPatternAvoidance/Programs&Notebooks&Packages.html"], None}, ButtonStyle->"Hyperlink"], ButtonBox["http://south.rotol.ramk.fi/keranen/StructuresGraphicsMusic.html", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/StructuresGraphicsMusic.html"], None}, ButtonStyle->"Hyperlink"], ".\n[Ma1] V. Mattila. Creating permutation repetition-free words and \ testing permutation repetition-freeness of morphisms (in Finnish). B.Sc. \ Thesis. Rovaniemi Polytechnic, 2002.\n[Ma2] V. Mattila. Constructing abelian \ square-free words and testing morphisms with ", StyleBox["Mathematica. ", FontSlant->"Italic"], "In V. Demidov and V. Ker\[ADoubleDot]nen, editors, Proc. IAS 2002. \ Murmansk State Pedagogical Institute and Rovaniemi Polytechnic, Murmansk \ 2002. Available online at\n", ButtonBox["http://south.rotol.ramk.fi/keranen/ias2002/ias2002papers.html.", ButtonData:>{ URL[ "http://south.rotol.ramk.fi/keranen/ias2002/ias2002papers.html"], None}, ButtonStyle->"Hyperlink"], "\n[M\[ADoubleDot]] S. M\[ADoubleDot]kel\[ADoubleDot]. Patterns in Words \ (in Finnish). M.Sc. Thesis. Univ. Turku, 2002.\n[Pl]", StyleBox[" ", FontWeight->"Bold"], "P.A.B. Pleasants. Non-repetitive sequences, Proc. Cambridge Phil. Soc., \ 68:267", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "274, 1970.\n[Pr] M. E. Prouhet. M\[EAcute]moire sur quelques relations \ entre les puissances des nombres. C. R. Acad. Sci. Paris, 33:225, 1851.\n[R] \ R.L. Rivest. Abelian square-free dithering for iterated hash functions. MIT. \ Draft; to appear, 2005. Available online at\n", ButtonBox["http://theory.lcs.mit.edu/~rivest/publications.html.", ButtonData:>{ URL[ "http://theory.lcs.mit.edu/~rivest/publications.html"], None}, ButtonStyle->"Hyperlink"], "\n[RS]", StyleBox[" ", FontWeight->"Bold"], "G. Rozenberg and A. Salomaa. The Mathematical Theory of L-systems. \ Academic Press, New York, London, Toronto, Sydney, San Fransisco, 1980.\n\ [S]", StyleBox[" ", FontWeight->"Bold"], "A. Salomaa. Jewels of Formal Language Theory. Computer Science Press, \ Rockville, Maryland, 1981.\n[T1] A. Thue. \[CapitalUDoubleDot]ber unendliche \ Zeichenreihe. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 7:1", Cell[BoxData[ \(TraditionalForm\`\[Dash]\)]], "22, 1906.\n", StyleBox["[T2]", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox["A. Thue. \[CapitalUDoubleDot]ber die gegenseitige Lage gleicher \ Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. \ Christiania, 7:1\[Dash]67, 1912.", FontFamily->"Times New Roman"], "\n[W] S. Wolfram. A New Kind of Science. Wolfram Media, 2002." }], "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell["Automatically Activated Code for the Case of 4 Letters", "Section", CellAutoOverwrite->False], Cell["\<\ Part of the code below is in a kind of preliminary form, though, most likely, \ it will be sufficient for all experimental purposes. \ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(\( (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \[IndentingNewLine]\)\(This\)\)\ file\ was\ generated\ automatically\ by\ the\ \ Mathematica\ front\ end . It\ contains\ Initialization\ cells\ from\ a\ Notebook\ file, which\ typically\ will\ have\ the\ same\ name\ as\ this\ file\ except\ \ ending\ in\ "\<.nb\>"\ instead\ of\ \(\("\<.m\>"\)\(.\)\)\[IndentingNewLine]\ *******************************************************************) \)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\(Off[General::spell];\)\n \(Off[General::spell1];\)\n\[IndentingNewLine] \(SetDirectory["\"];\)\[IndentingNewLine] (*Write\ here\ \ your\ own\ directory\ for\ reading\ and\ saving\ files*) \[IndentingNewLine]\n\ (*\ all\ structures, \ variables\ and\ constants\ starting\ with\ \[Euro]\ are\ global\ \ *) \[IndentingNewLine] \(\[Euro]alphTwoLet = {"\", "\"};\)\[IndentingNewLine] \(\[Euro]alphTwoInt = {0, 1};\)\[IndentingNewLine] \(\[Euro]alphThreeLet = {"\", "\", "\"};\)\[IndentingNewLine] \[Euro]alphThreeInt = {0, 1, 2\^16}; \ \ \ (*\ Words\ of\ length\ \[LessEqual] \ \((2\^16 - 1)\)*4/3\ = \ 87380\ \[IndentingNewLine]are\ safe\ to\ use\ - \ provided\ that\ they\ do\ not\ contain\ xxxx\ for\ a\ letter\ x\ in\ \ \[Euro]alphThreeLet\ *) \[IndentingNewLine]\[Euro]alphFourLet = {"\", "\ \", "\", "\"};\[IndentingNewLine] \[Euro]alphFourInt = {0, 1, 2\^10, 2\^20}; \ \ \ (*\ Words\ of\ length\ \[LessEqual] \ \((2\^10 - 1)\)*2\ = \ 2046\ \[IndentingNewLine]are\ safe\ to\ use\ - \ provided\ that\ they\ do\ not\ contain\ xx\ for\ a\ letter\ x\ in\ \ \[Euro]alphFourLet\ *) \[IndentingNewLine]\[Euro]alphThreeLetInt = \ {\[Euro]alphTwoLet, \[Euro]alphTwoInt};\[IndentingNewLine] \(\[Euro]alphThreeLetInt = {\[Euro]alphThreeLet, \[Euro]alphThreeInt};\)\ \[IndentingNewLine] \(\[Euro]alphFourLetInt = {\[Euro]alphFourLet, \[Euro]alphFourInt};\)\ \[IndentingNewLine]\n \(Clear[convertStrToCumulIntList, convertCumulIntListToStr, convertSuffNCumulIntListToStr, testA2Ffor1LetterLeft, testA2Ffor1LetterRight];\)\n\[IndentingNewLine] \(convertStrToCumulIntList[x_String, \ {alphLet_List, alphInt_List}] := Drop[FoldList[Plus, 0, ReplaceAll[Characters[x], Thread[Rule[alphLet, alphInt]]]], 1];\)\n\[IndentingNewLine] \(convertCumulIntListToStr[x_List, {alphLet_List, alphInt_List}] := ReplaceAll[ Map[#[\([2]\)] - #[\([1]\)] &, Partition[Flatten[{0, x}], 2, 1]], Thread[Rule[alphInt, alphLet]]] // StringJoin;\)\n\[IndentingNewLine] \(convertSuffNCumulIntListToStr[x_List, {alphLet_List, alphInt_List}, n_] := \((ReplaceAll[ Map[#[\([2]\)] - #[\([1]\)] &, Partition[Take[x, \(-n\) - 1], 2, 1]], Thread[Rule[alphInt, alphLet]]] // StringJoin\ )\) /; \ n\ < \ Length[x];\)\n\[IndentingNewLine] \(convertSuffNCumulIntListToStr[x_List, {alphLet_List, alphInt_List}, n_]\ := \ convertCumulIntListToStr[ x, {alphLet, alphInt}]\ /; \ \ n\ \[GreaterEqual] \ Length[x];\)\[IndentingNewLine]\n \ (*\[IndentingNewLine]convertStrToCumulIntList[x_String, 4] := Drop[FoldList[Plus, 0, ReplaceAll[ Characters[ x], {"\" \[Rule] 0, "\" \[Rule] 1, "\" \[Rule] 2\^10, "\" \[Rule] 2\^20}]], 1]\ \ (*\ alphSize = 4\ *) ; \[IndentingNewLine]\[IndentingNewLine]\ convertCumulIntListToStr[x_List, 4] := ReplaceAll[ Map[#[\([2]\)] - #[\([1]\)] &, Partition[Flatten[{0, x}], 2, 1]], {0 \[Rule] "\", 1 \[Rule] "\", 2\^10 \[Rule] "\", 2\^20 \[Rule] "\"}] // StringJoin;\ *) \[IndentingNewLine]\[IndentingNewLine] \(Clear[selectSuffixes];\)\n \(selectSuffixes[pref_String, word_List] := {pref, \(StringCases[#, StartOfString ~~ \(pref ~~ \(x___ ~~ EndOfString\)\) \[Rule] x] &\) /@ word // Flatten};\)\[IndentingNewLine]\n \(Clear[cutPref];\)\n \(cutPref[{x_String, y_String}] := \(StringCases[ x <> "\<.\>" <> y, \((p___ ~~ \(s1__ ~~ \("\<.\>" ~~ \(p___ ~~ s2__\)\)\) \[Rule] s2)\)]\)[\([1]\)];\)\[IndentingNewLine]\n \(Clear[selectSuffixes];\)\n \(selectSuffixes[pref_String, word_List] := {pref, \(StringCases[#, StartOfString ~~ \(pref ~~ \(x___ ~~ EndOfString\)\) \[Rule] x] &\) /@ word // Flatten};\)\n (*Save["\", reducedAddWordList8]; \ \[IndentingNewLine]Save["\", extendWordList4by4];*) \[IndentingNewLine]\[IndentingNewLine] \(Clear[cutPref];\)\n \(cutPref[{x_String, y_String}] := \(StringCases[ x <> "\<.\>" <> y, \((p___ ~~ \(s1__ ~~ \("\<.\>" ~~ \(p___ ~~ s2__\)\)\) \[Rule] s2)\)]\)[\([1]\)];\)\[IndentingNewLine]\n \(Clear[formPrefSuffPairs];\)\n \(formPrefSuffPairs[{x_String, y_String}] := \(StringCases[ x, {p__ ~~ y \[Rule] {p, y}, y \[Rule] {x, "\<\>"}}]\)[\([1]\)];\)\n\[IndentingNewLine] \(Clear[selectSuffixes];\)\n selectSuffixes[pref_String, word_List] := {pref, \(StringCases[#, StartOfString ~~ \(pref ~~ \(x___ ~~ EndOfString\)\) \[Rule] x] &\) /@ word // Flatten}\n\[IndentingNewLine] \(howManyToAppendToRight := Length[\[Euro]reducedWordListSuffNAddNToExpressions[\([\ ordinalIndexForCumulIntegerListN[ Take[\[Euro]cumulIntListRight, \ \(-\[Euro]suffLengthForWhichTryProperExtension\)]], 2]\)]];\)\n \(howManyToAppendToLeft := Length[\[Euro]reducedWordListSuffNAddNToExpressions[\([\ ordinalIndexForCumulIntegerListN[ Take[\[Euro]cumulIntListLeft, \ \(-\[Euro]suffLengthForWhichTryProperExtension\)]], 2]\)]];\)\n\[IndentingNewLine] Clear[ whichToCatenateRL]; (*cumulIntList\ must\ be\ of\ length \ \[GreaterEqual] \[Euro]suffLengthForWhichTryProperExtension + 1*) whichToCatenateRL[cumulIntList_, pointerExtendRL_] := tryProperExtension[ Take[cumulIntList, \(-\[Euro]suffLengthForWhichTryProperExtension\)] \ - cumulIntList[\([Length[ cumulIntList] - \ \[Euro]suffLengthForWhichTryProperExtension]\)], pointerExtendRL];\[IndentingNewLine] \(Clear[whichToCatenateWithKnownSuffRL];\)\n \(whichToCatenateWithKnownSuffRL[suff_, pointerExtendRL_] := tryProperExtension[suff, pointerExtendRL];\)\n\[IndentingNewLine] Clear[ catenateRL]; (*cumulIntList\ must\ be\ of\ length \[GreaterEqual] \ \[Euro]suffLengthForWhichTryProperExtension + 1*) catenateRL[cumulIntList_, pointerExtendRL_] := Join[cumulIntList, Last[cumulIntList] + whichToCatenateRL[cumulIntList, pointerExtendRL]];\[IndentingNewLine] \(Clear[catenateWithKnownAddRL];\)\n \(catenateWithKnownAddRL[cumulIntList_, whichToCatenateRL_] := Join[cumulIntList, Last[cumulIntList] + whichToCatenateRL];\)\n\[IndentingNewLine] (*Clear[ nextProperTrialForNewExtension]; \ \[IndentingNewLine]nextProperTrialForNewExtension[suff_, previousCatenationRL_, howFarExtendedRight_, pointerExtendRL_] := Module[{localExtendRL = pointerExtendRL, prefToCheck = Take[previousCatenationRL, howFarExtendedRight + 1], whatToCatenate, prefOfwhatToCatenate, jjPointerForTest, successTF = True}, While[If[ localExtendRL \[LessEqual] Length[tryProperExtension[suff]], True, successTF = False] && \((prefToCheck === \((prefOfwhatToCatenate = Take[whatToCatenate = whichToCatenateWithKnownSuffRL[suff, localExtendRL], howFarExtendedRight + 1])\))\), \(localExtendRL++\)]; \ \[Euro]pointerExtendRight = localExtendRL; \[IndentingNewLine]If[successTF, jjPointerForTest = \(Position[ prefOfwhatToCatenate - prefToCheck, _?\((# \[NotEqual] 0 &)\)]\)[\([1, 1]\)], jjPointerForTest = 0]; {successTF, whatToCatenate, jjPointerForTest}];*) \[IndentingNewLine]\[IndentingNewLine] Clear[ nextProperTrialForNewExtension2nd]; (*here\ the\ pointerExtendRL\ \ refers\ to\ present\ place - not\ the\ next*) nextProperTrialForNewExtension2nd[suff_, 0, 0] := {True, whichToCatenateWithKnownSuffRL[suff, 1], 1, 1};\[IndentingNewLine] \(nextProperTrialForNewExtension2nd[suff_, pointerExtendRL_, _] := {False, {}, 0, 0} /; pointerExtendRL \[GreaterEqual] Length[tryProperExtension[suff]];\)\n\[IndentingNewLine] \(nextProperTrialForNewExtension2nd[suff_, pointerExtendRL_, 0] := {True, whichToCatenateWithKnownSuffRL[suff, pointerExtendRL + 1], pointerExtendRL + 1, 1} /; pointerExtendRL > 0;\)\n\[IndentingNewLine] \(nextProperTrialForNewExtension2nd[suff_, pointerExtendRL_, howFarExtendedRight_] := Module[{localExtendRL = pointerExtendRL + 1, prefToCheck = Take[whichToCatenateWithKnownSuffRL[suff, If[pointerExtendRL \[Equal] 0, 1, pointerExtendRL]], howFarExtendedRight + 1], whatToCatenate, prefOfwhatToCatenate, jjPointerForTest, successTF = True}, While[If[ localExtendRL \[LessEqual] Length[tryProperExtension[suff]], True, successTF = False] && \((prefToCheck === \((prefOfwhatToCatenate = Take[whatToCatenate = whichToCatenateWithKnownSuffRL[suff, localExtendRL], howFarExtendedRight + 1])\))\), \(localExtendRL++\)]; \ \[IndentingNewLine]If[successTF, jjPointerForTest = \(Position[ prefOfwhatToCatenate - prefToCheck, _?\((# \[NotEqual] 0 &)\)]\)[\([1, 1]\)], jjPointerForTest = 0]; {successTF, whatToCatenate, localExtendRL, jjPointerForTest}];\)\n\[IndentingNewLine] \(Clear[nextProperTrialForNewExtensionFin];\)\n nextProperTrialForNewExtensionFin[cumulIntList_, pointerExtendRL_, howFarExtendedRight_] := nextProperTrialForNewExtension2nd[ Take[cumulIntList, \(-\[Euro]suffLengthForWhichTryProperExtension\)] \ - cumulIntList[\([Length[ cumulIntList] - \ \[Euro]suffLengthForWhichTryProperExtension]\)], pointerExtendRL, howFarExtendedRight]; Clear[ testA2RLpairCumulIntList];\[IndentingNewLine] \ (*testA2RLpairCumulIntList\ works\ also\ in\ the\ case\ of\ \ \[Euro]alphFourLet, if\ preCheckedSuffLength - addedOrReplacedWordLength > 1. \ Below\ x\ refers\ to\ \[Euro]cumulIntListRight\ and\ y\ refers\ \ to\ \[Euro]cumulIntListLeft, or\ the\ other\ way\ \(\(round\)\(.\)\)*) \[IndentingNewLine]\ \[IndentingNewLine] \(testA2RLpairCumulIntList[x_List, y_List, preCheckedSuffLength_, addedOrReplacedWordLength_, jj_: 1] := Module[{rLen = Length[x] - addedOrReplacedWordLength, lLen = Length[y], j = jj, res = True}, While[j \[LessEqual] addedOrReplacedWordLength, i = Floor[\((preCheckedSuffLength - addedOrReplacedWordLength + j)\)/2] + 1; \[IndentingNewLine]While[\((i < Floor[\((rLen + j)\)/ 2])\) && \((res = \((x[\([rLen + j]\)] - x[\([rLen + j - i]\)] === x[\([rLen + j - i]\)] - x[\([rLen + j - 2 i]\)] // Not)\))\), \(i++\)]; \[IndentingNewLine]If[\((2 i \[LessEqual] rLen + j)\) && res, If[OddQ[rLen + j], res = \((x[\([rLen + j]\)] - x[\([Ceiling[\((rLen + j)\)/2]]\)] === x[\([Ceiling[\((rLen + j)\)/2]]\)] - x[\([1]\)] // Not)\); \(i++\), res = \((x[\([rLen + j]\)] - x[\([\((rLen + j)\)/2]\)] === x[\([\((rLen + j)\)/2]\)] // Not)\); \(i++\)]]; \[IndentingNewLine]If[res // Not, Return[{res, j}]]; \[IndentingNewLine]While[\((i < rLen + j)\) && \((2 i \[LessEqual] lLen + rLen + j)\) && \((res = \((x[\([rLen + j]\)] - x[\([rLen + j - i]\)] === x[\([rLen + j - i]\)] + y[\([2 i - rLen - j]\)] // Not)\))\), \(i++\)]; \[IndentingNewLine]If[res, If[\((2 i \[LessEqual] lLen + rLen + j)\), res = \((x[\([rLen + j]\)] === y[\([2 i - rLen - j]\)] // Not)\)]]; \[IndentingNewLine]If[res // Not, Return[{res, j}]]; \[IndentingNewLine]\(i++\); \[IndentingNewLine]While[\ \((2 i \[LessEqual] lLen + rLen + j)\) && \((res = \((x[\([rLen + j]\)] + y[\([i - rLen - j]\)] === y[\([2 i - rLen - j]\)] - y[\([i - rLen - j]\)] // Not)\))\), \(i++\)]; \[IndentingNewLine]If[ res // Not, Return[{res, j}]]; \[IndentingNewLine]\(j++\)]; \[IndentingNewLine]{res, j - 1}];\)\[IndentingNewLine]\n \ (*\[Euro]bottomBoundaryLengthRight\ and\ \[Euro]bottomBoundaryLengthLeft\ \ will\ be\ computed\ in\ generateRLthree["\"] . The\ maximum\ length\ \((total)\)\ is\ \ \[Euro]bottomBoundaryLengthRight + \[Euro]extensionBoundaryLengthRight + \ \[Euro]bottomBoundaryLengthLeft + \[Euro]extensionBoundaryLengthLeft\ *) \[IndentingNewLine]\[IndentingNewLine] (*for\ example; \ \[Euro]preCheckedSuffLength - \[Euro]addWordLength\ and\ \ \[Euro]preCheckedSuffLength - \[Euro]addedOrReplacedWordLength \((which\ is\ \ kind\ of\ dynamically\ changing\ variable)\)\ represent\ the\ length\ of\ the\ \ a2free\ factor\ that\ does\ not\ require\ testing*) \[IndentingNewLine]\ \[IndentingNewLine] \(Clear[generateRLthree];\)\n \(generateRLthree["\"] := \((\[Euro]wordLeft = StringTake[\[Euro]givenWordForTest, Ceiling[StringLength[\[Euro]givenWordForTest]/ 2]]; \[Euro]wordRight = StringTake[\[Euro]givenWordForTest, \(-Floor[ StringLength[\[Euro]givenWordForTest]/ 2]\)]; \[IndentingNewLine]\[Euro]cumulIntListLeft = convertStrToCumulIntList[ StringReverse[\[Euro]wordLeft], \[Euro]alphLetInt]; \ (*\[Euro]cumulIntListLeft\ also\ grows\ from\ left\ to\ right*) \ \[Euro]cumulIntListRight = convertStrToCumulIntList[\[Euro]wordRight, \[Euro]alphLetInt]; \ \[IndentingNewLine]\[Euro]bottomBoundaryLengthRight = Length[\[Euro]cumulIntListRight]; \ \[IndentingNewLine]\[Euro]bottomBoundaryLengthLeft = Length[\[Euro]cumulIntListLeft]; \[IndentingNewLine]\[Euro]state \ = "\";)\);\)\[IndentingNewLine]\n \(generateRLthree["\"] := If[Length[\[Euro]cumulIntListLeft] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthLeft + \[Euro]extensionBoundaryLengthLeft, \ \[Euro]howFarExtendedRight = 0; \[Euro]state = "\", \[Euro]howFarExtendedRight = 0; \[IndentingNewLine]\[Euro]indexListLeft = Append[\[Euro]indexListLeft, 0]; \[Euro]state = "\"] /; Length[\[Euro]cumulIntListRight] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthRight + \[Euro]extensionBoundaryLengthRight;\)\ \[IndentingNewLine]\n \(generateRLthree["\"] := If[\(({\[Euro]successTF, \[Euro]whatToCatenateRight, \ \[Euro]pointerExtendRight, \[Euro]howFarExtendedRight} = nextProperTrialForNewExtensionFin[\[Euro]cumulIntListRight, Last[\[Euro]indexListRight], \ \[Euro]howFarExtendedRight])\)[\([1]\)], \[Euro]cumulIntListRight = catenateWithKnownAddRL[\[Euro]cumulIntListRight, \ \[Euro]whatToCatenateRight]; \[Euro]indexListRight = ReplacePart[\[Euro]indexListRight, \[Euro]pointerExtendRight, Length[\[Euro]indexListRight]]; \[Euro]state = "\", \ \[Euro]indexListRight = Take[\[Euro]indexListRight, Length[\[Euro]indexListRight] - 1]; \[Euro]state = "\"];\)\[IndentingNewLine]\n \(generateRLthree["\"] := Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListRight, \ \[Euro]cumulIntListLeft, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedRight]}, If[testResult[\([1]\)], \[Euro]howFarExtendedRight = 0; \[Euro]indexListLeft = Append[\[Euro]indexListLeft, 0]; \[Euro]state = "\", \ \[Euro]howFarExtendedRight = testResult[\([2]\)] - 1; \[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \[Euro]addWordLength]; \ \[Euro]state = "\"]];\)\[IndentingNewLine]\n \(generateRLthree["\"] := If[Length[\[Euro]cumulIntListRight] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthRight + \[Euro]extensionBoundaryLengthRight, \ \[Euro]howFarExtendedLeft = 0; \[Euro]state = "\", \[Euro]howFarExtendedLeft = 0; \[Euro]indexListRight = Append[\[Euro]indexListRight, 0]; \[Euro]state = "\"] /; Length[\[Euro]cumulIntListLeft] \[GreaterEqual] \ \[Euro]bottomBoundaryLengthLeft + \[Euro]extensionBoundaryLengthLeft;\)\ \[IndentingNewLine]\n \(generateRLthree["\"] := If[\(({\[Euro]successTF, \[Euro]whatToCatenateLeft, \ \[Euro]pointerExtendLeft, \[Euro]howFarExtendedLeft} = nextProperTrialForNewExtensionFin[\[Euro]cumulIntListLeft, Last[\[Euro]indexListLeft], \ \[Euro]howFarExtendedLeft])\)[\([1]\)], \[Euro]cumulIntListLeft = catenateWithKnownAddRL[\[Euro]cumulIntListLeft, \ \[Euro]whatToCatenateLeft]; \[Euro]indexListLeft = ReplacePart[\[Euro]indexListLeft, \[Euro]pointerExtendLeft, Length[\[Euro]indexListLeft]]; \[Euro]state = "\", \ \[Euro]indexListLeft = Take[\[Euro]indexListLeft, Length[\[Euro]indexListLeft] - 1]; \[Euro]state = "\"];\)\[IndentingNewLine]\n \(generateRLthree["\"] := Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListLeft, \ \[Euro]cumulIntListRight, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedLeft]}, If[testResult[\([1]\)], \[Euro]howFarExtendedLeft = 0; \[Euro]indexListRight = Append[\[Euro]indexListRight, 0]; \[Euro]state = "\", \ \[Euro]howFarExtendedLeft = testResult[\([2]\)] - 1; \[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \[Euro]addWordLength]; \ \[Euro]state = "\"]];\)\[IndentingNewLine]\n \(generateRLthree["\"] := \((\[Euro]state = "\")\ \) /; \((\((Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight)\) \[And] \((Length[\[Euro]cumulIntListLeft] \ \[Equal] \[Euro]bottomBoundaryLengthLeft)\))\);\)\n \(generateRLthree["\"] := \((\[Euro]state = \ "\")\) /; \((\((Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight)\) \[And] \((Length[\[Euro]cumulIntListLeft] \ > \[Euro]bottomBoundaryLengthLeft)\))\);\)\[IndentingNewLine]\n \(generateRLthree["\"] := \((\[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \[Euro]addWordLength]; \ \[Euro]howFarExtendedRight = 0; \[Euro]state = "\")\) /; Length[\[Euro]cumulIntListRight] \[Equal] \ \[Euro]bottomBoundaryLengthRight + \[Euro]addWordLength;\)\n\ \[IndentingNewLine] \(generateRLthree["\"] := \((\[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \[Euro]addWordLength]; \ \[Euro]howFarExtendedRight = 0; \[Euro]state = "\")\) /; Length[\[Euro]cumulIntListRight] > \[Euro]bottomBoundaryLengthRight \ + \[Euro]addWordLength;\)\[IndentingNewLine]\[IndentingNewLine] \(generateRLthree["\"] := If[\(({\[Euro]successTF, \[Euro]whatToCatenateRight, \ \[Euro]pointerExtendRight, \[Euro]howFarExtendedRight} = nextProperTrialForNewExtensionFin[\[Euro]\ cumulIntListOfGivenWordForTest, Last[\[Euro]indexListRight], \ \[Euro]howFarExtendedRight])\)[\([1]\)], \[Euro]cumulIntListRight = catenateWithKnownAddRL[\[Euro]cumulIntListRight, \ \[Euro]whatToCatenateRight]; \[Euro]indexListRight = ReplacePart[\[Euro]indexListRight, \[Euro]pointerExtendRight, Length[\[Euro]indexListRight]]; \[Euro]state = \ "\", \[Euro]indexListRight = Take[\[Euro]indexListRight, Length[\[Euro]indexListRight] - 1]; \[Euro]state = "\"];\)\n\[IndentingNewLine] \(generateRLthree["\"] := Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListRight, \ \[Euro]cumulIntListLeft, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedRight]}, If[testResult[\([1]\)], \[Euro]howFarExtendedRight = 0; \[Euro]indexListLeft = Append[\[Euro]indexListLeft, 0]; \[Euro]state = "\", \ \[Euro]howFarExtendedRight = testResult[\([2]\)] - 1; \[Euro]cumulIntListRight = Take[\[Euro]cumulIntListRight, Length[\[Euro]cumulIntListRight] - \[Euro]addWordLength]; \ \[Euro]state = "\"]];\)\n\[IndentingNewLine] \(generateRLthree["\"] := \((\[Euro]state = \ "\")\) /; \((\((Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft)\) \[And] \((Length[\[Euro]cumulIntListRight] \ \[Equal] \[Euro]bottomBoundaryLengthRight)\))\);\)\n \(generateRLthree["\"] := \((\[Euro]state = \ "\")\) /; \((\((Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft)\) \[And] \((Length[\[Euro]cumulIntListRight] \ > \[Euro]bottomBoundaryLengthRight)\))\);\)\n\[IndentingNewLine] \(\(generateRLthree["\"] := \((\[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \[Euro]addWordLength]; \ \[Euro]howFarExtendedLeft = 0; \[Euro]state = "\")\) /; Length[\[Euro]cumulIntListLeft] \[Equal] \ \[Euro]bottomBoundaryLengthLeft + \[Euro]addWordLength;\);\)\n\ \[IndentingNewLine] \(generateRLthree["\"] := \((\[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \[Euro]addWordLength]; \ \[Euro]howFarExtendedLeft = 0; \[Euro]state = "\")\) /; Length[\[Euro]cumulIntListLeft] > \[Euro]bottomBoundaryLengthLeft + \ \[Euro]addWordLength;\)\n\[IndentingNewLine] \(generateRLthree["\"] := If[\(({\[Euro]successTF, \[Euro]whatToCatenateLeft, \ \[Euro]pointerExtendLeft, \[Euro]howFarExtendedLeft} = nextProperTrialForNewExtensionFin[\[Euro]\ cumulIntListReverseOfGivenWordForTest, Last[\[Euro]indexListLeft], \ \[Euro]howFarExtendedLeft])\)[\([1]\)], \[Euro]cumulIntListLeft = catenateWithKnownAddRL[\[Euro]cumulIntListLeft, \ \[Euro]whatToCatenateLeft]; \[Euro]indexListLeft = ReplacePart[\[Euro]indexListLeft, \[Euro]pointerExtendLeft, Length[\[Euro]indexListLeft]]; \[Euro]state = \ "\", \[Euro]indexListLeft = Take[\[Euro]indexListLeft, Length[\[Euro]indexListLeft] - 1]; \[Euro]state = "\"];\)\n\[IndentingNewLine] \(generateRLthree["\"] := Module[{testResult = testA2RLpairCumulIntList[\[Euro]cumulIntListLeft, \ \[Euro]cumulIntListRight, \[Euro]preCheckedSuffLength, \[Euro]addWordLength, \ \[Euro]howFarExtendedLeft]}, If[testResult[\([1]\)], \[Euro]howFarExtendedLeft = 0; \[Euro]indexListRight = Append[\[Euro]indexListRight, 0]; \[Euro]state = "\", \ \[Euro]howFarExtendedLeft = testResult[\([2]\)] - 1; \[Euro]cumulIntListLeft = Take[\[Euro]cumulIntListLeft, Length[\[Euro]cumulIntListLeft] - \[Euro]addWordLength]; \ \[Euro]state = "\"]];\)\n\[IndentingNewLine] \(generateRLthree["\"] := Print["\", \[Euro]givenWordForTest, "\< the extension \ failed!\>"];\)\n \(generateRLthree["\"] := Print["\", \[Euro]givenWordForTest, "\< the extension was \ successful!\>"];\)\[IndentingNewLine]\n generateRL = generateRLthree\ ; (*\ Indeed, also\ the\ four\ letter\ case, in\ which\ no\ abelian\ squares\ are\ allowed, will\ work\ all\ right\ in\ our\ \(setting!\)\ *) \[IndentingNewLine]\n Clear[initialise];\n \(initialise[howManyStepsLeftAndRight_, givenWordForTest_, alphLetInt_: \[Euro]alphThreeLetInt] := \((\[Euro]alphLetInt = alphLetInt; (*\[Euro]alphLetInt\ needs\ to\ be\ set\ to\ \ \[Euro]alphThreeLetInt\ or\ to\ \[Euro]alphFourLetInt*) If[ alphLetInt \[Equal] \[Euro]alphThreeLetInt, << \ "\", << "\"]; \ \[IndentingNewLine]\[Euro]addWordLength = 4; \[IndentingNewLine]\[Euro]preCheckedSuffLength = 8; \[IndentingNewLine]\[Euro]suffLengthForWhichTryProperExtension \ = \[Euro]preCheckedSuffLength - \[Euro]addWordLength; \[IndentingNewLine]\ \[Euro]reducedWordListSuffNAddNToExpressions = \ \[Euro]reducedWordListSuff4Add4ToExpressions; \ \[IndentingNewLine]ordinalIndexForCumulIntegerListN = ordinalIndexForCumulIntegerList4; (*function\ names*) \ \[Euro]pointerExtendRight = 0; \[IndentingNewLine]\[Euro]pointerExtendLeft = 0; \[IndentingNewLine]\[Euro]extensionBoundaryLengthRight = howManyStepsLeftAndRight*\[Euro]addWordLength; \ \[IndentingNewLine]\[Euro]extensionBoundaryLengthLeft = howManyStepsLeftAndRight*\[Euro]addWordLength; (*Indeed, both\ of\ these\ should\ be\ equal\ and\ of\ the\ form\ \ k*\[Euro]addWordLength\ for\ an\ positive\ integer\ k*) \ \[Euro]howFarExtendedRight = 0; \[IndentingNewLine]\[Euro]howFarExtendedLeft = 0; \[IndentingNewLine]\[Euro]indexListRight = {0}; \ \[IndentingNewLine]\[Euro]indexListLeft = {}; \ \[IndentingNewLine]\[Euro]givenWordForTest = givenWordForTest; \ \[IndentingNewLine]\[Euro]cumulIntListOfGivenWordForTest = convertStrToCumulIntList[\[Euro]givenWordForTest, \ \[Euro]alphLetInt]; \ \[IndentingNewLine]\[Euro]cumulIntListReverseOfGivenWordForTest = convertStrToCumulIntList[ StringReverse[\[Euro]givenWordForTest], \ \[Euro]alphLetInt])\);\)\n\[IndentingNewLine] \(Clear[s];\)\n \(s[x_List] := Apply[Plus, x];\)\n \(s[x_String] := Apply[Plus, Characters[x]];\)\n\[IndentingNewLine] \(Clear[test3Suff];\)\n \(test3Suff[x_] := Module[{sl = StringLength[x], i = 1, res = True}, While[i < Floor[sl/2] && \((res = s[StringTake[x, {sl - 1 - 2 i, sl - 1 - i}]] === s[StringTake[x, {sl - i, sl}]] // Not)\), \(i++\)]; \[IndentingNewLine]res];\)\n\ \[IndentingNewLine] \(Clear[test3A2];\)\n \(test3A2[x_, a2FreePrefLen_: 4] := If[StringLength[x] < 4, "\", Module[{sl = StringLength[x], j = 0, res = True}, While[a2FreePrefLen + j \[LessEqual] sl && \((res = test3Suff[ StringTake[x, a2FreePrefLen + j]])\), \(j++\)]; \[IndentingNewLine]\((asp = a2FreePrefLen + j; res)\)]];\)\n\[IndentingNewLine] \(Clear[test3SuffVis];\)\n \(test3SuffVis[x_] := Module[{sl = StringLength[x], i = 1, res}, While[i < Floor[sl/2] && \((res = s[StringTake[x, {sl - 1 - 2 i, sl - 1 - i}]] === s[StringTake[x, {sl - i, sl}]] // Not)\), \(i++\)]; \[IndentingNewLine]If[res, True, {StringTake[x, {sl - 1 - 2 i, sl - 1 - i}], StringTake[x, {sl - i, sl}], 2 \((i + 1)\)}]];\)\n\[IndentingNewLine] \(Clear[test3A2Vis];\)\n \(test3A2Vis[x_, a2FreePrefLen_: 4] := If[StringLength[x] < 2, "\", Module[{sl = StringLength[x], i = 0, res}, While[a2FreePrefLen + i \[LessEqual] sl, If[\((res = test3SuffVis[StringTake[x, a2FreePrefLen + i]])\) === True, True, Print[res]]; \(i++\)]; \[IndentingNewLine]"\"]];\)\n\ \[IndentingNewLine] \(Clear[testSuff];\)\n \(testSuff[x_] := Module[{sl = StringLength[x], i = 0, res = True}, While[i < Floor[sl/2] && \((res = s[StringTake[x, {sl - 1 - 2 i, sl - 1 - i}]] === s[StringTake[x, {sl - i, sl}]] // Not)\), \(i++\)]; \[IndentingNewLine]res];\)\n\ \[IndentingNewLine] \(Clear[testSuffVis];\)\n \(testSuffVis[x_] := Module[{sl = StringLength[x], i = 0, res}, While[i < Floor[sl/2] && \((res = s[StringTake[x, {sl - 1 - 2 i, sl - 1 - i}]] === s[StringTake[x, {sl - i, sl}]] // Not)\), \(i++\)]; \[IndentingNewLine]If[res, True, {StringTake[x, {sl - 1 - 2 i, sl - 1 - i}], StringTake[x, {sl - i, sl}], 2 \((i + 1)\)}]];\)\n\[IndentingNewLine] \(Clear[testA2];\)\n \(testA2[x_, a2FreePrefLen_: 2] := If[StringLength[x] < 2, "\", Module[{sl = StringLength[x], i = 0, res}, While[a2FreePrefLen + i \[LessEqual] sl && \((res = testSuff[ StringTake[x, a2FreePrefLen + i]])\), \(i++\)]; \[IndentingNewLine]\((asp = a2FreePrefLen + i; res)\)]];\)\n\[IndentingNewLine] \(Clear[testA2Mir];\)\n \(testA2Mir[x_, a2FreePrefLen_: 2] := testA2[StringReverse[x], a2FreePrefLen];\)\n\[IndentingNewLine] \(Clear[testA2Vis];\)\n testA2Vis[x_, a2FreePrefLen_: 2] := If[StringLength[x] < 2, "\", Module[{sl = StringLength[x], i = 0, res}, While[a2FreePrefLen + i \[LessEqual] sl, If[\((res = testSuffVis[StringTake[x, a2FreePrefLen + i]])\) === True, True, Print[res]]; \(i++\)]; \[IndentingNewLine]"\"]]\n\ \[IndentingNewLine] \(Clear[testA2MirVis];\)\n \(testA2MirVis[x_, a2FreePrefLen_: 2] := testA2Vis[StringReverse[x], a2FreePrefLen];\)\n\[IndentingNewLine] \(Clear[myPrint];\)\n \(myPrint["\<3 letters\>"] := \((Print["\<\[Euro]cumulIntListLeft as \ reverse string | \[Euro]cumulIntListRight as string = \>", convertCumulIntListToStr[\[Euro]cumulIntListLeft, \ \[Euro]alphThreeLetInt] // StringReverse, "\<|\>", convertCumulIntListToStr[\[Euro]cumulIntListRight, \ \[Euro]alphThreeLetInt]]; \ \[IndentingNewLine]Print["\<\[Euro]howFarExtendedLeft = \>", \ \[Euro]howFarExtendedLeft]; \[IndentingNewLine]Print["\<\[Euro]indexListLeft \ = \>", \[Euro]indexListLeft]; \ \[IndentingNewLine]Print["\<\[Euro]howFarExtendedRight = \>", \ \[Euro]howFarExtendedRight]; \ \[IndentingNewLine]Print["\<\[Euro]indexListRight = \>", \ \[Euro]indexListRight]; \[IndentingNewLine]Print["\<\[Euro]state = \>", \ \[Euro]state])\);\)\n\[IndentingNewLine] myPrint["\<4 letters\>"] := \((Print["\<\[Euro]cumulIntListLeft as \ reverse string | \[Euro]cumulIntListRight as string = \>", convertCumulIntListToStr[\[Euro]cumulIntListLeft, \ \[Euro]alphFourLetInt] // StringReverse, "\<|\>", convertCumulIntListToStr[\[Euro]cumulIntListRight, \ \[Euro]alphFourLetInt]]; \ \[IndentingNewLine]Print["\<\[Euro]howFarExtendedLeft = \>", \ \[Euro]howFarExtendedLeft]; \[IndentingNewLine]Print["\<\[Euro]indexListLeft \ = \>", \[Euro]indexListLeft]; \ \[IndentingNewLine]Print["\<\[Euro]howFarExtendedRight = \>", \ \[Euro]howFarExtendedRight]; \ \[IndentingNewLine]Print["\<\[Euro]indexListRight = \>", \ \[Euro]indexListRight]; \[IndentingNewLine]Print["\<\[Euro]state = \>", \ \[Euro]state])\)\n\[IndentingNewLine] \(Clear[s];\)\n \(s[x_List] := Apply[Plus, x];\)\n \(s[x_String] := Apply[Plus, Characters[x]];\)\n \(Clear[testSuff];\)\n testSuff[x_] := Module[{sl = StringLength[x], i = 0, res = True}, While[i < Floor[sl/2] && \((res = s[StringTake[x, {sl - 1 - 2 i, sl - 1 - i}]] === s[StringTake[x, {sl - i, sl}]] // Not)\), \(i++\)]; \[IndentingNewLine]res]\n\ \[IndentingNewLine] \(Clear[testSuffVis];\)\n testSuffVis[x_] := Module[{sl = StringLength[x], i = 0, res}, While[i < Floor[sl/2] && \((res = s[StringTake[x, {sl - 1 - 2 i, sl - 1 - i}]] === s[StringTake[x, {sl - i, sl}]] // Not)\), \(i++\)]; \[IndentingNewLine]If[res, True, {StringTake[x, {sl - 1 - 2 i, sl - 1 - i}], StringTake[x, {sl - i, sl}], 2 \((i + 1)\)}]]\n\[IndentingNewLine] \(Clear[testA2];\)\n testA2[x_, a2FreePrefLen_: 2] := If[StringLength[x] < 2, "\", Module[{sl = StringLength[x], i = 0, res}, While[a2FreePrefLen + i \[LessEqual] sl && \((res = testSuff[ StringTake[x, a2FreePrefLen + i]])\), \(i++\)]; \[IndentingNewLine]\((asp = a2FreePrefLen + i; res)\)]]\[IndentingNewLine]\n\[IndentingNewLine] \(Clear[testA2Mir];\)\n testA2Mir[x_, a2FreePrefLen_: 2] := testA2[StringReverse[x], a2FreePrefLen]\n\[IndentingNewLine] \(Clear[testA2Vis];\)\n testA2Vis[x_, a2FreePrefLen_: 2] := If[StringLength[x] < 2, "\", Module[{sl = StringLength[x], i = 0, res}, While[a2FreePrefLen + i \[LessEqual] sl, If[\((res = testSuffVis[StringTake[x, a2FreePrefLen + i]])\) === True, True, Print[res]]; \(i++\)]; \[IndentingNewLine]"\"]]\n\ \[IndentingNewLine] \(Clear[testA2MirVis];\)\n testA2MirVis[x_, a2FreePrefLen_: 2] := testA2Vis[StringReverse[x], a2FreePrefLen]\n\[IndentingNewLine] \(Clear[testAllPattern];\)\n \(testAllPattern[{___, x_, _, x_, _, x_, _, x_, _, x_, _, x_, ___}] := False;\)\n \(testAllPattern[{___, a_, b_, c_, a_, d_, b_, a_, d_, ___}] := False;\)\n \(testAllPattern[{___, d_, a_, b_, d_, a_, c_, b_, a_, ___}] := False;\)\n \(testAllPattern[{___, a_, b_, a_, c_, d_, a_, b_, a_, ___}] := False;\)\n \(testAllPattern[{___, a_, b_, a_, c_, d_, b_, a_, b_, ___}] := False;\)\n \(testAllPattern[{___, a_, b_, c_, a_, b_, d_, a_, b_, ___}] := False;\)\n \(testAllPattern[_] := True;\)\n\[IndentingNewLine] \(Clear[allPerm];\)\n allPerm[ x_String] := \(StringReplace[x, Thread[Rule[{"\", "\", "\", "\"}, #]]] &\) /@ Permutations[{"\", "\", "\", "\"}]\n\ \[IndentingNewLine] \(Clear[deletePermutations];\)\n deletePermutations[{x_List, y_List}] := {Append[x, y[\([1]\)]], Complement[Rest[y], allPerm[y[\([1]\)]]]}\n\[IndentingNewLine] \(Clear[deleteMirrorPermutations];\)\n deleteMirrorPermutations[{x_List, y_List}] := {Append[x, y[\([1]\)]], Complement[Rest[y], allPerm[StringReverse[y[\([1]\)]]]]}\n\[IndentingNewLine] \(Clear[prod, myRepl1, myRepl2, myRepl];\)\n \(prod = {"\" \[Rule] "\", "\" \[Rule] "\", "\" \ \[Rule] "\", "\" \[Rule] "\"};\)\n \(myRepl1[x_String] := StringReplace[x, prod];\)\n \(myRepl2[x_String] := "\<{___,\>" <> x <> "\<___}\>";\)\n \(myRepl[x_String] := myRepl2[myRepl1[x]];\)\n\[IndentingNewLine] \(Clear[abacXabcaXabcbXabcdPREF];\)\n \(abacXabcaXabcbXabcdPREF[{a_, b_, a_, c_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", \(Complement[{"\", "\", "\", \ "\"}, {a, b, c}]\)[\([1]\)] \[Rule] "\"};\)\n \(abacXabcaXabcbXabcdPREF[{a_, b_, c_, a_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", \(Complement[{"\", "\", "\", \ "\"}, {a, b, c}]\)[\([1]\)] \[Rule] "\"};\)\n \(abacXabcaXabcbXabcdPREF[{a_, b_, c_, b_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", \(Complement[{"\", "\", "\", \ "\"}, {a, b, c}]\)[\([1]\)] \[Rule] "\"};\)\n \(abacXabcaXabcbXabcdPREF[{a_, b_, c_, d_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", d \[Rule] "\"} /; Sort[{a, b, c, d}] === {"\", "\", "\", "\"};\)\n\ \[IndentingNewLine] Clear[abacXabcaXabcbXabcdRule]; (*same\ as\ abacXabcaXabcbXabcdPREF*) abacXabcaXabcbXabcdRule[{a_, b_, a_, c_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", \(Complement[{"\", "\", "\", "\"}, \ {a, b, c}]\)[\([1]\)] \[Rule] "\"};\[IndentingNewLine] \(abacXabcaXabcbXabcdRule[{a_, b_, c_, a_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", \(Complement[{"\", "\", "\", \ "\"}, {a, b, c}]\)[\([1]\)] \[Rule] "\"};\)\n \(abacXabcaXabcbXabcdRule[{a_, b_, c_, b_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", \(Complement[{"\", "\", "\", \ "\"}, {a, b, c}]\)[\([1]\)] \[Rule] "\"};\)\n \(abacXabcaXabcbXabcdRule[{a_, b_, c_, d_}] := {a \[Rule] "\", b \[Rule] "\", c \[Rule] "\", d \[Rule] "\"} /; Sort[{a, b, c, d}] === {"\", "\", "\", "\"};\)\n\ \[IndentingNewLine] \(Clear[abacXabcaXabcbXabcdFORM];\)\n \(abacXabcaXabcbXabcdFORM[x_String] := StringReplace[x, abacXabcaXabcbXabcdPREF[ Characters[StringTake[x, 4]]]];\)\n\[IndentingNewLine] \(Clear[abacXabcaXabcbXabcdFactorFORM];\)\n abacXabcaXabcbXabcdFactorFORM[x_String, factorPointer_] := StringReplace[x, abacXabcaXabcbXabcdRule[ Characters[StringTake[x, {factorPointer, factorPointer + 3}]]]]; Clear[myR1Select, myL1Select, myRL];\[IndentingNewLine] \(myR1Select[x_String] := If[testSuff[x], x, {}];\)\n \(myL1Select[x_String] := If[testSuff[StringReverse[x]], x, {}];\)\n \(myRL[{w_List, len_, state_}] := Module[{wList = \((\(\((\(myR1Select[#] &\) /@ catenate1Right[#])\) &\) /@ w)\) // Flatten}, {wList, Length[wList], "\"}] /; state \[Equal] "\";\)\n \(myRL[{w_List, len_, state_}] := Module[{wList = \((\(\((\(myL1Select[#] &\) /@ catenate1Left[#])\) &\) /@ w)\) // Flatten}, {wList, Length[wList], "\"}] /; state \[Equal] "\";\)\n\[IndentingNewLine] \(words2 = {"\", "\", "\", "\", "\", "\", "\ \", "\", "\", "\", "\", "\"};\)\n\ \[IndentingNewLine] \(words3 = {"\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\", "\"};\)\n\[IndentingNewLine] \(words4 = {"\", "\", "\", "\", "\", "\ \", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\"};\)\)\)\)], "Input", CellLabel->"In[196]:=", InitializationCell->True, CellAutoOverwrite->False], Cell[TextData[{ "The code in the package ", StyleBox[" ", FontWeight->"Bold"], StyleBox["alphFourLetIntCase.m", FontFamily->"Courier New", FontWeight->"Bold"], " (partly in a preliminary form) follows:" }], "Text", CellAutoOverwrite->False], Cell[BoxData[ \(\(\( (*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ \[IndentingNewLine]\)\(This\)\)\ file\ was\ generated\ automatically\ by\ the\ \ Mathematica\ front\ end . It\ contains\ Initialization\ cells\ from\ a\ Notebook\ file, which\ typically\ will\ have\ the\ same\ name\ as\ this\ file\ except\ \ ending\ in\ "\<.nb\>"\ instead\ of \(\("\<.m\>"\)\(.\)\)\[IndentingNewLine]\ ***********************************************************************) \)\(\ \[IndentingNewLine]\)\(\(Off[General::spell];\)\n \(Off[General::spell1];\)\n\[IndentingNewLine] \(Clear[cutPref];\)\n \(cutPref[{x_String, y_String}] := \(StringCases[ x <> "\<.\>" <> y, \((p___ ~~ \(s1__ ~~ \("\<.\>" ~~ \(p___ ~~ s2__\)\)\) \[Rule] s2)\)]\)[\([1]\)];\)\n\[IndentingNewLine] \(Clear[formPrefSuffPairs];\)\n \(formPrefSuffPairs[{x_String, y_String}] := \(StringCases[ x, {p__ ~~ y \[Rule] {p, y}, y \[Rule] {x, "\<\>"}}]\)[\([1]\)];\)\n\[IndentingNewLine] \(Clear[selectSuffixes];\)\n selectSuffixes[pref_String, word_List] := {pref, \(StringCases[#, StartOfString ~~ \(pref ~~ \(x___ ~~ EndOfString\)\) \[Rule] x] &\) /@ word // Flatten}\n\[IndentingNewLine] \(\[Euro]reducedWordListSuff4Add4ToExpressions = {{abac, {abad, abda, abdb, abdc, adab, adac, adba, adbc, adbd, adca, adcb, adcd, babd, bada, badb, badc, bcda, bcdb, bcdc, bdab, bdac, bdad, bdba, bdbc, bdca, bdcb, bdcd, daba, dabc, dabd, daca, dacb, dadb, dbab, dbac, dbad, dbca, dbcb, dbcd, dbda, dbdc, dcab, dcac, dcba, dcbc, dcbd}}, {abca, {bada, badb, badc, bdab, bdac, bdad, bdba, bdbc, bdca, bdcb, bdcd, cdab, cdac, cdad, cdba, cdbc, cdbd, cdca, cdcb, daba, dabc, dabd, daca, dacb, dbab, dbac, dbad, dbca, dbcb, dbcd, dbda, dbdc, dcab, dcac, dcba, dcbc, dcbd, dcdb}}, {abcb, {abcd, abda, abdb, abdc, adab, adac, adba, adbc, adbd, adca, adcb, adcd, daba, dabc, dabd, daca, dacb, dacd, dadb, dadc, dbab, dbac, dbad, dbca, dbcb, dcab, dcac, dcad, dcba, dcbc, dcda}}, {abcd, {abac, abad, abca, abcb, abda, abdb, acab, acba, acbc, acdc, adba, adbd, adcd, babc, babd, baca, bacb, bada, badb, bcab, bcac, bcba, bdab, bdad, caba, cabc, cacb, cacd, cada, cadc, cbab, cbac, cbca, cbcd}}};\)\n\[IndentingNewLine] \(\[Euro]ruleSymbolsToCumulIntegerLists = {abac \[Rule] {0, 1, 1, 1025}, abca \[Rule] {0, 1, 1025, 1025}, abcb \[Rule] {0, 1, 1025, 1026}, abcd \[Rule] {0, 1, 1025, 1049601}, abad \[Rule] {0, 1, 1, 1048577}, abda \[Rule] {0, 1, 1048577, 1048577}, abdb \[Rule] {0, 1, 1048577, 1048578}, abdc \[Rule] {0, 1, 1048577, 1049601}, acab \[Rule] {0, 1024, 1024, 1025}, acba \[Rule] {0, 1024, 1025, 1025}, acbc \[Rule] {0, 1024, 1025, 2049}, acbd \[Rule] {0, 1024, 1025, 1049601}, acad \[Rule] {0, 1024, 1024, 1049600}, acda \[Rule] {0, 1024, 1049600, 1049600}, acdc \[Rule] {0, 1024, 1049600, 1050624}, acdb \[Rule] {0, 1024, 1049600, 1049601}, adab \[Rule] {0, 1048576, 1048576, 1048577}, adba \[Rule] {0, 1048576, 1048577, 1048577}, adbd \[Rule] {0, 1048576, 1048577, 2097153}, adbc \[Rule] {0, 1048576, 1048577, 1049601}, adac \[Rule] {0, 1048576, 1048576, 1049600}, adca \[Rule] {0, 1048576, 1049600, 1049600}, adcd \[Rule] {0, 1048576, 1049600, 2098176}, adcb \[Rule] {0, 1048576, 1049600, 1049601}, babc \[Rule] {1, 1, 2, 1026}, bacb \[Rule] {1, 1, 1025, 1026}, baca \[Rule] {1, 1, 1025, 1025}, bacd \[Rule] {1, 1, 1025, 1049601}, babd \[Rule] {1, 1, 2, 1048578}, badb \[Rule] {1, 1, 1048577, 1048578}, bada \[Rule] {1, 1, 1048577, 1048577}, badc \[Rule] {1, 1, 1048577, 1049601}, bcba \[Rule] {1, 1025, 1026, 1026}, bcab \[Rule] {1, 1025, 1025, 1026}, bcac \[Rule] {1, 1025, 1025, 2049}, bcad \[Rule] {1, 1025, 1025, 1049601}, bcbd \[Rule] {1, 1025, 1026, 1049602}, bcdb \[Rule] {1, 1025, 1049601, 1049602}, bcdc \[Rule] {1, 1025, 1049601, 1050625}, bcda \[Rule] {1, 1025, 1049601, 1049601}, bdba \[Rule] {1, 1048577, 1048578, 1048578}, bdab \[Rule] {1, 1048577, 1048577, 1048578}, bdad \[Rule] {1, 1048577, 1048577, 2097153}, bdac \[Rule] {1, 1048577, 1048577, 1049601}, bdbc \[Rule] {1, 1048577, 1048578, 1049602}, bdcb \[Rule] {1, 1048577, 1049601, 1049602}, bdcd \[Rule] {1, 1048577, 1049601, 2098177}, bdca \[Rule] {1, 1048577, 1049601, 1049601}, cacb \[Rule] {1024, 1024, 2048, 2049}, cabc \[Rule] {1024, 1024, 1025, 2049}, caba \[Rule] {1024, 1024, 1025, 1025}, cabd \[Rule] {1024, 1024, 1025, 1049601}, cacd \[Rule] {1024, 1024, 2048, 1050624}, cadc \[Rule] {1024, 1024, 1049600, 1050624}, cada \[Rule] {1024, 1024, 1049600, 1049600}, cadb \[Rule] {1024, 1024, 1049600, 1049601}, cbca \[Rule] {1024, 1025, 2049, 2049}, cbac \[Rule] {1024, 1025, 1025, 2049}, cbab \[Rule] {1024, 1025, 1025, 1026}, cbad \[Rule] {1024, 1025, 1025, 1049601}, cbcd \[Rule] {1024, 1025, 2049, 1050625}, cbdc \[Rule] {1024, 1025, 1049601, 1050625}, cbdb \[Rule] {1024, 1025, 1049601, 1049602}, cbda \[Rule] {1024, 1025, 1049601, 1049601}, cdca \[Rule] {1024, 1049600, 1050624, 1050624}, cdac \[Rule] {1024, 1049600, 1049600, 1050624}, cdad \[Rule] {1024, 1049600, 1049600, 2098176}, cdab \[Rule] {1024, 1049600, 1049600, 1049601}, cdcb \[Rule] {1024, 1049600, 1050624, 1050625}, cdbc \[Rule] {1024, 1049600, 1049601, 1050625}, cdbd \[Rule] {1024, 1049600, 1049601, 2098177}, cdba \[Rule] {1024, 1049600, 1049601, 1049601}, dadb \[Rule] {1048576, 1048576, 2097152, 2097153}, dabd \[Rule] {1048576, 1048576, 1048577, 2097153}, daba \[Rule] {1048576, 1048576, 1048577, 1048577}, dabc \[Rule] {1048576, 1048576, 1048577, 1049601}, dadc \[Rule] {1048576, 1048576, 2097152, 2098176}, dacd \[Rule] {1048576, 1048576, 1049600, 2098176}, daca \[Rule] {1048576, 1048576, 1049600, 1049600}, dacb \[Rule] {1048576, 1048576, 1049600, 1049601}, dbda \[Rule] {1048576, 1048577, 2097153, 2097153}, dbad \[Rule] {1048576, 1048577, 1048577, 2097153}, dbab \[Rule] {1048576, 1048577, 1048577, 1048578}, dbac \[Rule] {1048576, 1048577, 1048577, 1049601}, dbdc \[Rule] {1048576, 1048577, 2097153, 2098177}, dbcd \[Rule] {1048576, 1048577, 1049601, 2098177}, dbcb \[Rule] {1048576, 1048577, 1049601, 1049602}, dbca \[Rule] {1048576, 1048577, 1049601, 1049601}, dcda \[Rule] {1048576, 1049600, 2098176, 2098176}, dcad \[Rule] {1048576, 1049600, 1049600, 2098176}, dcac \[Rule] {1048576, 1049600, 1049600, 1050624}, dcab \[Rule] {1048576, 1049600, 1049600, 1049601}, dcdb \[Rule] {1048576, 1049600, 2098176, 2098177}, dcbd \[Rule] {1048576, 1049600, 1049601, 2098177}, dcbc \[Rule] {1048576, 1049600, 1049601, 1050625}, dcba \[Rule] {1048576, 1049600, 1049601, 1049601}};\)\n\[IndentingNewLine] \(listOfRules = {{"\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, \ {"\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, {"\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, \ {"\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, {"\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, \ {"\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, {"\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, \ {"\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\"}, {"\" \[Rule] "\", "\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\"}, {"\" \[Rule] "\", "\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\"}, {"\ \" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\"}};\)\n\[IndentingNewLine] \(allWords4ForRules = {{"\", "\", "\", "\", "\ \", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\"}, {"\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\