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Freeness of partial words

Florin Manea, Robert Mercaş
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
This result seems interesting because any partial word containing two holes with less than two symbols between them is not cube-free.  ...  In the final section of the paper, we show that there exists an infinite word, over a four-symbol alphabet, in which we can substitute randomly one symbol with a hole, and still obtain a cube-free word  ...  It is clear that any overlap-free word w is k-free, for k ≥ 3, and, as well, any 2-free word is overlap-free.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2007.09.028">doi:10.1016/j.tcs.2007.09.028</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s3ydasynd5ehzptlm2aefi2bka">fatcat:s3ydasynd5ehzptlm2aefi2bka</a> </span>
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Overlap-free symmetric D0L words

Anna Frid
<span title="2001-01-01">2001</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
Shallit: if all the symbols in \varphi(0) are distinct (i.e., if t_i ≠q t_j for i ≠q j), then the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ^* and a  ...  International audience A D0L word on an alphabet Σ =\0,1,\ldots,q-1\ is called symmetric if it is a fixed point w=\varphi(w) of a morphism \varphi:Σ ^* → Σ ^* defined by \varphi(i)=øverlinet_1 + i øverlinet  ...  Thue gave the first example of an overlap-free infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.46298/dmtcs.293">doi:10.46298/dmtcs.293</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/62wf7o2toretznqc62r2ci4pgy">fatcat:62wf7o2toretznqc62r2ci4pgy</a> </span>
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A rewriting of Fife's theorem about overlap-free words [chapter]

J. Berstel
<span title="">1994</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
After some preliminaries, we give two general, basic lemmas on overlap-free words. In the next section, we present the result of Fife. The last section is devoted to the proof.  ...  Recently, two results have given new insights in this topic. J. Cassaigne [2] and A. Carpi [1] have presented encodings of finite overlap-free words that are similar to Fife's.  ...  Introduction One of the first results about avoidable regularities in words was Axel Thue's proof of the existence of an infinite overlap-free words over two letters.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-58131-6_34">doi:10.1007/3-540-58131-6_34</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ijmxgumkx5ezjamyytuzqxp52m">fatcat:ijmxgumkx5ezjamyytuzqxp52m</a> </span>
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Retrograde codes and bounded synchronization delay

Hollie L. Buchanan, Michael E. Mays
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/joe2ngto45hbnl3pncnesnq344" style="color: black;">Information and Computation</a> </i> &nbsp;
Retrograde codes are a subclass of comma-free codes in which the dictionary of code words excludes not only overlaps of code words but also reversals of overlaps.  ...  We give a general upper bound on the maximum size of a retrograde comma-free distionary, provide a construction for a bounded synchronization delay retrograde code which attains this maximum size, and  ...  An (n-2)'-' solution: Devote two spaces per word (say the last two) and two alphabet symbols (say a and 6) to marking breakpoints.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0890-5401(92)90054-j">doi:10.1016/0890-5401(92)90054-j</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/55jgnzaymbdfhbrjjqhfqfv5hi">fatcat:55jgnzaymbdfhbrjjqhfqfv5hi</a> </span>
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An Early Completion Algorithm: Thue's 1914 Paper on the Transformation of Symbol Sequences [chapter]

James F. Power
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
References to Thue's 1914 paper on string transformation systems are based mainly on a small section of that work defining Thue systems.  ...  Thue proves that the morphism µ preserves this property: it will always map overlap-free words to overlap-free words.  ...  Start with the given identity word R. Form equations C = D based on the remainder from the overlaps within R.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-08019-2_35">doi:10.1007/978-3-319-08019-2_35</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zji223d4y5f47dbct2jddl4ygm">fatcat:zji223d4y5f47dbct2jddl4ygm</a> </span>
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A linear-time algorithm to decide whether a binary word contains an overlap

A. J. Kfoury
<span title="">1988</span> <i title="EDP Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/foywj2stzfcwdln5izx2cxkroe" style="color: black;">RAIRO - Theoretical Informatics and Applications</a> </i> &nbsp;
-We present two new resuits. First, we give a new Hnear-time aigorithm to test whether an arbitrary binary word contains an overlap.  ...  A linear-time algorithm to decide whether a binary word contains an overlap Informatique théorique et applications, tome 22, n o 2 (1988), p. 135-145. © AFCET, 1988, tous droits réservés.  ...  Jean Berstel guided me through the recent literature on square-free and overlap-free words, in particular [5] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1051/ita/1988220201351">doi:10.1051/ita/1988220201351</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cihinppbyvbdbljup3q4rnlhtq">fatcat:cihinppbyvbdbljup3q4rnlhtq</a> </span>
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Algebraic Irrational Binary Numbers Cannot Be Fixed Points of Non-trivial Constant Length or Primitive Morphisms

Jean-Paul Allouche, Luca Q Zamboni
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bevk4sowfzcstelrgkfq275p4e" style="color: black;">Journal of Number Theory</a> </i> &nbsp;
We prove that a positive real number whose binary expansion is a fixed point of a morphism on the alphabet [0, 1] that is either of constant length 2 or primitive is either rational or transcendental.  ...  Two other corollaries are, first that a binary morphism h is overlap-free if and only if the word h(01101001) is overlap-free, second Se e bold's result.  ...  Define a morphism to be overlap-free if and only if the image of any overlap-free word is itself overlap-free.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jnth.1997.2207">doi:10.1006/jnth.1997.2207</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zdvp3oc7rve3nfxms2su6lagpi">fatcat:zdvp3oc7rve3nfxms2su6lagpi</a> </span>
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The Morphisms With Unstackable Image Words [article]

C. Robinson Tompkins
<span title="2010-06-07">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We generalize the combinatorial properties of the Leech square-free morphism to gain insights on a larger class of both overlap-free morphisms and square-free morphisms.  ...  In an attempt to classify all of the overlap-free morphisms constructively using the Latin-square morphism, we came across an interesting counterexample, the Leech square-free morphism.  ...  Surprisingly it is known that µ and its natural complement are the only non-trivial overlap-free morphisms on the two letter alphabet {−0, 1} [2] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1006.1273v1">arXiv:1006.1273v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7diecq6krba6daw5y74mumsipy">fatcat:7diecq6krba6daw5y74mumsipy</a> </span>
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Sums of Digits, Overlaps, and Palindromes

Jean-Paul Allouche, Jeffrey Shallit
<span title="2000-01-01">2000</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
In this paper, generalizing Thue's result, we prove that the infinite word t_k,m := (s_k(n) \bmod m)_n≥ 0 is overlap-free if and only if m≥ k.  ...  In a celebrated paper, Thue showed that the infinite word (s_2(n) \bmod 2)_n≥ 0 is \emphoverlap-free, i.e., contains no subword of the form axaxa where x is any finite word and a is a single symbol.  ...  Thue explicitly constructed an infinite word on two symbols that is overlapfree, that is, contains no subword that is an overlap.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.46298/dmtcs.282">doi:10.46298/dmtcs.282</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/npjoev47grg3xhokfc4nwlotme">fatcat:npjoev47grg3xhokfc4nwlotme</a> </span>
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On the overlap assembly of strings and languages

Srujan Kumar Enaganti, Oscar H. Ibarra, Lila Kari, Steffen Kopecki
<span title="2016-01-23">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6zolbg3dcjdt3edqtvzmiiuuim" style="color: black;">Natural Computing</a> </i> &nbsp;
strands: The overlap assembly of two strings, xy and yz, which share an "overlap" y, results in the string xyz.  ...  In this context, we investigate overlap assembly and its properties: closure properties of basic language families under this operation, decision problems, as well as the possible use of iterated overlap  ...  Acknowledgements We thank Giuditta Franco for useful suggestions and discussions on experimental aspects of XPCR, as well as Sepinoud Azimi and Florin Manea for pointing out important references.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11047-015-9538-x">doi:10.1007/s11047-015-9538-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vhgjnqjbtzbcbjzgazd7yzdcoi">fatcat:vhgjnqjbtzbcbjzgazd7yzdcoi</a> </span>
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Further remarks on DNA overlap assembly

Srujan Kumar Enaganti, Oscar H. Ibarra, Lila Kari, Steffen Kopecki
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/joe2ngto45hbnl3pncnesnq344" style="color: black;">Information and Computation</a> </i> &nbsp;
of two given others.  ...  The operation of overlap assembly was defined by Csuhaj-Varju, Petre, and Vaszil as a formal model of the linear self-assembly of DNA strands: The overlap assembly of two strings, xy and yz, which share  ...  NPDAs accept exactly the context-free languages (CFLs). • DCA (NCA) is a DPDA (NPDA) which uses only one stack symbol in addition to the bottom stack symbol which is never erased.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ic.2017.01.009">doi:10.1016/j.ic.2017.01.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vfo472uabrhv3hq2qhhxh2kp74">fatcat:vfo472uabrhv3hq2qhhxh2kp74</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922003117/http://www.csd.uwo.ca/%7Elila/pdfs/Advanced_OC.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f8/90/f890b8afe2681ed29ceb603f00d525c57bbf2986.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ic.2017.01.009"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Lexicon-Free, Novel Segmentation of Online Handwritten Indic Words

Suresh Sundaram, A. G. Ramakrishan
<span title="">2011</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bytixui4szbgzkqzgwf65jphty" style="color: black;">2011 International Conference on Document Analysis and Recognition</a> </i> &nbsp;
In the context of lexicon-free recognition of words, one of the primary issues to be addressed is that of segmentation.  ...  As a preliminary attempt, this paper proposes a novel script-independent, lexicon-free method for segmenting online handwritten words to their constituent symbols.  ...  (a) Computing the degree of overlap O c k between two contiguous strokes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icdar.2011.237">doi:10.1109/icdar.2011.237</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icdar/SundaramR11.html">dblp:conf/icdar/SundaramR11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jrxktneiivhm3j43apfb7hsb4a">fatcat:jrxktneiivhm3j43apfb7hsb4a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170829181453/http://mile.ee.iisc.ernet.in/mile/publications/softCopy/DocumentAnalysis/Suresh_ICDAR2011.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ca/f8/caf83648d5ec33523860c81b0547d109bec76fd4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icdar.2011.237"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Codes with bounded synchronization delay

Solomon W. Golomb, Basil Gordon
<span title="">1965</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbkexx2qg5anljkxraobuswdq4" style="color: black;">Information and Control</a> </i> &nbsp;
COMMA-FREE CODES A subset S of the n ~ words of length k over an n-symbol alphabet which satisfies the condition that whenever ala2 ... ak and blb~ ... b~ are in S, then none of the "overlaps" a2 ... akbi  ...  For if it did, then the message ~..(1)~..(1)~..(1). (1)~ (1)~ (1) .... cu a ¢u~ cua cuf~ ~ua tuff "*" would have two decodings (one into one-letter words and one into two-_ . (1)~ (1) letter words).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0019-9958(65)90300-1">doi:10.1016/s0019-9958(65)90300-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/unwcdbqt45eudhmdef3ozsukjq">fatcat:unwcdbqt45eudhmdef3ozsukjq</a> </span>
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A Frameless 2-Coloring of the Plane Lattice [article]

Craig S. Kaplan, Jeffrey Shallit
<span title="2020-05-19">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A picture frame in two dimensions is a rectangular array of symbols, with at least two rows and columns, where the first and last rows are identical, and the first and last columns are identical.  ...  Introduction A classic problem from combinatorics on words is the avoidance of overlaps. An overlap is a word of the form axaxa, where a is a single symbol and x is a (possibly empty) word.  ...  The two-sided infinite word u := t R t = · · · t(3)t(2)t(1)t(0)t(0)t(1)t(2)t(3) · · · is overlap-free. Proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2005.09585v1">arXiv:2005.09585v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pa2mv5aqezge7fyxwg6nrospuy">fatcat:pa2mv5aqezge7fyxwg6nrospuy</a> </span>
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Fife's Theorem Revisited [article]

Jeffrey Shallit
<span title="2011-07-27">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As an application we give a complete characterization of the overlap-free words that are 2-automatic.  ...  We give another proof of a theorem of Fife - understood broadly as providing a finite automaton that gives a complete description of all infinite binary overlap-free words.  ...  An example of an overlap in English is the word alfalfa. A finite or infinite word is overlap-free if it contains no finite factor that is an overlap. From now on we fix Σ = {0, 1}.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1102.3932v3">arXiv:1102.3932v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o3pldfrq4fc2tam5eznk6yy2yu">fatcat:o3pldfrq4fc2tam5eznk6yy2yu</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1102.3932/1102.3932.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1102.3932v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>
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