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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Gamard, G"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.04522">arXiv:2302.04522</a> <span> [<a href="https://arxiv.org/pdf/2302.04522">pdf</a>, <a href="https://arxiv.org/ps/2302.04522">ps</a>, <a href="https://arxiv.org/format/2302.04522">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Complexity">cs.CC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> </div> </div> <p class="title is-5 mathjax"> Hardness of monadic second-order formulae over succinct graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a>, <a href="/search/cs?searchtype=author&query=Guillon%2C+P">Pierre Guillon</a>, <a href="/search/cs?searchtype=author&query=Perrot%2C+K">K茅vin Perrot</a>, <a href="/search/cs?searchtype=author&query=Theyssier%2C+G">Guillaume Theyssier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.04522v2-abstract-short" style="display: inline;"> Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every arborescent monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Arborescent properties are those which have infinitely many models and countermodels with bounded treewi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.04522v2-abstract-full').style.display = 'inline'; document.getElementById('2302.04522v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.04522v2-abstract-full" style="display: none;"> Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every arborescent monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Arborescent properties are those which have infinitely many models and countermodels with bounded treewidth. Moreover, we explore what happens when the arborescence condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.04522v2-abstract-full').style.display = 'none'; document.getElementById('2302.04522v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.08022">arXiv:1812.08022</a> <span> [<a href="https://arxiv.org/pdf/1812.08022">pdf</a>, <a href="https://arxiv.org/ps/1812.08022">ps</a>, <a href="https://arxiv.org/format/1812.08022">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On the entropy of coverable subshifts </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.08022v1-abstract-short" style="display: inline;"> A coloration w of Z^2 is said to be coverable if there exists a rectangular block q such that w is covered with occurrences of q, possibly overlapping. In this case, q is a cover of w. A subshift is said to have the cover q if each of its points has the cover q. In a previous article, we characterized the covers that force subshifts to be finite (in particular, all configurations are periodic). We… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.08022v1-abstract-full').style.display = 'inline'; document.getElementById('1812.08022v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.08022v1-abstract-full" style="display: none;"> A coloration w of Z^2 is said to be coverable if there exists a rectangular block q such that w is covered with occurrences of q, possibly overlapping. In this case, q is a cover of w. A subshift is said to have the cover q if each of its points has the cover q. In a previous article, we characterized the covers that force subshifts to be finite (in particular, all configurations are periodic). We also noticed that some covers force subshifts to have zero topological entropy while not forcing them to be finite. In the current paper we work towards characterizing precisely covers which force a subshift to have zero entropy, but not necessarily periodicity. We give a necessary condition and a sufficient condition which are close, but not quite identical. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.08022v1-abstract-full').style.display = 'none'; document.getElementById('1812.08022v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.02643">arXiv:1803.02643</a> <span> [<a href="https://arxiv.org/pdf/1803.02643">pdf</a>, <a href="https://arxiv.org/ps/1803.02643">ps</a>, <a href="https://arxiv.org/format/1803.02643">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> Quasiperiods of biinfinite Sturmian words </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Barbero%2C+F">Florian Barbero</a>, <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a>, <a href="/search/cs?searchtype=author&query=Grandjean%2C+A">Ana毛l Grandjean</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1803.02643v1-abstract-short" style="display: inline;"> We study the notion of quasiperiodicity, in the sense of "coverability", for biinfinite words. All previous work about quasiperiodicity focused on right infinite words, but the passage to the biinfinite case could help to prove stronger results about quasiperiods of Sturmian words. We demonstrate this by showing that all biinfinite Sturmian words have infinitely many quasiperiods, which is not qui… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.02643v1-abstract-full').style.display = 'inline'; document.getElementById('1803.02643v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.02643v1-abstract-full" style="display: none;"> We study the notion of quasiperiodicity, in the sense of "coverability", for biinfinite words. All previous work about quasiperiodicity focused on right infinite words, but the passage to the biinfinite case could help to prove stronger results about quasiperiods of Sturmian words. We demonstrate this by showing that all biinfinite Sturmian words have infinitely many quasiperiods, which is not quite (but almost) true in the right infinite case, and giving a characterization of those quasiperiods. The main difference between right infinite and the biinfinite words is that, in the latter case, we might have several quasiperiods of the same length. This is not possible with right infinite words because a quasiperiod has to be a prefix of the word. We study in depth the relations between quasiperiods of the same length in a given biinfinite quasiperiodic word. This study gives enough information to allow to determine the set of quasiperiods of an arbitrary word. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.02643v1-abstract-full').style.display = 'none'; document.getElementById('1803.02643v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1610.04439">arXiv:1610.04439</a> <span> [<a href="https://arxiv.org/pdf/1610.04439">pdf</a>, <a href="https://arxiv.org/ps/1610.04439">ps</a>, <a href="https://arxiv.org/format/1610.04439">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> Avoidability of circular formulas </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a>, <a href="/search/cs?searchtype=author&query=Ochem%2C+P">Pascal Ochem</a>, <a href="/search/cs?searchtype=author&query=Richomme%2C+G">Gwena毛l Richomme</a>, <a href="/search/cs?searchtype=author&query=S%C3%A9%C3%A9bold%2C+P">Patrice S茅茅bold</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1610.04439v1-abstract-short" style="display: inline;"> Clark has defined the notion of $n$-avoidance basis which contains the avoidable formulas with at most $n$ variables that are closest to be unavoidable in some sense. The family $C_i$ of circular formulas is such that $C_1=AA$, $C_2=ABA.BAB$, $C_3=ABCA.BCAB.CABC$ and so on. For every $i\le n$, the $n$-avoidance basis contains $C_i$. Clark showed that the avoidability index of every circular formul… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1610.04439v1-abstract-full').style.display = 'inline'; document.getElementById('1610.04439v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1610.04439v1-abstract-full" style="display: none;"> Clark has defined the notion of $n$-avoidance basis which contains the avoidable formulas with at most $n$ variables that are closest to be unavoidable in some sense. The family $C_i$ of circular formulas is such that $C_1=AA$, $C_2=ABA.BAB$, $C_3=ABCA.BCAB.CABC$ and so on. For every $i\le n$, the $n$-avoidance basis contains $C_i$. Clark showed that the avoidability index of every circular formula and of every formula in the $3$-avoidance basis (and thus of every avoidable formula containing at most 3 variables) is at most 4. We determine exactly the avoidability index of these formulas. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1610.04439v1-abstract-full').style.display = 'none'; document.getElementById('1610.04439v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 October, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.06915">arXiv:1602.06915</a> <span> [<a href="https://arxiv.org/pdf/1602.06915">pdf</a>, <a href="https://arxiv.org/format/1602.06915">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Periodicity in Rectangular Arrays </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a>, <a href="/search/cs?searchtype=author&query=Richomme%2C+G">Gwena毛l Richomme</a>, <a href="/search/cs?searchtype=author&query=Shallit%2C+J">Jeffrey Shallit</a>, <a href="/search/cs?searchtype=author&query=Smith%2C+T+J">Taylor J. Smith</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1602.06915v2-abstract-short" style="display: inline;"> We discuss several two-dimensional generalizations of the familiar Lyndon-Schutzenberger periodicity theorem for words. We consider the notion of primitive array (as one that cannot be expressed as the repetition of smaller arrays). We count the number of m x n arrays that are primitive. Finally, we show that one can test primitivity and compute the primitive root of an array in linear time. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.06915v2-abstract-full" style="display: none;"> We discuss several two-dimensional generalizations of the familiar Lyndon-Schutzenberger periodicity theorem for words. We consider the notion of primitive array (as one that cannot be expressed as the repetition of smaller arrays). We count the number of m x n arrays that are primitive. Finally, we show that one can test primitivity and compute the primitive root of an array in linear time. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.06915v2-abstract-full').style.display = 'none'; document.getElementById('1602.06915v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 68R15 (Primary); 68W32; 68W40; 05A15 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.08375">arXiv:1506.08375</a> <span> [<a href="https://arxiv.org/pdf/1506.08375">pdf</a>, <a href="https://arxiv.org/format/1506.08375">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Comparison of Coverability and Multi-Scale Coverability in One and Two Dimensions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a>, <a href="/search/cs?searchtype=author&query=Richomme%2C+G">Gwena毛l Richomme</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.08375v1-abstract-short" style="display: inline;"> A word is quasiperiodic (or coverable) if it can be covered with occurrences of another finite word, called its quasiperiod. A word is multi-scale quasiperiodic (or multi-scale coverable) if it has infinitely many different quasiperiods. These notions were previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend them to infinite pictures (two-dime… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.08375v1-abstract-full').style.display = 'inline'; document.getElementById('1506.08375v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.08375v1-abstract-full" style="display: none;"> A word is quasiperiodic (or coverable) if it can be covered with occurrences of another finite word, called its quasiperiod. A word is multi-scale quasiperiodic (or multi-scale coverable) if it has infinitely many different quasiperiods. These notions were previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend them to infinite pictures (two-dimensional words). Then we compare the regularity properties (uniform recurrence, uniform frequencies, topological entropy) of quasiperiodicity with multi-scale quasiperiodicity, and we also compare each of them with its one-dimensional counterpart. We also study which properties of quasiperiods enforce properties on the quasiperiodic words. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.08375v1-abstract-full').style.display = 'none'; document.getElementById('1506.08375v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1312.4126">arXiv:1312.4126</a> <span> [<a href="https://arxiv.org/pdf/1312.4126">pdf</a>, <a href="https://arxiv.org/ps/1312.4126">ps</a>, <a href="https://arxiv.org/format/1312.4126">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/978-3-319-09698-8_15">10.1007/978-3-319-09698-8_15 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Aperiodic tilings and entropy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Durand%2C+B">Bruno Durand</a>, <a href="/search/cs?searchtype=author&query=Gamard%2C+G">Guilhem Gamard</a>, <a href="/search/cs?searchtype=author&query=Grandjean%2C+A">Anael Grandjean</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1312.4126v2-abstract-short" style="display: inline;"> In this paper we present a construction of Kari-Culik aperiodic tile set - the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy. We also explain why this result was not expected. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1312.4126v2-abstract-full" style="display: none;"> In this paper we present a construction of Kari-Culik aperiodic tile set - the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy. We also explain why this result was not expected. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.4126v2-abstract-full').style.display = 'none'; document.getElementById('1312.4126v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2013. </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> 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