{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:48:18Z","timestamp":1759063698078,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Shallit and Wang studied deterministic automatic complexity of words.\u00a0 They showed that the automatic Hausdorff dimension $I(\\mathbf t)$ of the infinite Thue word satisfies $1\/3\\le I(\\mathbf t)\\le 1\/2$.\u00a0\u00a0 We improve that result by showing that $I(\\mathbf t)= 1\/2$.\u00a0 We prove that the nondeterministic automatic complexity $A_N(x)$ of a word $x$ of length $n$ is bounded by $b(n):=\\lfloor n\/2\\rfloor + 1$.\u00a0 This enables us to define the complexity deficiency $D(x)=b(n)-A_N(x)$.\u00a0 If $x$ is square-free then $D(x)=0$. If $x$ is almost square-free in the sense of Fraenkel and Simpson, or if $x$ is a overlap-free binary word such as the infinite Thue--Morse word, then $D(x)\\le 1$.\u00a0 On the other hand, there is no constant upper bound on $D$ for overlap-free words over a ternary alphabet, nor for cube-free words over a binary alphabet.The decision problem whether $D(x)\\ge d$ for given $x$, $d$ belongs to $\\mathrm{NP}\\cap \\mathrm{E}$.<\/jats:p>","DOI":"10.37236\/4851","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:26:41Z","timestamp":1578670001000},"source":"Crossref","is-referenced-by-count":7,"title":["Nondeterministic Automatic Complexity of Overlap-Free and Almost Square-Free Words"],"prefix":"10.37236","volume":"22","author":[{"given":"Kayleigh K.","family":"Hyde","sequence":"first","affiliation":[]},{"given":"Bj\u00f8rn","family":"Kjos-Hanssen","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,8,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i3p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i3p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:13:35Z","timestamp":1579256015000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i3p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,14]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2015,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/4851","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,8,14]]},"article-number":"P3.22"}}