Palindromes in finite and infinite words

Abstract

<p>In the thesis we are concerned with actual problems on palindromic subwords and palindromic factors of finite and infinite words. The main course of the research are the ways of determining which of two given words is &ldquo;more palindromic&rdquo; than the other one, that is, defining a measure for the degree of &ldquo;palindromicity&rdquo; of a word. Particularly, we pay attention to two actual approaches: the so-called MP-ratio and the so-called <em>palindromic defect</em>, and answer several open questions about them.<br /><br />Namely, concerning the MP-ratio, a few plausible-looking question have been asked in the literature, which would have, if answered positively, made computations of MP-ratios significantly simpler. We add one more related question to these ones, and then show that, rather unexpectedly, all these questions have negative answer.<br /><br />Concerning the palindromic defect, the main result of this work is a construction of an infinite class of infinite words that have several properties that were sought after in some recent works in this area. Among the most interesting facts is that that all these words are aperiodic words of a finite positive defect, having the set of factors closed under reversal---in some recent works, the construction of even a single word having these properties turned out to be quite hard. Using these words, which we are calling <em>highly potential words</em>, we check the validity of several open&nbsp; conjectures, and for several of them we find out that they are false.</p>

<p> U tezi razmatramo aktuelne probleme u vezi s palindromskim podrečima i palindromskim faktorima konačnih i beskonačnih reči. Glavni pravac istraživanja jesu kriterijumi za određivanje koja od dve date reči je &bdquo;palindromičnija&ldquo; od druge, tj. određivanje stepena &bdquo;palindromičnosti&ldquo; date reči. Akcenat stavljamo na dva aktuelna pristupa: tzv. <em>MP-razmeru</em> i tzv. <em>palindromski defekt</em>, i odgovaramo na vi&scaron;e otvorenih pitanja u vezi s njima.<br /> <br /> Naime, u vezi sa MP-razmerom u literaturi je postavljeno vi&scaron;e pitanja, intuitivno uverljivih, koja bi, u slučaju pozitivnog razre&scaron;enja, znatno pojednostavila izračunavanje MP-razmere. Ovim pitanjima dodajemo jo&scaron; jedno srodno, a zatim pokazujemo da, prilično neočekivano, sva ova pitanja imaju negativan odgovor.<br /> <br /> U vezi s palindromskim defektom, glavni rezultat rada je konstrukcija beskonačne klase beskonačnih reči koje imaju vi&scaron;e osobina za kojima je iskazana potreba u skora&scaron;njim radovima iz ove oblasti. Među najzanimljivije spada činjenica da su sve aperiodične reči konačnog pozitivnog defekta, i da im je skup faktora zatvoren za preokretanje &ndash; u nekim skora&scaron;njim radovima konstrukcija makar jedne reči s ovim osobinama pokazala se kao prilično te&scaron;ka. Pomoću ovih reči, koje nazivamo <em>visokopotencijalne reči</em>, ispitujemo validnost vi&scaron;e otvorenih hipoteza, i za vi&scaron;e njih ustanovljavamo da nisu validne.</p>