Representations of Circular Words

In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The latter form can also be used as a graphical representation of periodic properties of finite (in some cases, infinite) words. We also define iterative representations which can be seen as an encoding utilizing the flexible properties of circular words. Every word over the two letter alphabet can be constructed starting from ab by applying the fractional power and the cyclic shift operators one after the other, iteratively.Comment: In Proceedings AFL 2014, arXiv:1405.527

Absztrakt automaták és formális nyelvek = Abstract Automata and Formal Languages

Monográfiában foglaltuk össze a véges automata hálózatok elméletének alapvető eredményeit . Megadtuk a Leticsevszkij kritérium nélküli automata-hálózatok egy új jellemzését. Megadtunk bizonyos szimbólum osztályokkal ellátott többszalagos automatákat. Megadtuk és vizsgáltuk a kutatásaink során felfedezett automataelméleti elvű új titkosítási rendszert. Új bizonyítást adtunk a Lyndon-Schützenberger tételre és a Shyr-Yu tételre. Általánosítottuk a szavak primitivitásának, illetve periodicitásának fogalmát, s megadtuk, hogy melyek azok a Marcus nyelvtanok, amelyek az adott típusú szavakból álló nyelveket képesek generálni. Sikerült találni egy iterációs lemmát azon környezetfüggetlen nyelvekre, melyek nem lineárisak. A contextuális sztringnyelvek általánosításaként bevezettük és vizsgáltuk a hypergráf contextuális nyelvek és nyelvtanok fogalmát. Automaták segítségével jellemeztük az uniómentes nyelveket. Leírtuk a különféle logikai kalkulusok és valamely levezetési rendszer szabályai szerint megadott levezetések kapcsolatait, s a különféle kalkulusok normalizálhatósági tulajdonságait. Új elvű számítási kutatásainkban az intervallum-értékű számításokat, mint új számítási modellt írtuk le. Digitális geometriai kutatásainkban digitális távolság alapján értelmezett szakaszokat, köröket, hiperbolákat és parabolákat vizsgáltunk. | We summarized in monograph the fundamental results of theory of finite automata networks. We gave a new characterization of automata-networks having no Letichevsky criteria. We gave multi-tape automata supplied by certain symbol classes. We gave and investigated a novel cryptosystem based on automata theory discovered during our research.. We gave a new proof of Lyndon-Schützenberger Theorem and Shyr-Yu Theorem. We generalized the concept of primitivity and periodicity of words and we give the Marcus gramars which are able to generate languages consisting of given type words. It succeed in finding an iteration lemma for non-linear context-free languages. As a generalization of contextual string languages, we introduced and investigated the concept of hypergraph contextual grammars and languages. We characterized the union-free languages by automata. We described the connections of derivations given by various logical calculi and certain derivation system, moreover the normalizable properties of various calculi. In our new computation principle researches we described the intervallum-value computations as new computation model In our digital geometric researches we investigated the sessions, circles, hyperbolas and parabolas defined by a digital distance

Two-Dimensional Digitized Picture Arrays and Parikh Matrices

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained

Repetitive subwords

The central notionof thisthesisis repetitionsin words. We studyproblemsrelated to contiguous repetitions. More specifically we will consider repeating scattered subwords of non-primitive words, i.e. words which are complete repetitions of other words. We will present inequalities concerning these occurrences as well as giving apartial solutionto an openproblemposedby Salomaaet al. We will characterize languages, whichare closed under the operation ofduplication, thatis repeating any factor of a word. We alsogive newbounds onthe number of occurrencesof certain types of repetitions of words. We give a solution to an open problem posed by Calbrix and Nivat concerning regular languages consisting of non-primitive words. We alsopresentsomeresultsregarding theduplication closureoflanguages,among which a new proof to a problem of Bovet and Varricchio