Avoidability of formulas with two variables

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words

Avoiding conjugacy classes on the 5-letter alphabet

We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [TCS 2018

Theoretical and Practical Aspects Related to the Avoidability of Patterns in Words

This thesis concerns repetitive structures in words. More precisely, it contributes to studying appearance and absence of such repetitions in words. In the first and major part of this thesis, we study avoidability of unary patterns with permutations. The second part of this thesis deals with modeling and solving several avoidability problems as constraint satisfaction problems, using the framework of MiniZinc. Solving avoidability problems like the one mentioned in the past paragraph required, the construction, via a computer program, of a very long word that does not contain any word that matches a given pattern. This gave us the idea of using SAT solvers. Representing the problem-based SAT solvers seemed to be a standardised, and usually very optimised approach to formulate and solve the well-known avoidability problems like avoidability of formulas with reversal and avoidability of patterns in the abelian sense too. The final part is concerned with a variation on a classical avoidance problem from combinatorics on words. Considering the concatenation of i different factors of the word w, pexp_i(w) is the supremum of powers that can be constructed by concatenation of such factors, and RTi(k) is then the infimum of pexp_i(w). Again, by checking infinite ternary words that satisfy some properties, we calculate the value RT_i(3) for even and odd values of i

Pattern avoidance: themes and variations

AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems

Combinatorics and Algorithmics of Strings

Edited in cooperation with Robert MercaşStrings (aka sequences or words) form the most basic and natural data structure. They occur whenever information is electronically transmitted (as bit streams), when natural language text is spoken or written down (as words over, for example, the Latin alphabet), in the process of heredity transmission in living cells (through DNA sequences) or the protein synthesis (as sequence of amino acids), and in many more different contexts. Given this universal form of representing information, the need to process strings is apparent and is actually a core purpose of computer use. Algorithms to efficiently search through, analyze, (de-)compress, match, encode and decode strings are therefore of chief interest. Combinatorial problems about strings lie at the core of such algorithmic questions. Many such combinatorial problems are common in the string processing efforts in the different fields of application.http://drops.dagstuhl.de/opus/volltexte/2014/4552

De bruijn partial words

In a kn-complex word over an alphabet Σ of size k each of the kn words of length n appear as a subword at least once. Such a word is said to have maximum subword complexity. De Bruijn sequences of order n over Σ are the shortest words of maximum subword complexity and are well known to have length kn+n-1. They are efficiently constructed by finding Eulerian cycles in so-called de Bruijn graphs. In this thesis, we investigate partial words, or sequences with wildcard symbols or hole symbols, of maximum subword complexity. The subword complexity function of a partial word w over a given alphabet of size k assigns to each positive integer n, the number pw(n) of distinct full words over the alphabet that are compatible with factors of length n of w. For positive integers h, k and n, a de Bruijn partial word of order n with h holes over an alphabet Σ of size k is a partial word w with h holes over Σ of minimal length with the property that pw(n)=kn. In some cases, they are efficiently constructed by finding Eulerian paths in modified de Bruijn graphs. We are concerned with the following three questions: (1) What is the length of k-ary de Bruijn partial words of order n with h holes? (2) What is an efficient method for generating such partial words? (3) How many such partial words are there

How do Motifs Endure and Perform? Motif Theory for the Study of Biblical Narratives

Motifs are often confused with and overshadowed by themes. This article argues for the unique identity, functions, and rhetorical force of motifs. Synchronically, motifs are recurrent performers having progressive and cumulative force in a singular narrative. They often build on and acquire strength diachronically through motifs within a given literary tradition. Literary theory on motifs is supplemented here through the intertextual and performative values of motifs as a part of three phases of narrative writing and experience: emplotment, plot, and explotment. Thus, these interrelated aspects of motifs show how they survive through time, reappearing and performing in narratives. A discussion on the motif of light in Luke-Acts demonstrates the value of this approach