On Rough Approximations of Languages under Infinite Index Indiscernibility Relations

In the paper [13] Păun, Polkowski and Skowron introduce several indiscernibility relations among strings that are infinite index equivalence or tolerance relations, and study lower and upper rough approximations of languages defined by them. In this paper we develop a further study of some of these indiscernibility relations among strings. We characterize the classes defined by them, and the rough approximations of general and context free languages under them. We also compare some of the rough approximations these relations produce to the ones given by the congruences defining testable, reverse testable, locally testable, piecewise testable and commutative languages. Those yield languages belonging to that families. Next, we modify some of the relations to obtain congruences, and study the families of languages the rough approximations under them give rise to. One of these modificated relations turns out to be the k-abelian congruence, that was defined by J. Karhumäki in [7], in the context of combinatorics on words. We show that it defines a pseudo-principal +-variety, a term defined in [9]. Our results in that work are then applied to determine when a given language has a best upper approximation in that family. Finally, we make some comments on the accuracy of the rough approximations obtained in each case.

On universal partial words for word-patterns and set partitions

Universal words are words containing exactly once each element from a given set of combinatorial structures admitting encoding by words. Universal partial words (u-p-words) contain, in addition to the letters from the alphabet in question, any number of occurrences of a special “joker” symbol. We initiate the study of u-p-words for word-patterns (essentially, surjective functions) and (2-)set partitions by proving a number of existence/non-existence results and thus extending the results in the literature on u-p-words and u-p-cycles for words and permutations. We apply methods of graph theory and combinatorics on words to obtain our results.

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.

On winning shifts of marked uniform substitutions

The second author introduced with I. Törmä a two-player word-building game [Fund. Inform. 132 (2014) 131–152]. The game has a predetermined (possibly finite) choice sequence α1, α2, … of integers such that on round n the player A chooses a subset Sn of size αn of some fixed finite alphabet and the player B picks a letter from the set Sn. The outcome is determined by whether the word obtained by concatenating the letters B picked lies in a prescribed target set X (a win for player A) or not (a win for player B). Typically, we consider X to be a subshift. The winning shift W(X) of a subshift X is defined as the set of choice sequences for which A has a winning strategy when the target set is the language of X. The winning shift W(X) mirrors some properties of X. For instance, W(X) and X have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue–Morse substitutions. It is known that W(X) and X have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue–Morse substitutions.

Palindromic Decompositions with Gaps and Errors

Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes. We first present an on-line algorithm for obtaining a palindromic decomposition of a string of length [Formula: see text] with the minimal total gap length in time [Formula: see text] and space [Formula: see text], where [Formula: see text] is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal [Formula: see text]-palindromes (i.e. palindromes with [Formula: see text] errors under the edit or Hamming distance) and [Formula: see text] allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time [Formula: see text] and space [Formula: see text]. Finally, we provide an implementation of our algorithms.