A universal word for a finite alphabet A and some integer n ≥ 1 is a word over A such that every word in An appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any A and n. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from A may contain an arbitr...

We prove several decidability and undecidability results for the satisfiability/validity problem of formulas over a language of finite-length strings and integers (interpreted as lengths of strings). The atomic formulas over this language are equality over string terms (word equations), linear inequality over length function (length constraints), and membership predicate over regular expression...

In [2], Aleksi Saarela has introduced a new approach to word equations that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equ...

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors...

The study of word equations (or the existential theory of equations over free monoids) is a central topic in mathematics and theoretical computer science. The problem of deciding whether a given word equation has a solution was shown to be decidable by Makanin in the late 1970s, and since then considerable work has been done on this topic. In recent years, this decidability question has gained ...

One of the most intricate algorithms related to words is Makanin's algorithm solving word equations. The algorithm is very complicated and the complexity of the problem of solving word equations is not well understood. Word equations can be used to deene various properties of strings, e.g. general versions of pattern-matching with variables. This paper is devoted to introduce a new approach and...

We investigate the class of regular-ordered word equations. The sides of such an equation contain both exactly the same variables, occurring in the same order (but separated by potentially distinct constant factors). Surprisingly, we obtain that solving even such simple equations is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general qua...