Hmelevskii proved in 1971 that every constant-free three-variable word equation has a parametric solution. We prove an improved version of this result by showing that every such equation has a parametric solution using only three numerical parameters and with only two levels of nesting. This means that the structure of the solution sets of these equations is considerably simpler than has been known before.

Abstract It is known that the set of solutions of any constant-free three-variable word equation can be represented using parametric words, and the number of numerical parameters and the level of nesting in these parametric words is at most logarithmic with respect to the length of the equation. We show that this result can be significantly improved in the case of unbalanced equations, that is, equations where at least one variable has a different number of occurrences on the left-hand side and on the right-hand side. More specifically, it is sufficient to have two numerical parameters and one level of nesting in this case. We also discuss the possibility of proving a similar result for balanced equations in the future.

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system has at most 17 equations. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open.

For a given language L, we study the languages X such that for all distinct words u, v ∈ L, there exists a word x ∈ X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.

We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism α such that α(xy) ≠ α(yx) for given variables x and y.

We briefly survey some results and open problems on word equations, especially on those equations where the right-hand side is a power of a variable. We discuss a method that was recently used to prove one of the results, and we prove improved versions of some lemmas that are related to the method and can be used as tools when studying word equations. We use the method and the tools to give new, simple proofs for several old results.

We solve a long-standing open problem on word equations by proving that if the words x_0, ..., x_n satisfy the equation x_0^k = x_1^k ... x_n^k for three positive values of k, then the words commute. One of our methods is to assign numerical values for the letters, and then study the sums of the letters of words and their prefixes. We also give a geometric interpretation of our methods.

We study the question of what can be said about a word based on the numbers of occurrences of certain factors in it. We do this by defining a family of equivalence relations that generalize the so called k-abelian equivalence. The characterizations and answers we obtain are linear algebraic. We als o use these equivalence relations to help us in solving some problems related to repetitions and palindromes, and to point out that some previous results about Sturmian words and k-abelian equivalence hold in a more general form.

We show by a simple reduction that the unique decipherability problem in the language monoid of regular languages over a non-unary alphabet is undecidable.

Most contemporary security mechanisms and protocols include exchange of random or time-variant nonces as an essential means of protection against replay and other threats or as a seed for randomness. In many cases, it would be beneficial to have such nonces available from a trusted common source, such as a satellite. The goal of this paper is to present a protocol by which a loosely connected network of devices can agree on a common piece of randomness, and show how it can be applied to improve efficiency of a privacy protection system and group session key exchange for PAN/LAN devices.