Two-variable word equations

Two-Variable Word Equations

We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of the same length. Speci cally, we prove that there are only ve types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizat...

Word Equations with Two Variables

The problem whether the set of all equations that are satisfiable in some free semigroup or, equivalently, in an algebra of words with concatenation is recursive (usually called the satisfiability problem for semigroup equations) was first formulated by A.A. Markov in early sixties (see [3]). Special cases of the problem were solved affirmatively by A.A. Markov (see [3]), Yu.I. Khmelevskiı̆ [8],...

Symmetric Word Equations in Two Positive Definite Letters

For every symmetric (“palindromic”) word S(A,B) in two positive definite letters and for each fixed n-by-n positive definite B and P , it is shown that the symmetric word equation S(A,B) = P has an n-by-n positive definite solution A. Moreover, if B and P are real, there is a real solution A. The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. I...

Quadrati Word Equations

Towards parametrizing word equations

Classically, in order to resolve an equation u ≈ v over a free monoid X∗, we reduce it by a suitable family F of substitutions to a family of equations uf ≈ vf , f ∈ F , each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get F in a handy parametrized form. The method we propose consists in parametrizing the path traces in t...