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...

Satisfiability of word equations problem is: Given two sequences consisting letters and variables decide whether there is a substitution for the that turns this equation into true equality. The exact computational complexity remains unknown, with best lower upper bounds being, respectively, NP PSPACE . Recently, novel technique recompression was applied to problem, simplifying known proofs lowe...

We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions to word equations with length constraints. The atomic formulas over this language are equality over string terms (word equations), linear inequality over the length function (length constraints), and membership in regular sets. These questions are import...

We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions to word equations with length constraints. The atomic formulas over this language are equality over string terms (word equations), linear inequality over the length function (length constraints), and membership in regular sets. These questions are import...

Word equations (in free semigroup) are an important problem on the intersection of formal languages and algebra. Given two sequences consisting of letters and variables we are to decide whether there is a substitution for the variables that turns this formal equation into true equality of strings. The exact computational complexity of this problem remains unknown, with the best known lower and ...

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],...