Decidability of Word Equations with Linear Length Constraints

Determine whether the satisfiability problem for word equations with linear length constraints in a free monoid is decidable: given a word equation U = V over a finite alphabet A with variables X, together with a finite system of linear equations over the integers constraining the lengths |σ(X)| of the variable images, decide whether there exists a morphism σ: (A ∪ X)* → A* fixing A pointwise such that σ(U) = σ(V) and all the specified linear length constraints hold.

Background

In the paper of word equations over free monoids, adding non-rational constraints such as relations among lengths, exponent-sums, or Parikh images can change decidability behavior. Büchi and Senger showed that certain counting constraints make the problem undecidable, and further work established equivalences between some types of counting constraints.

Despite these advances, the case of linear length constraints remains unresolved. Establishing decidability (or undecidability) of word equations with linear length constraints would have significant implications, including connections to Hilbert’s 10th problem and practical consequences for string solvers in program analysis.