Abstract
For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations – those for which each variable occurs at most once on each side of the equation – we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP.
Highlights
A word equation is a tuple (α, β), which we shall usually write as α =. β, such that α and β are words comprised of letters from a terminal alphabet Σ = {a, b, . . .} and variables from a set X = {x, y, z, . . .}
Due to Proposition 3.4, we may infer directly from Theorem 8.11 that the satisfiability problem for regular word equations is in NP
A famous algorithm for solving quadratic word equations can be used to produce a graph containing all solutions to the equation
Summary
It is natural to represent this relation as a directed graph G ⇒NT in which the vertices are word equations and the edges are the rewriting transformations This has the advantage that the set of all solutions to. We give an example class of equations for which the DAG-width is unbounded, as well as a class for which the DAG-width is at most two The latter includes the class of regular-ordered equations which is the most general subclass of QWEs for which it is known that the satisfiability problem with length constraints is decidable [20], and we expect that both cases will be interesting classes to consider in the context of this problem
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