An Algorithm for the Solution of Tree Equations

We consider the problem of solving equations over k-ary trees. Here an equation is a pair of labeled-ary trees, where is a function associating an arity to each label. A solution to an equation is a morphism from-ary trees to k-ary trees that maps the left and right hand side of the equation to the same k-ary tree. This problem is a generalization of the word uniication problem posed by A. Markov in the fties, which corresponds to the case k=1, (in this case also the arity function must be identically equal to 1, and equations are pairs of words). The word uniication problem was solved in two steps. First in 1976 Makanin proved the decidability of the existence of a solution to a word equation, and more recently in 1990 Jaaar gave an algorithm that nds the set of all principal solutions to a word equation when this set is nite. In this paper we solve the-ary tree equation problem for all other k > 1. We describe an eecient uniication algorithm that on input an-ary tree equation, computes a most general (-ary) solution to the equation if the equation is solvable and reports failure otherwise. This also proves that any satissable-ary tree equation has a most general solution. All k-ary solutions to the equation can be easily obtained from the-ary solution output by our algorithm.