Uniication in Extensions of Shallow Equational Theories

We show that uniication in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove de-cidability of uniication in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is eeective on sorted shallow equational theories. So called semi-linear equational theories can be eeectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.