The Unification Hierarchy is Undecidable

In unification theory, equational theories can be classified according to the existence and cardinality of minimal complete solution sets for equation systems. For unitary, finitary, and infinitary theories minimal complete solution sets always exist and are singletons, finite, or possibly infinite sets, respectively. In nullary theories, minimal complete sets do not exist for some equation systems. These classes form the unification hierarchy. We show that it is not possible to decide where a given equational theory resides in the unification hierarchy. Moreover, it is proved that for some classes this problem is not even recursively enumerable