# A classification of many-sorted unification problems

## Abstract

Many-sorted unification is considered, i.e. unification in the many-sorted free algebras of terms, where variables as well as the domains and ranges of functions are restricted to certain subsets of the universe, given as a potentially infinite hierarchy of sorts. Many-sorted unification is the same as solving an equation in the corresponding heterogeneous ‘order-sorted’ algebra /7/ rather than in a homogeneous algebra. It is proved that many-sorted unification can be classified completely by conditions imposed on the structure of the sort hierarchy. It is shown that complete and minimal sets of unifiers may not always exist for many-sorted unification. Conditions for sort hierarchies which are equivalent to the existence of these sets with one, finitely many or infinitely many elements are presented. It is also proved that being a forest-structured sort hierarchy is a necessary and sufficient criterion for the Robinson Unification Theorem to hold for many-sorted unification, i.e. a criterion for the existence of a most general unifier without ‘auxiliary variables’. An algorithm for many-sorted unification is given. This paper generalizes and extends the results presented in /22/. It is a shortened version of the technical report /24/, to which the reader is referred for any omitted proofs.

## Keywords

Auxiliary Variable Forest Structure Equational Theory Unification Algorithm Variable Symbol## Preview

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