We present a generalisation of first-order unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #-equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a `nominal' approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution that respects #-equivalence and possesses good algorithmic properties. We achieve this by adapting an existing idea and introducing a key new idea. The existing idea is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The key new idea is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of the classical first-order unification algorithm to this setting which retains the latter's pleasant properties: unification problems involving #-equivalence and freshness are decidable; and solvable problems possess most general solutions.

For representing high level knowledge, such as the mathematical knowledge used in interactive theorem provers and verification systems, it is desirable to extend Prolog's concept of data object. A basic reason is that Prolog data objects --- Herbrand objects --- are terms of a minimal object language, which does not include its own object variables, or quantification over those variables. Qu-Prolog (Quantifier Prolog) is an extended logic programming concept which takes as its data objects, object terms which may include object level variables, free or bound, and arbitrary quantifiers to bind those variables. Qu-Prolog is unique in allowing its data objects to include free occurrences of object variables. This paper describes Qu-Prolog's data objects and its facilities for computing on those objects. The Qu-Prolog unification algorithm unifies QuProlog terms up to changes of bound variables. The power of Qu-Prolog unification is greatly increased by the inclusion in the language of a...