Nominal uni cation

We present a generalisation of rst-order uni cation 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 two existing ideas. The rst one is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The second one is that the uni cation algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of classical rst-order uni cation problems to this setting which retains the latter’s pleasant properties: uni cation problems involving � -equivalence and freshness are decidable; and solvable problems possess most general solutions. c 2004 Elsevier B.V. All rights reserved.