Abstract. The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of 'name-abstraction' and 'fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science.

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We present a generalisation of ÿrst-order uniÿcation 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 ÿrst 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 uniÿcation algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of classical ÿrst-order uniÿcation problems to this setting which retains the latter's pleasant properties: uniÿcation problems involving -equivalence and freshness are decidable; and solvable problems possess most general solutions.

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