A Computer Science Application of Permutation Models of Set Theory

Andrew M. Pitts, Cambridge University Computer Laboratory, Pembroke Street Cambridge CB2 3QG, UK

Joint work with Murdoch J. Gabbay, DPMMS, Cambridge University, Cambridge CB2 1SB, UK


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.