Constraint solving in non-permutative nominal abstract syntax

Nominal abstract syntax is a popular first-order technique for encoding, and reasoning about, abstract syntax involving binders. Many of its applications involve constraint solving. The most commonly used constraint solving algorithm over nominal abstract syntax is the Urban-Pitts-Gabbay nominal unification algorithm, which is well-behaved, has a well-developed theory and is applicable in many cases. However, certain problems require a constraint solver which respects the equivariance property of nominal logic, such as Cheney's equivariant unification algorithm. This is more powerful but is more complicated and computationally hard. In this paper we present a novel algorithm for solving constraints over a simple variant of nominal abstract syntax which we call non-permutative. This constraint problem has similar complexity to equivariant unification but without many of the additional complications of the equivariant unification term language. We prove our algorithm correct, paying particular attention to issues of termination, and present an explicit translation of name-name equivariant unification problems into non-permutative constraints

A dependent nominal type theory

Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles