Constraint solving in non-permutative nominal abstract syntax

TL;DR

This paper introduces a new algorithm for solving constraints in non-permutative nominal abstract syntax, offering a simpler alternative to equivariant unification with comparable complexity.

Contribution

It presents a novel constraint solving algorithm for non-permutative nominal syntax, simplifying equivariant unification while maintaining similar computational complexity.

Findings

Algorithm is correct and terminates reliably.

Explicit translation from equivariant to non-permutative constraints.

Comparable complexity to equivariant unification.

Abstract

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.