Equational Reasoning Modulo Commutativity in Languages with Binders (Extended Version)

TL;DR

This paper introduces a novel nominal algebra framework that integrates commutativity into reasoning with binders, providing sound semantics and a terminating unification algorithm, advancing formal reasoning in languages with binders and equational axioms.

Contribution

It presents a new nominal algebra with permutation fixed-point constraints, establishing proof-theoretical properties and a complete semantics, along with a terminating unification algorithm for reasoning modulo commutativity.

Findings

Established proof-theoretical properties of the new algebra

Provided a sound and complete semantics in nominal sets

Developed a terminating and correct unification algorithm

Abstract

Many formal languages include binders as well as operators that satisfy equational axioms, such as commutativity. Here we consider the nominal language, a general formal framework which provides support for the representation of binders, freshness conditions and $\alpha$-renaming. Rather than relying on the usual freshness constraints, we introduce a nominal algebra which employs permutation fixed-point constraints in $\alpha$-equivalence judgements, seamlessly integrating commutativity into the reasoning process. We establish its proof-theoretical properties and provide a sound and complete semantics in the setting of nominal sets. Additionally, we propose a novel algorithm for nominal unification modulo commutativity, which we prove terminating and correct. By leveraging fixed-point constraints, our approach ensures a finitary unification theory, unlike standard methods relying on freshness constraints. This framework offers a robust foundation for structural induction and recursion over syntax with binders and commutative operators, enabling reasoning in settings such as first-order logic and the $\pi$-calculus.