# On Nominal Syntax and Permutation Fixed Points

## TL;DR

This paper introduces a new axiomatisation of alpha-equivalence for nominal terms using fixed-point constraints, improving unification behavior under equational theories like commutativity.

## Contribution

It proposes a fixed-point based notion of alpha-equivalence and unification, enhancing finitary properties in the presence of equational theories such as commutativity.

## Key findings

- Alpha-equivalence can be derived from permutation fixed-points.
- Unification remains finitary with fixed-point constraints under certain theories.
- Provides a sound and complete C-unification algorithm.

## Abstract

We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new $\alpha$-equivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. Although it may seem less natural than the standard notion of nominal unifier based on freshness constraints, the notion of unifier based on fixed-point constraints behaves better when equational theories are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts. We provide a definition of $\alpha$-equivalence modulo equational theories that take into account A, C and AC theories. Based on this notion of equivalence, we show that C-unification is finitary and we provide a sound and complete C-unification algorithm, as a first step towards the development of nominal unification modulo AC and other equational theories with permutative properties.