Checking Overlaps of Nominal Rewriting Rules

Rewriting with generalized nominal unification

“…(Suzuki et al 2015) (3) Can these properties be checked efficiently? (Ayala-Rincón et al 2016) A first necessary condition is uniformity, which prevents rules from introducing fresh unbound atoms (see Definition 4.2) However, this is not a sufficient condition, as an example from Suzuki et al (2015) shows.…”

“…To get around this, Suzuki et al (2015) defined the notion abstract skeleton preserving, which implies α-stability but can be efficiently checked. Ayala-Rincón et al (2016) extended this research further by introducing the notion closed rewrite rules, which again are α-stable but can be reduced to nominal matching and therefore be efficiently checked. The relation of these concept to the formalism in this paper is subject to future research.…”

“…Therefore, we investigate rewriting matching and critical pairs a la Knuth-Bendix (Knuth and Bendix 1970) in a higher-order language with alpha-equivalence and nominal modelling, where in the nominal unification and matching algorithm also atom-variables are permitted in addition to expression-variables and where the rewriting is done using a corresponding form of nominal matching with atom-variables. Previous approaches (Aoto and Kikuchi 2016;Ayala-Rincón et al 2016;Fernández and Rubio 2012;Kikuchi et al 2017;Suzuki et al 2016) to nominal rewriting and checking local confluence in order to apply the Knuth-Bendix confluence check usually proceed as follows: Rewrite rules are formulated such that atoms represent names, and in order to make the formulation independent of atom names, equivariance of the rewriting relation is implicitly assumed. This is the same as using atom-variables together with the implicit (extra) assumption that different atom-variables can only be instantiated with different atoms.…”