Relating Nominal and Higher-Order Rewriting

From nominal to higher-order rewriting and back again

“…Although in this paper we focus on the relationship between NRSs and CRSs, thanks to the existing translations between CRSs and other higher-order rewriting formalisms, this is sufficient to obtain a bridge between nominal and higher-order rewriting.Our work is closely related to the work reported in [6,27]: Cheney [6] represented higherorder unification as nominal unification, and Levy and Villaret [27] transformed nominal unification into higher-order unification, providing a translation that preserves unifiers. Our translation differs from [6,27] in that our requirement is to have a mapping of NRS ground terms and rules to CRS terms and rules in such a way that reductions are preserved.This paper is an updated and extended version of [11]. We have included here, in addition to the translation from NRSs to CRSs given in [11], all the proofs previously omitted due to space constraints as well as a translation from CRSs to NRSs, improving on a previous result given in [16].…”

“…Our translation differs from [6,27] in that our requirement is to have a mapping of NRS ground terms and rules to CRS terms and rules in such a way that reductions are preserved.This paper is an updated and extended version of [11]. We have included here, in addition to the translation from NRSs to CRSs given in [11], all the proofs previously omitted due to space constraints as well as a translation from CRSs to NRSs, improving on a previous result given in [16]. We provide detailed explanations, and illustrate the translations with examples.Overview of the paper.…”

Unificação, confluência e tipos com interseção para sistemas de reescrita nominal

“…Closed rules roughly correspond to rules without free atoms, where rewriting cannot change the binding status of an atom. They are the counterpart of rules in standard higher-order rewriting formats (see [Domínguez and Fernández, 2014]). Below we first recall the definition of nominal matching and then give a structural definition and an operational characterisation of closed terms.…”

“…The following structural definition of closedness follows [Clouston, 2007] and [Domínguez and Fernández, 2014].…”

“…We could: these are equivalent-in a nominal context. The version above directly translates the corresponding CRS rule (see [Domínguez and Fernández, 2014]) which, following Barendregt's convention, must use different names for bound variables in a rule. Theorem 4.14 tells us that the closed rewriting relation generated by the theory in Example 4.15 is confluent.…”