Nominal Completion for Rewrite Systems with Binders

Fernández, Maribel; Rubio, Albert

doi:10.1007/978-3-642-31585-5_21

Cited by 7 publications

(4 citation statements)

References 23 publications

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“…If the theory under consideration is not orthogonal, then the alternative is to check termination and to check that all critical pairs are joinable (which is a sufficient condition for convergence, see [FGM04]). Reduction orderings (to check termination) and completion procedures (to ensure that all critical pairs are joinable) are available for closed nominal rules [FR10].…”

Section: Discussionmentioning

confidence: 99%

Closed nominal rewriting and efficiently computable nominal algebra equality

Fernández

Gabbay²

2010

Electron. Proc. Theor. Comput. Sci.

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We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of lambda-calculus and first-order logic.

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Section: Discussionmentioning

confidence: 99%

Closed nominal rewriting and efficiently computable nominal algebra equality

Fernández

Gabbay²

2010

Electron. Proc. Theor. Comput. Sci.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Rewriting with generalized nominal unification

Kutz

Schmidt-Schauß

2020

Math. Struct. Comp. Sci.

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We consider matching, rewriting, critical pairs and the Knuth–Bendix confluence test on rewrite rules in a nominal setting extended by atom-variables. We utilize atom-variables instead of atoms to formulate and rewrite rules on constrained expressions, which is an improvement of expressiveness over previous approaches. Nominal unification and nominal matching are correspondingly extended. Rewriting is performed using nominal matching, and computing critical pairs is done using nominal unification. We determine the complexity of several problems in a quantified freshness logic. In particular we show that nominal matching is $$\prod _2^p$$ -complete. We prove that the adapted Knuth–Bendix confluence test is applicable to a nominal rewrite system with atom-variables, and thus that there is a decidable test whether confluence of the ground instance of the abstract rewrite system holds. We apply the nominal Knuth–Bendix confluence criterion to the theory of monads and compute a convergent nominal rewrite system modulo alpha-equivalence.

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“…Since we now have reduction-preserving translations in both directions, properties and techniques developed for one formalism can be exported to the other (e.g., termination techniques based on the construction of a well-founded reduction ordering). A Haskell implementation of the translation functions, along with a tool to prove termination using the nominal recursive path ordering [17], are available from [10,9].Related work. CRSs, HRSs and ERSs are well-known examples of higher-order rewriting formalisms.…”

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confidence: 99%

From nominal to higher-order rewriting and back again

Domínguez¹,

Fernández²

2015

Logical Methods in Computer Science

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Abstract. We present a translation function from nominal rewriting systems (NRSs) to combinatory reduction systems (CRSs), transforming closed nominal rules and ground nominal terms to CRSs rules and terms, respectively, while preserving the rewriting relation. We also provide a reduction-preserving translation in the other direction, from CRSs to NRSs, improving over a previously defined translation. These tools, together with existing translations between CRSs and other higher-order rewriting formalisms, open up the path for a transfer of results between higher-order and nominal rewriting. In particular, techniques and properties of the rewriting relation, such as termination, can be exported from one formalism to the other.

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Nominal Completion for Rewrite Systems with Binders

Cited by 7 publications

References 23 publications

Closed nominal rewriting and efficiently computable nominal algebra equality

Closed nominal rewriting and efficiently computable nominal algebra equality

Rewriting with generalized nominal unification

From nominal to higher-order rewriting and back again

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