Encoding rewriting strategies in λ-calculi with patterns

We propose a patch to the pure pattern calculus: we claim that this is strictly more powerful to define the application of the match fail as the pure λ-term defining the boolean false instead of the identity function as it is done in the original version of the pure pattern calculus [JK09]. We show that using non algebraic patterns we are able to encode in a natural way any rewriting strategies as well as the branching construct | used in functional programming languages. We close the open question (raised in [Cir00, CK01]) whether rewriting strategies can be directly encoded in λ-calculi with patterns. Key-words: Rewriting strategies, lambda-calculus with patterns, rewriting calculus, pure pattern calculus, higher-order encodings. in ria -0 04 13 17 8, v er si on 2 3 Se p 20 09 Encodage des stratégies de réécritures dans les λ-calculs avec motifs Résumé : Nous proposons une alternative au calcul pur de motifs: nous affirmons qu’il est strictement plus puissant de définir l’application du filtre d’échec comme le terme du λ-calcul pur définissant la constante false plutôt que de le définir comme la fonction identité comme cela a été fait dans la version originale du calcul pur de motifs [JK09]. Nous montrons qu’en utilisant des motifs non algébriques nous pouvons obtenir un encodage naturel des stratégies de réécriture ainsi que du constructeur de branchement | des langages fonctionnels. Nous clôturons la question ouverte (formulée dans [Cir00, CK01]) si les stratégies de réécriture sont directement encodables dans les λ-calculs avec motifs. Mots-clés : Stratégies de réécriture, lambda-calcul de motifs, calcul de réécriture, calcul pur de motifs, encodage d’ordre supérieur. in ria -0 04 13 17 8, v er si on 2 3 Se p 20 09 Encoding rewriting strategies in λ-calculi with patterns 3 1 Preliminaries We first recall some classical encoding of pairs, boolean etc. in the pure λcalculus (see for example [Bar84]) 〈t, u〉 = λx. x t u π1t = t (λxy. x) π2t = t (λxy. y) true = λxy. x false = λxy. y if t then u else v = t u v let x = t in u = (λx. u) t fix = (λx. λf. f(x x f)) (λx. λf. f(x x f)) If C[] is a context then we will write fun = fix(C[fun]) for the definition of a recursive function called fun and defined by fix (λs.C[s]). 2 Yet a more general framework We consider the general framework given in Section 2 of [JK09] and use the same notations. We first begin by a remark. The application of a match μ to a term can be defined in a more general way as follows: If μ is a substitution, then the application of the match to a term is obtained by applying the substitution to variables of the term as explained in [JK09]. If μ is fail, we define fail t = u where u is an arbitrary term [the calculus is parametrized by this term u]. Theorem 2.1 (Confluence) The pure pattern calculus, as defined in Section 3 of [JK09] but with the above application of a match, is confluent when u is a pure λ-term in normal form.