Abstract
This paper proposes and analyses a monadic translation of an intuitionistic sequent calculus. The source of the translation is a typed λ-calculus previously introduced by the authors, corresponding to the intuitionistic fragment of the call-by-name variant of \(\overline{\lambda}\mu\tilde{\mu}\) of Curien and Herbelin, and the target is a variant of Moggi’s monadic meta-language, where the rewrite relation includes extra permutation rules that may be seen as variations of the “associativity” of bind (the Kleisli extension operation of the monad).
The main result is that the monadic translation simulates reduction strictly, so that strong normalisation (which is enjoyed at the target, as we show) can be lifted from the target to the source. A variant translation, obtained by adding an extra monad application in the translation of types, still enjoys strict simulation, while requiring one fewer extra permutation rule from the target.
Finally we instantiate, for the cases of the identity monad and the continuations monad, the meta-language into the simply-typed λ-calculus. By this means, we give a generic account of translations of sequent calculus into natural deduction, which encompasses the traditional mapping studied by Zucker and Pottinger, and CPS translations of intuitionistic sequent calculus.
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References
Curien, P.-L., Herbelin, H.: The duality of computation. In: Proc. of 5th ACM SIGPLAN Int. Conf. on Functional Programming (ICFP 2000), Montréal, pp. 233–243. IEEE, Los Alamitos (2000)
Espírito Santo, J.: Completing Herbelin’s programme. In: Ronchi Della Rocca, S. (ed.) TLCA 2007. LNCS, vol. 4583, pp. 118–132. Springer, Heidelberg (2007)
Espírito Santo, J.: Delayed substitutions. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 169–183. Springer, Heidelberg (2007)
Espírito Santo, J.: Addenda to Delayed Substitutions (2008) (manuscript available from the author’s web page)
Espírito Santo, J., Matthes, R., Pinto, L.: Continuation-passing style and strong normalisation for intuitionistic sequent calculi. In: Ronchi Della Rocca, S. (ed.) TLCA 2007. LNCS, vol. 4583, pp. 133–147. Springer, Heidelberg (2007)
Espírito Santo, J., Matthes, R., Pinto, L.: Continuation-passing style and strong normalisation for intuitionistic sequent calculi. Logical Methods in Computer Science (to appear, 2009)
Hatcliff, J., Danvy, O.: A generic account of continuation-passing styles. In: POPL 1994: Proceedings of the 21st ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pp. 458–471. ACM, New York (1994)
Ikeda, S., Nakazawa, K.: Strong normalization proofs by CPS-translations. Information Processing Letters 99, 163–170 (2006)
Kfoury, A.J., Wells, J.B.: New notions of reduction and non-semantic proofs of beta-strong normalisation in typed lambda-calculi. In: Proceedings of LICS 1995, pp. 311–321 (1995)
Lengrand, S.: Call-by-value, call-by-name, and strong normalization for the classical sequent calculus. In: Gramlich, B., Lucas, S. (eds.) Post-proc. of the 3rd Workshop on Reduction Strategies in Rewriting and Programming (WRS 2003). Electronic Notes in Theoretical Computer Science, vol. 86, Elsevier, Amsterdam (2003)
Lengrand, S.: Temination of lambda-calculus with the extra call-by-value rule known as assoc. arXiv:0806.4859v2 (2007)
Moggi, E.: Notions of computation and monads. Inf. Comput. 93(1), 55–92 (1991)
Polonovski, E.: Strong normalization of \(\overline{\lambda}\mu\tilde{\mu}\) with explicit substitutions. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 423–437. Springer, Heidelberg (2004)
Pottinger, G.: Normalization as a homomorphic image of cut-elimination. Annals of Mathematical Logic 12(3), 323–357 (1977)
Regnier, L.: Une équivalence sur les lambda-termes. Theoretical Computer Science 126(2), 281–292 (1994)
Sabry, A., Wadler, P.: A reflection on call-by-value. ACM Trans. Program. Lang. Syst. 19(6), 916–941 (1997)
Zucker, J.: The correspondence between cut-elimination and normalization. Annals of Mathematical Logic 7(1), 1–112 (1974)
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Espírito Santo, J., Matthes, R., Pinto, L. (2009). Monadic Translation of Intuitionistic Sequent Calculus. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds) Types for Proofs and Programs. TYPES 2008. Lecture Notes in Computer Science, vol 5497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02444-3_7
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DOI: https://doi.org/10.1007/978-3-642-02444-3_7
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