Abstract
This paper studies a new classical natural deduction system, presented as a typed calculus named \(\underline{\lambda}\mu let\). It is designed to be isomorphic to Curien-Herbelin’s \(\underline{\lambda}\mu \widetilde{\mu}\)-calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot’s λμ-calculus with the idea of ”coercion calculus” due to Cervesato-Pfenning, accommodating let-expressions in a surprising way: they expand Parigot’s syntactic class of named terms.
This calculus aims to be the simultaneous answer to three problems. The first problem is the lack of a canonical natural deduction system for classical logic. \(\underline{\lambda}\mu let\) is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus. The second problem is the lack of a formalization of the usual semantics of \(\underline{\lambda}\mu \widetilde{\mu}\)-calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. The mentioned isomorphism is the required formalization, based on the precise notions of context and hole-expression offered by \(\underline{\lambda}\mu let\). The third problem is the lack of a robust process of ”read-back” into natural deduction syntax of calculi in the sequent calculus format, that affects mainly the recent proof-theoretic efforts of derivation of λ-calculi for call-by-value. An isomorphic counterpart to the Q-subsystem of \(\underline{\lambda}\mu \widetilde{\mu}\)-calculus is derived, obtaining a new λ-calculus for call-by-value, combining control and let-expressions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abadi, M., Cardelli, L., Curien, P.-L., Lévy, J.-J.: Explicit substitutions. Journal of Functional Programming 1(4), 375–416 (1991)
Cervesato, I., Pfenning, F.: A linear spine calculus. Journal of Logic and Computation 13(5), 639–688 (2003)
Curien, P.-L., Herbelin, H.: The duality of computation. In: Proceedings of the Fifth ACM SIGPLAN International Conference on Functional Programming (ICFP 2000), Montreal, Canada, September 18-21. SIGPLAN Notices, vol. 35(9), pp. 233–243. ACM, New York (2000)
Dyckhoff, R., Lengrand, S.: Call-by-value lambda calculus and LJQ. Journal of Logic and Computation 17, 1109–1134 (2007)
Espírito Santo, J.: Revisiting the correspondence between cut-elimination and normalisation. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 600–611. Springer, Heidelberg (2000)
Espírito Santo, J.: An isomorphism between a fragment of sequent calculus and an extension of natural deduction. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, pp. 354–366. Springer, Heidelberg (2002)
Espírito Santo, J.: The λ-calculus and the unity of structural proof theory. Theory of Computing Systems 45, 963–994 (2009)
Felleisen, M., Friedman, D., Kohlbecker, E., Duba, B.: Reasoning with continuations. In: 1st Symposium on Logic and Computer Science. IEEE, Los Alamitos (1986)
Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The collected papers of Gerhard Gentzen, pp. 68–131. North Holland, Amsterdam (1969)
Griffin, T.: A formulae-as-types notion of control. In: ACM Conf. Principles of Programming Languages. ACM Press, New York (1990)
Herbelin, H.: C’est maintenant qu’on calcule, Habilitation Thesis (2005)
Herbelin, H., Zimmermann, S.: An operational account of call-by-value minimal and classical lambda-calculus in “natural deduction” form. In: Curien, P.-L. (ed.) TLCA 2009. LNCS, vol. 5608, pp. 142–156. Springer, Heidelberg (2009)
Kikuchi, K.: Call-by-name reduction and cut-elimination in classical logic. Annals of Pure and Applied Logic 153, 38–65 (2008)
Moggi, E.: Computational lambda-calculus and monads. Technical Report ECS-LFCS-88-86, University of Edinburgh (1988)
Negri, S., von Plato, J.: Structural Proof Theory, Cambridge (2001)
Ong, C.-H.L., Stewart, C.A.: A Curry-Howard foundation for functional computation with control. In: Proc. of Symposium on Principles of Programming Languages (POPL 1997), pp. 215–217. ACM Press, New York (1997)
Parigot, M.: λμ-calculus: an algorithmic interpretation of classic natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992)
Plotkin, G.: Call-by-name, call-by-value and the λ-calculus. Theoretical Computer Science 1, 125–159 (1975)
Polonovski, E.: Strong normalization of lambda-mu-mu-tilde with explicit substitutions. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 423–437. Springer, Heidelberg (2004)
Prawitz, D.: Natural Deduction. A Proof-Theoretical Study. Almquist and Wiksell, Stockholm (1965)
Rehof, N., Sorensen, M.: The λ Δ-calculus. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789. Springer, Heidelberg (1994)
Rocheteau, J.: λμ-calculus and duality: call-by-name and call-by-value. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 204–218. Springer, Heidelberg (2005)
Sabry, A., Felleisen, M.: Reasoning about programms in continuation-passing-style. LISP and Symbolic Computation 6(3/4), 289–360 (1993)
Sabry, A., Wadler, P.: A reflection on call-by-value. ACM Transactions on Programming Languages and Systems 19(6), 916–941 (1997)
Troelstra, A., Schwitchtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)
von Plato, J.: Natural deduction with general elimination rules. Annals of Mathematical Logic 40(7), 541–567 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Santo, J.E. (2010). Towards a Canonical Classical Natural Deduction System. In: Dawar, A., Veith, H. (eds) Computer Science Logic. CSL 2010. Lecture Notes in Computer Science, vol 6247. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15205-4_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-15205-4_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15204-7
Online ISBN: 978-3-642-15205-4
eBook Packages: Computer ScienceComputer Science (R0)