Abstract
Originating from automated theorem proving, deduction modulo removes computational arguments from proofs by interleaving rewriting with the deduction process. From a proof-theoretic point of view, deduction modulo defines a generic notion of cut that applies to any first-order theory presented as a rewrite system. In such a setting, one can prove cut-elimination theorems that apply to many theories, provided they verify some generic criterion. Pre-Heyting algebras are a generalization of Heyting algebras which are used by Dowek to provide a semantic intuitionistic criterion called superconsistency for generic cut-elimination. This paper uses pre-Boolean algebras (generalizing Boolean algebras) and biorthogonality to prove a generic cut-elimination theorem for the classical sequent calculus modulo. It gives this way a novel application of reducibility candidates techniques, avoiding the use of proof-terms and simplifying the arguments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abrusci, V.M.: Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic 56(4), 1403–1451 (1991)
Cousineau, D.: Complete reducibility candidates. In: PSTT 2009 (2009)
Ciabattoni, A., Terui, K.: Towards a semantic characterization of cut-elimination. Studia Logica 82(1), 95–119 (2006)
Dowek, G., Hermant, O.: A simple proof that super-consistency implies cut elimination. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 93–106. Springer, Heidelberg (2007)
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. Journal of Automated Reasoning 31(1), 33–72 (2003)
Dowek, G.: Truth values algebras and proof normalization. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 110–124. Springer, Heidelberg (2007)
Dowek, G.: Fondements des systèmes de preuve. Course notes (2010)
Dowek, G., Werner, B.: Proof normalization modulo. Journal of Symbolic Logic 68(4), 1289–1316 (2003)
Gimenez, S.: Programmer, Calculer et Raisonner avec les Réseaux de la Logique Linéaire. PhD thesis, Université Paris 7 (2009)
Girard, J.-Y.: Interprétation Fonctionnelle et Élimination des Coupures de l’Arithmétique dOrdre Supérieur. PhD thesis, Université Paris 7 (1972)
Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)
Krivine, J.-L.: Realizability in classical logic. Panoramas et synthèses 27, 197–229 (2009)
Lengrand, S., Miquel, A.: Classical F [omega], orthogonality and symmetric candidates. Annals of Pure and Applied Logic 153(1-3), 3–20 (2008)
Okada, M.: Phase semantic cut-elimination and normalization proofs of first-and higher-order linear logic. Theoretical Computer Science 227(1-2), 333–396 (1999)
Okada, M.: A uniform semantic proof for cut-elimination and completeness of various first and higher order logics. Theoretical Computer Science 281(1-2), 471–498 (2002)
Tait, W.W.: A realizability interpretation of the theory of species. In: Parikh, R.J. (ed.) Logic Colloquium, pp. 240–251. Springer, Heidelberg (1975)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brunel, A., Hermant, O., Houtmann, C. (2011). Orthogonality and Boolean Algebras for Deduction Modulo. In: Ong, L. (eds) Typed Lambda Calculi and Applications. TLCA 2011. Lecture Notes in Computer Science, vol 6690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21691-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-21691-6_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21690-9
Online ISBN: 978-3-642-21691-6
eBook Packages: Computer ScienceComputer Science (R0)