Abstract
Double-negation translations are used to encode and decode classical proofs in intuitionistic logic. We show that, in the cut-free fragment, we can simplify the translations and introduce fewer negations. To achieve this, we consider the polarization of the formulæ and adapt those translation to the different connectives and quantifiers. We show that the embedding results still hold, using a customized version of the focused classical sequent calculus. We also prove the latter equivalent to more usual versions of the sequent calculus. This polarization process allows lighter embeddings, and sheds some light on the relationship between intuitionistic and classical connectives.
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References
Cohen, P.J.: Set Theory and the Continuum Hypothesis. W.A. Benjamin, New York (1966)
Kolmogorov, A.: On the principle of the excluded middle. Mat. Sb. 32, 646–667 (1925)
Gödel, K.: Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse Eines Mathematischen Kolloquiums 4, 34–38 (1933)
Gentzen, G.: Die widerspruchsfreiheit der reinen zahlentheorie. Mathematische Annalen 112, 493–565 (1936)
Schwichtenberg, H., Senjak, C.: Minimal from classical proofs. Annals of Pure and Applied Logic 164, 740–748 (2013)
Schwichtenberg, H., Troelstra, A.S.: Basic Proof Theory. Cambridge University Press (1996)
Ferreira, G., Oliva, P.: On various negative translations. In: CL&C, pp. 21–33 (2010)
Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, An Introduction. North-Holland (1988)
Kuroda, S.: Intuitionistische untersuchungen der formalistischen logik. Nagoya Mathematical Journal 3, 35–47 (1951)
Krivine, J.L.: Opérateurs de mise en mémoire et traduction de Gödel. Arch. Math. Logic 30, 241–267 (1990)
Girard, J.Y.: On the unity of logic. Ann. Pure Appl. Logic 59, 201–217 (1993)
Curien, P.L., Herbelin, H.: The duality of computation. In: Odersky, M., Wadler, P. (eds.) ICFP, pp. 233–243. ACM (2000)
Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theor. Comput. Sci. 410, 4747–4768 (2009)
Kleene, S.C.: Permutability of inferences in Gentzen’s calculi LK and LJ. Memoirs of the American Mathematical Society 10, 1–26, 27–68 (1952)
Hermant, O.: Resolution is cut-free. Journal of Automated Reasoning 44, 245–276 (2010)
Dowek, G.: Polarized deduction modulo. In: IFIP Theoretical Computer Science. (2010)
Burel, G.: Embedding deduction modulo into a prover. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 155–169. Springer, Heidelberg (2010)
Burel, G.: Experimenting with deduction modulo. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 162–176. Springer, Heidelberg (2011)
Dowek, G., Werner, B.: Proof normalization modulo. The Journal of Symbolic Logic 68, 1289–1316 (2003)
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Boudard, M., Hermant, O. (2013). Polarizing Double-Negation Translations. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2013. Lecture Notes in Computer Science, vol 8312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45221-5_14
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DOI: https://doi.org/10.1007/978-3-642-45221-5_14
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