Abstract
Starting from Laurent’s work on Polarized Linear Logic, we define a new polarized linear deduction system which handles recursion. This is achieved by extending the cut-rule, in such a way that iteration unrolling is achieved by cut-elimination. The proof nets counterpart of this extension is obtained by allowing oriented cycles, which had no meaning in usual polarized linear logic. We also free proof nets from additional constraints, leading up to a correctness criterion as straightforward as possible (since almost all proof structures are correct). Our system has a sound semantics expressed in traced models.
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References
Z. M. Ariola and J.W. Klop, Cyclic Lambda Graph Rewriting. In Proc. of the Eight IEEE Symposium on Logic in Computer Science, Paris, July 1994.
N. Benton and M. Hyland, Traced Premonoidal Categories (Extended Abstract). Workshop on Fixed Points in Computer Science (FICS 2002). July 2002.
V. Danos, Une Application de la Logique Linéaire à l’étude des Processus de Normalisation (principalement du λ-calcul). Thèse de Doctorat, Université Paris 7, 1990
J-Y. Girard, Linear Logic. Theoretical Computer Science 50:1–102, 1987.
M. Hasegawa, Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi. In Proc. 3rd International Conference on Typed Lambda Calculi and Applications, volume LNCS 1210. Springer, 1997.
M. Hasegawa and Y. Kakutani, Axioms for recursion in call-by-value. Higher-Order and Symbolic Computation 15(2/3):235–264, September 2002
M. Hofmann and T. Streicher, Continuation models are universal for lambda-mucalculus. In Proc. LICS’ 97, Warsaw, IEEE, pp. 387–397.
A. Joyal, R. Street and D. Verity, Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society, 119(3), pages 447–468, 1996.
O. Laurent, Etude de la polarisation en logique. PHD thesis, Université Aix-Marseille II, March 2002.
O. Laurent, Polarized games (extended abstract). Seventeenth annual IEEE symposium on Logic In Computer Science, p. 265–274. IEEE Computer Society, 2002.
O. Laurent, Polarized proof nets and λμ-calculus. Theoretical Computer Science, 290(1):161–188, Dec. 2002.
P-A. Melliès and P. Selinger, Polarized categories and polarized games. Forthcoming paper, 2003.
R. Montelatici, Présentation axiomatique de théorèmes de complétude forte en sémantique des jeux et en logique classique. Mémoire de DEA, Univ. Paris 7, 2001.
M. Parigot, λμ-calculus: an algorithmic interpretation of classical natural deduction. In Proc. of International Conference on Logic Programming and Automated Deduction, volume LNCS 624. Springer, 1992.
P. Selinger, Control Categories and Duality: on the Categorical Semantics of the Lambda-Mu Calculus. Mathematical Structures in Computer Science, 2001.
A. K. Simpson and G. D. Plotkin, Complete Axioms for Categorical Fixed-Point Operators. In Logic in Computer Science, pages 30–41, 2000.
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Montelatici, R. (2003). Polarized Proof Nets with Cycles and Fixpoints Semantics. In: Hofmann, M. (eds) Typed Lambda Calculi and Applications. TLCA 2003. Lecture Notes in Computer Science, vol 2701. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44904-3_18
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DOI: https://doi.org/10.1007/3-540-44904-3_18
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