Abstract
In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the relational semantics with a notion of color or priority in the sense of parity games. This extension enables us to define a new fixpoint operator Y combining both inductive and coinductive policies. We conclude the paper by mentionning a connection between the resulting model of λ-calculus with recursion and higher-order model-checking.
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Baelde, D.: Least and greatest fixed points in linear logic. ACM Trans. Comput. Log. 13(1), 2 (2012)
Bierman, G.M.: What is a categorical model of intuitionistic linear logic? In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 78–93. Springer, Heidelberg (1995)
Bloom, S.L., Ésik, Z.: Iteration theories: the equational logic of iterative processes. EATCS monographs on theoretical computer science. Springer (1993)
Bloom, S.L., Ésik, Z.: Fixed-point operations on ccc’s. part i. Theoretical Computer Science 155(1), 1–38 (1996)
Carraro, A., Ehrhard, T., Salibra, A.: Exponentials with infinite multiplicities. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 170–184. Springer, Heidelberg (2010)
Fortier, J., Santocanale, L.: Cuts for circular proofs: semantics and cut-elimination. In: Rocca, S.R.D. (ed.) CSL. LIPIcs, vol. 23, pp. 248–262. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)
Grellois, C., Melliès, P.-A.: Tensorial logic with colours and higher-order model checking (submitted, 2015), http://arxiv.org/abs/1501.04789
Hasegawa, M.: Models of Sharing Graphs: A Categorical Semantics of Let and Letrec. Distinguished dissertations series, vol. 1192. Springer (1999)
Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society 119, 447–468 (1996)
Kobayashi, N., Luke Ong, C.-H.: A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In: LICS, pp. 179–188. IEEE Computer Society (2009)
Melliès, P.-A.: Categorical semantics of linear logic. In: Interactive models of computation and program behaviour, pp. 1–196 (2009)
Miquel, A.: Le calcul des constructions implicites: syntaxe et sémantique. PhD thesis, Université Paris 7 (2001)
Montelatici, R.: Polarized proof nets with cycles and fixpoints semantics. In: Hofmann, M.O. (ed.) TLCA 2003. LNCS, vol. 2701, pp. 256–270. Springer, Heidelberg (2003)
Salvati, S., Walukiewicz, I.: Evaluation is msol-compatible. In: Seth, A., Vishnoi, N.K. (eds.) FSTTCS. LIPIcs, vol. 24, pp. 103–114. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)
Salvati, S., Walukiewicz, I.: Using models to model-check recursive schemes. In: Hasegawa, M. (ed.) TLCA 2013. LNCS, vol. 7941, pp. 189–204. Springer, Heidelberg (2013)
Salvati, S., Walukiewicz, I.: Typing weak MSOL properties (September 2014)
Santocanale, L.: A calculus of circular proofs and its categorical semantics. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 357–371. Springer, Heidelberg (2002)
Santocanale, L.: μ-bicomplete categories and parity games. ITA 36(2), 195–227 (2002)
Seely, R.A.G.: Linear logic, *-autonomous categories and cofree coalgebras. In: Categories in Computer Science and Logic, pp. 371–382. American Mathematical Society (1989)
Simpson, A.K., Plotkin, G.D.: Complete axioms for categorical fixed-point operators. In: LICS 2000, USA, June 26-29, pp. 30–41. IEEE Computer Society (2000)
Terui, K.: Semantic evaluation, intersection types and complexity of simply typed lambda calculus. In: Tiwari, A. (ed.) RTA. LIPIcs, vol. 15, pp. 323–338. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)
Tsukada, T., Luke Ong, C.-H.: Compositional higher-order model checking via ω-regular games over böhm trees. In: Henzinger, T.A., Miller, D. (eds.) CSL-LICS 2014, Vienna, Austria, July 14-18, p. 78. ACM (2014)
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Grellois, C., Melliès, PA. (2015). An Infinitary Model of Linear Logic. In: Pitts, A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2015. Lecture Notes in Computer Science(), vol 9034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46678-0_3
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