Abstract
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are developed, the latter in terms of (strict) indexed symmetric monoidal categories with comprehension. Various optional type formers are treated in a modular way. In particular, we will see that the historically much-debated multiplicative quantifiers and identity types arise naturally from categorical considerations. These new multiplicative connectives are further characterised by several identities relating them to the usual connectives from dependent type theory and linear logic. Finally, one important class of models, given by families with values in some symmetric monoidal category, is investigated in detail.
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References
Martin-Löf, P.: An intuitionistic theory of types. Twenty-five Years of Constructive Type Theory 36, 127–172 (1998)
Girard, J.Y.: Linear logic. Theoretical Computer Science 50(1), 1–101 (1987)
Program, T.U.F.: Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study (2013), http://homotopytypetheory.org/book
Cervesato, I., Pfenning, F.: A linear logical framework. In: LICS 1996. Proceedings, pp. 264–275. IEEE (1996)
Dal Lago, U., Gaboardi, M.: Linear dependent types and relative completeness. In: LiCS 2011. Proceeding, pp. 133–142. IEEE (2011)
Petit, B., et al.: Linear dependent types in a call-by-value scenario. In: Proceedings of the 14th Symposium on Principles and Practice of Declarative Programming, pp. 115–126. ACM (2012)
Gaboardi, M., Haeberlen, A., Hsu, J., Narayan, A., Pierce, B.C.: Linear dependent types for differential privacy. ACM SIGPLAN Notices 48, 357–370 (2013)
Watkins, K., Cervesato, I., Pfenning, F., Walker, D.: A concurrent logical framework i: Judgments and properties. Technical report, DTIC Document (2003)
May, J.P., Sigurdsson, J.: Parametrized homotopy theory, vol. 132. American Mathematical Soc. (2006)
Shulman, M.: Enriched indexed categories. Theory and Applications of Categories 28(21), 616–695 (2013)
Ponto, K., Shulman, M.: Duality and traces for indexed monoidal categories. Theory and Applications of Categories 26(23), 582–659 (2012)
Schreiber, U.: Quantization via linear homotopy types. arXiv preprint arXiv:1402.7041 (2014)
Vákár, M.: Syntax and semantics of linear dependent types. arXiv preprint arXiv:1405.0033 (original preprint from April 2014)
Krishnaswami, N.R., Pradic, P., Benton, N.: Integrating dependent and linear types (July 2014), https://www.mpi-sws.org/~neelk/dlnl-paper.pdf
Barber, A.: Dual intuitionistic linear logic. Technical Report ECS-LFCS-96-347, University of Edinburgh, Edinburgh (1996)
Hofmann, M.: Syntax and semantics of dependent types. In: Extensional Constructs in Intensional Type Theory, pp. 13–54. Springer (1997)
Jacobs, B.: Comprehension categories and the semantics of type dependency. Theoretical Computer Science 107(2), 169–207 (1993)
Pitts, A.M.: Categorical logic. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science. Algebraic and Logical Structures, vol. 5, pp. 39–128. Oxford University Press (2000)
Lawvere, F.W.: Equality in hyperdoctrines and comprehension schema as an adjoint functor. Applications of Categorical Algebra 17, 1–14 (1970)
Abramsky, S., Duncan, R.: A categorical quantum logic. Mathematical Structures in Computer Science 16(3), 469–489 (2006), Preprint available at http://arxiv.org/abs/quant-ph/0512114
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Vákár, M. (2015). A Categorical Semantics for Linear Logical Frameworks. 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_7
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