2000 character limit reached
A linear linear lambda-calculus (2201.11221v7)
Published 26 Jan 2022 in cs.LO
Abstract: We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part of a broader research program aiming to define a logic with a proof language that forms a quantum programming language.
- T. Altenkirch and J. Grattage. A functional quantum programming language. In Proceedings of LICS 2005, pages 249–258. IEEE, 2005.
- P. Arrighi and A. Díaz-Caro. A System F accounting for scalars. Logical Methods in Computer Science, 8(1:11), 2012.
- The vectorial lambda-calculus. Information and Computation, 254(1):105–139, 2017.
- P. Arrighi and G. Dowek. Lineal: A linear-algebraic lambda-calculus. Logical Methods in Computer Science, 13(1), 2017.
- R. Blute. Hopf algebras and linear logic. Mathematical Structures in Computer Science, 6(2):189–217, 1996.
- K. Chardonnet. Towards a Curry-Howard Correspondence for Quantum Computation. PhD thesis, Université Paris-Saclay, 2023.
- B. Coecke and A. Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017.
- A. Díaz-Caro and G. Dowek. A new connective in natural deduction, and its application to quantum computing. Theoretical Computer Science, 957:113840, 2023.
- Two linearities for quantum computing in the lambda calculus. BioSystems, 186:104012, 2019. Postproceedings of TPNC 2017.
- Realizability in the unitary sphere. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2019), pages 1–13, 2019.
- A. Díaz-Caro and O. Malherbe. Quantum control in the unitary sphere: Lambda-𝒮1subscript𝒮1{\mathcal{S}}_{1}caligraphic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and its categorical model. Logical Methods in Computer Science, 18(3:32), 2022.
- A. Díaz-Caro and O. Malherbe. Semimodules and the (syntactically-)linear lambda calculus. Draft at arXiv:2205.02142, 2022.
- A. Díaz-Caro and B. Petit. Linearity in the non-deterministic call-by-value setting. In L. Ong and R. de Queiroz, editors, Proceedings of WoLLIC 2012, volume 7456 of LNCS, pages 216–231, 2012.
- T. Ehrhard. On Köthe sequence spaces and linear logic. Mathematical Structures in Computer Science, 12(5):579–623, 2002.
- J.-Y. Girard. Interprétation fonctionnelle et élimination des coupures dans l’arithmétique d’ordre supérieur. PhD thesis, Université de Paris 7, 1972.
- J.-Y. Girard. Linear logic. Theoreoretical Computer Science, 50:1–102, 1987.
- J.-Y. Girard. Coherent Banach spaces: A continuous denotational semantics. Theoretical Computer Science, 227(1-2):275–297, 1999.
- R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3–29, 1998.
- P. Selinger and B. Valiron. A lambda calculus for quantum computation with classical control. Mathematical Structures in Computer Science, 16(3):527–552, 2006.
- L. Vaux. The algebraic lambda calculus. Mathematical Structures in Computer Science, 19(5):1029–1059, 2009.
- M. Zorzi. On quantum lambda calculi: a foundational perspective. Mathematical Structures in Computer Science, 26(7):1107–1195, 2016.