Quantum coherent spaces and linear logic
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 44 (2010) no. 4, p. 419-441

Quantum Coherent Spaces were introduced by Girard as a quantum framework where to interpret the exponential-free fragment of Linear Logic. Aim of this paper is to extend Girard's interpretation to a subsystem of linear logic with bounded exponentials. We provide deduction rules for the bounded exponentials and, correspondingly, we introduce the novel notion of bounded exponentials of Quantum Coherent Spaces. We show that the latter provide a categorical model of our system. In order to do that, we first study properties of the category of Quantum Coherent Spaces.

DOI : https://doi.org/10.1051/ita/2010021
Classification:  68Q55,  03F52
Keywords: quantum coherent spaces, linear logic, bounded exponentials, denotational semantics, normalization
@article{ITA_2010__44_4_419_0,
author = {Baratella, Stefano},
title = {Quantum coherent spaces and linear logic},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
publisher = {EDP-Sciences},
volume = {44},
number = {4},
year = {2010},
pages = {419-441},
doi = {10.1051/ita/2010021},
mrnumber = {2775405},
language = {en},
url = {http://www.numdam.org/item/ITA_2010__44_4_419_0}
}

Baratella, Stefano. Quantum coherent spaces and linear logic. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 44 (2010) no. 4, pp. 419-441. doi : 10.1051/ita/2010021. http://www.numdam.org/item/ITA_2010__44_4_419_0/

[1] S. Abramsky and R. Jagadeesan, Games and full completeness for multiplicative linear logic. J. Symb. Log. 2 (1994) 543-574. | Zbl 0822.03007

[2] J.M. Ansemil and K. Floret, The symmetric tensor product of a direct sum of locally convex spaces. Stud. Math. 129 (1998) 285-295. | Zbl 0931.46005

[3] M. Barr, $*$-autonomous categories and linear logic. Math. Struct. Comp. Sci. 1 (1991) 159-178. | Zbl 0777.18006

[4] J.R.B. Cockett and R.A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories. Theory and Applications of categories 3 (1997) 85-131. | Zbl 0879.03022

[5] J.-Y. Girard, Le Point Aveugle Ii, Cours de logique, Vers l'imperfection. Hermann, Paris (2007). | Zbl pre05382524

[6] J.-Y. Girard, Truth, modality and intersubjectivity. Math. Struct. Comp. Sci. 17 (2007) 1153-1167. | Zbl 1146.03003

[7] J.-Y. Girard, A. Scedrov and P. Scott. Bounded linear logic: a modular approach to polynomial-time computability. Theoret. Comput. Sci. 97 (1992) 1-66. | Zbl 0788.03005

[8] S. Mac Lane, Categories for the Working Mathematician. 2nd edition Springer, Berlin (1998). | Zbl 0232.18001

[9] R.E. Megginson, An Introduction to Banach Space Theory. Springer, Berlin (1998). | Zbl 0910.46008

[10] P.-A. Melliès, Categorical semantics of linear logic, available at http://www.pps.jussieu.fr/ mellies/.

[11] B.F. Redmond, Multiplexor categories and models of Soft Linear Logic. Logical foundations of computer science, Lecture Notes in Comput. Sci. 4514, Springer, Berlin (2007) 472-485. | Zbl 1132.03352

[12] P. Selinger, Towards a semantics for higher-order quantum computation. Proc. QPL (2004) 127-143.

[13] J. Weidmann, Linear Operators in Hilbert Spaces. Springer, Berlin (1980). | Zbl 0434.47001