Quantum coherent spaces and linear logic
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 4, pp. 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 : 10.1051/ita/2010021
Classification : 68Q55, 03F52
Mots-clés : 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},
     pages = {419--441},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {4},
     year = {2010},
     doi = {10.1051/ita/2010021},
     mrnumber = {2775405},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2010021/}
}
TY  - JOUR
AU  - Baratella, Stefano
TI  - Quantum coherent spaces and linear logic
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2010
SP  - 419
EP  - 441
VL  - 44
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2010021/
DO  - 10.1051/ita/2010021
LA  - en
ID  - ITA_2010__44_4_419_0
ER  - 
%0 Journal Article
%A Baratella, Stefano
%T Quantum coherent spaces and linear logic
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2010
%P 419-441
%V 44
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ita/2010021/
%R 10.1051/ita/2010021
%G en
%F ITA_2010__44_4_419_0
Baratella, Stefano. Quantum coherent spaces and linear logic. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 4, pp. 419-441. doi : 10.1051/ita/2010021. http://www.numdam.org/articles/10.1051/ita/2010021/

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

[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

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

[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

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

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

[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

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

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

[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

[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

Cité par Sources :