Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T23:37:47.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum coherent spaces and linear logic

Published online by Cambridge University Press:  28 October 2010

Stefano Baratella
Stefano Baratella*
Affiliation:
Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38050 Povo, Italy. [email protected]; [email protected]
Get access

Abstract

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.

Keywords

Quantum coherent spaceslinear logicbounded exponentialsdenotational semanticsnormalization
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 44 , Issue 4 , October 2010 , pp. 419 - 441
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. and Jagadeesan, R., Games and full completeness for multiplicative linear logic. J. Symb. Log. 2 (1994) 543–574. CrossRef
Ansemil, J.M. and Floret, K., The symmetric tensor product of a direct sum of locally convex spaces. Stud. Math. 129 (1998) 285–295.
Barr, M., $*$ -autonomous categories and linear logic. Math. Struct. Comp. Sci. 1 (1991) 159–178. CrossRef
Cockett, J.R.B. and Seely, R.A.G., Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories. Theory and Applications of categories 3 (1997) 85–131.
J.-Y. Girard, Le Point Aveugle II, Cours de logique, Vers l'imperfection. Hermann, Paris (2007).
Girard, J.-Y., Truth, modality and intersubjectivity. Math. Struct. Comp. Sci. 17 (2007) 1153–1167. CrossRef
Girard, J.-Y., Scedrov, A. and Scott, P.. Bounded linear logic: a modular approach to polynomial-time computability. Theoret. Comput. Sci. 97 (1992) 1–66. CrossRef
S. Mac Lane, Categories for the Working Mathematician. 2nd edition Springer, Berlin (1998).
R.E. Megginson, An Introduction to Banach Space Theory. Springer, Berlin (1998).
P.-A. Melliès, Categorical semantics of linear logic, available at http://www.pps.jussieu.fr/ mellies/.
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.
P. Selinger, Towards a semantics for higher-order quantum computation. Proc. QPL (2004) 127–143.
J. Weidmann, Linear Operators in Hilbert Spaces. Springer, Berlin (1980).