Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T17:10:09.688Z Has data issue: false hasContentIssue false
  • English
  • Français

A linear/producer/consumer model of classical linear logic

Published online by Cambridge University Press:  27 October 2016

JENNIFER PAYKIN  and
STEVE ZDANCEWIC
JENNIFER PAYKIN
Affiliation:
Department of Computer and Information Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Emails: [email protected], [email protected]
STEVE ZDANCEWIC
Affiliation:
Department of Computer and Information Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Emails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper defines a new proof- and category-theoretic framework for classical linear logic that separates reasoning into one linear regime and two persistent regimes corresponding to ! and ?. The resulting linear/producer/consumer (LPC) logic puts the three classes of propositions on the same semantic footing, following Benton's linear/non-linear formulation of intuitionistic linear logic. Semantically, LPC corresponds to a system of three categories connected by adjunctions reflecting the LPC structure. The paper's meta-theoretic results include admissibility theorems for the cut and duality rules, and a translation of the LPC logic into category theory. The work also presents several concrete instances of the LPC model.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Special Issue 5: Linearity 2014 , May 2018 , pp. 710 - 735
Copyright
Copyright © Cambridge University Press 2016 

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

Barber, A. (1996). Dual Intuitionistic Linear Logic. ECS-LFCS-96-347, University of Edinburgh, Department of Computer Science, Laboratory for Foundations of Computer Science.Google Scholar
Barr, M. (1991). *-autonomous categories and linear logic. Mathematical Structures in Computer Science 1 (02) 159.CrossRefGoogle Scholar
Benton, N., Bierman, G., de Paiva, V. and Hyland, M. (1993). A term calculus for intuitionistic linear logic. In: Bezem, M. and Groote, J. (eds.) Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, vol. 664, Springer, Berlin, Heidelberg, 75–90.Google Scholar
Benton, P. (1995). A mixed linear and non-linear logic: Proofs, terms and models. In: Pacholski, L. and Tiuryn, J. (eds.) Computer Science Logic, Lecture Notes in Computer Science, vol. 933, Springer, Berlin, Heidelberg, 121–135.Google Scholar
Bierman, G. (1995). What is a categorical model of intuitionistic linear logic? In: Dezani-Ciancaglini, M. and Plotkin, G. (eds.) Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, vol. 902, Springer, Berlin, Heidelberg, 78–93.Google Scholar
Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal 3 (3) 443–454.Google Scholar
Cockett, J. and Seely, R. (1997). Weakly distributive categories. Journal of Pure and Applied Algebra 114 (2) 133–173.Google Scholar
Cockett, J. and Seely, R. (1999). Linearly distributive functors. Journal of Pure and Applied Algebra 143 (13) 155–203.Google Scholar
Ehrhard, T. (2005). Finiteness spaces. Mathematical Structures in Computer Science 15 (4) 615–646.Google Scholar
Gaboardi, M. (2007). Linearity: An Analytic Tool in the study of Complexity and Semantics of Programming Languages. Phd Thesis, Università di Torino, Institut National Polytechnique de Lorraine.Google Scholar
Girard, J.-Y. (1986). The system F of variable types, fifteen years later. Theoretical Computer Science 45 (2) 159–192.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 (1) 1–101.Google Scholar
Girard, J.-Y. (1993). On the unity of logic. Annals of Pure and Applied Logic 59 (3) 201–217.Google Scholar
Hicks, M., Morrisett, G., Grossman, D. and Jim, T. (2004). Experience with safe manual memory-management in cyclone. In: Proceedings of the 4th International Symposium on Memory Management, ISMM '04, New York, NY, USA: ACM, 73–84.Google Scholar
Lafont, Y. (1988). The linear abstract machine. Theoretical Computer Science 59 157–180. Corrections in vol. 62, pp. 327–328.CrossRefGoogle Scholar
Liang, C. and Miller, D. (2007). Focusing and polarization in intuitionistic logic. In: Duparc, J. and Henzinger, T.A. (eds.) Computer Science Logic, Springer, Berlin, Heidelberg, 451–465.Google Scholar
Melliés, P.A. (2003). Categorical models of linear logic revisited. Laboratoire PPS a la Université Paris Denis Diderot, 22.Google Scholar
Melliès, P.-A. (2009). Categorical semantics of linear logic. In: Interactive Models of Computation and Program Behavior, Panoramas et synthèses, Société Mathématique de France.Google Scholar
Morrisett, G., Ahmed, A. and Fluet, M. (2005). L3: A linear language with locations. In: Urzyczyn, P. (ed.) Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, vol. 3461, Springer, Berlin, Heidelberg, 293–307.CrossRefGoogle Scholar
Pfenning, F. and Griffith, D. (2015). Polarized substructural session types. In: Pitts, A. (ed.) Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, vol. 9034, Springer, Berlin, Heidelberg, 3–22.Google Scholar
Pratt, V. (1994). Chu spaces: Complementarity and uncertainty in rational mechanics. Course notes, TEMPUS summer school, Budapest.Google Scholar
Reed, J. (2009). A judgmental deconstruction of modal logic. Available at http://www.cs.cmu.edu/~jcreed/papers/jdml.pdf.Google Scholar
Schalk, A. (2004). Whats is a categorical model of linear logic. Technical report, University of Manchester.Google Scholar
Seely, R.A.G. (1989). Linear Logic, *-Autonomous Categories and Cofree Coalgebras, American Mathematical Society (AMS).Google Scholar
Stanley, R.P. (2011). Enumerative combinatorics, vol. 49, Cambridge University Press.Google Scholar
Valiron, B. and Zdancewic, S. (2014). Finite vector spaces as model of simply-typed lambda-calculi. In: Ciobanu, G. and Mry, D. (eds.) Theoretical Aspects of Computing ICTAC 2014, Lecture Notes in Computer Science, vol. 8687, Springer International Publishing, 442–459.Google Scholar
Wadler, P. (1992). There's no substitute for linear logic. In: Proceedings of the 8th International Workshop on the Mathematical Foundations of Programming Semantics.Google Scholar
Wadler, P. (September 2012). Propositions as sessions. SIGPLAN Not. 47 (9) 273–286.Google Scholar
Zdancewic, S. and Myers, A.C. (2002). Secure information flow via linear continuations. Higher-Order and Symbolic Computation 15 (2–3) 209–234.Google Scholar