Skip to main content

On geometry of interaction for polarized linear logic


We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multi-points to various categorical models of GoI. Multi-points are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as being fundamental to our construction of concrete polarized GoI models.

Our approach to polarized GoI involves following two independent studies, based on different categorical perspectives of GoI: (i)

Inspired by the work of Abramsky, Haghverdi and Scott, a polarized GoI situation is defined in which multi-points are added to a traced monoidal category equipped with a reflexive object U. Using this framework, categorical versions of Girard's execution formula are defined, as well as the GoI interpretation of MLLP proofs. Running the execution formula is shown to characterize the focusing property (and thus polarities) as well as the dynamics of cut elimination.


The Int construction of Joyal–Street–Verity is another fundamental categorical structure for modelling GoI. Here, we investigate it in a multi-pointed setting. Our presentation yields a compact version of Hamano–Scott's polarized categories, and thus denotational models of MLLP. These arise from a contravariant duality between monoidal categories of positive and negative objects, along with an appropriate bimodule structure (representing ‘non-focused proofs’) between them.

Finally, as a special case of (ii) above, a compact model of MLLP is also presented based on Rel (the category of sets and relations) equipped with multi-points.

Hide All
Abramsky, S., Haghverdi, E. and Scott, P.J. (2002). Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12 (5) 140.
Andreoli, J.-M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3) 297347.
Andreoli, J.-M. (2001). Focussing and proof construction. Annals of Pure and Applied Logic 107 (1) 131163.
Blute, R.F. and Scott, P.J. (2004). Category theory for linear logicians. In: Ehrhard, T., Girard, J.-Y., Ruet, P., Scott, P. (eds.) Linear Logic and Computer Science, London Mathematical Society Lecture Note Series, vol. 316, C.U.P.
Chaudhuri, K. (2008). Focusing strategies in the sequent calculus of synthetic connectives. In: Logic for Programming, Artificial Intelligence and Reasoning (LPAR-15), Doha, Qatar, Lecture Notes in Computer Science, vol. 5330, Springer, 467481.
Chaudhuri, K., Hetzl, S. and Miller, D. A. (2013). Multi-focused proof system isomorphic to expansion proofs. Journal of Logic and Computation 26 (2) 577603.
Cockett, J.R.B., Hasegawa, M. and Seely, R.A.G. (2006). Coherence of the double involution on *-autonomous categories. TAC 17 1729.
Cockett, J.R.B. and Seely, R.A.G. (2007). Polarized category theory, modules and game semantics. TAC 18 4101.
Ehrhard, T. (2012). The Scott model of linear logic is the extensional collapse of its relational model. Theoretical Computer Science 424 (23) 2045.
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1102.
Girard, J.-Y. (1989). Geometry of interaction I: Interpretation of system F. In: Ferro, R. et al. (eds.) Logic Colloquium '88, North-Holland, 221260.
Girard, J.-Y. (1991). A new constructive logic: Classical logic. Mathematical Structures in Computer 1 (3) 255296.
Girard, J.-Y. (1995). Geometry of interaction III: Accommodating the additives. In: Advances in Linear Logic, Lecture Notes in Computer Science, vol. 222, CUP, 329389.
Girard, J.-Y. (1999). On the meaning of logical rules I: Syntax vs. semantics. In Berger, U. and Schwichtenberg, H. (eds.) Computational Logic, NATO ASI Series, vol. 165, Springer, 215272.
Girard, J.-Y. (2001). Locus solum: From the rules of logic to the logic of rules. Mathematical Structures in Computer Science 11 301506.
Girard, J.-Y. (2011). The Blind Spot: Lectures in Logic, European Mathematical Society, 550.
Haghverdi, E. (2000). A Categorical Approach to Linear Logic, Geometry of Proofs and Full Completeness. PhD Thesis, University of Ottawa, Canada.
Haghverdi, E. and Scott, P. (2006). A categorical model for the geometry of interaction. Theoretical Computer Science 350 (2–3) 252274.
Haghverdi, E. and Scott, P.J. (2010). Towards a typed geometry of interaction. Mathematical Structures in Computer Science, 20 473521.
Haghverdi, E. and Scott, P.J. (2011). Geometry of interaction and the dynamics of proof reduction: A tutorial, new structures in physics. In: Coecke, R. (ed.) Springer Lectures Notes in Physics, vol. 813, Oxford.
Hamano, M. and Scott, P. (2007). A categorical semantics for polarized MALL. Annals of Pure & Applied Logic 145 276313.
Hamano, M. and Takemura, R. (2008) An indexed system for multiplicative additive polarized linear logic. In: Proceedings of 17th Annual Conference on Computer Science Logic (CSL'08), Lecture Notes in Computer Science, vol. 5213, 262–277.
Hamano, M. and Takemura, R. (2010). A phase semantics for polarized linear logic and second order conservativity. The Journal of Symbolic Logic 75 (1) 77102.
Hasegawa, M. (2004). The uniformity principle on traced monoidal categories. Publications of the Research Institute for Mathematical Sciences 40 (3) 9911014.
Hyland, J.M.E. and Schalk, A. (2003). Glueing and orthogonality for models of linear logic. Theoretical Computer Science 294 183231.
Joyal, A., Street, R. and Verity, D. (1996). Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society 119 447468.
Joyal, A. (Nov. 2011) Joyal's CatLab, Distributors and Barrels. Available at:
Lambek, J. and Scott, P.J. (1986). Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics, vol. 7, Cambridge University Press.
Laurent, O. (1999). Polarized proof-nets: Proof-nets for LC (Extended Abstract). In: LNCS 1581 (TLCA '99), 213–217.
Laurent, O. (2001). A token machine for full geometry of interaction (Extended Abstract). In: LNCS 2044 (TLCA '01), 283–297.
Laurent, O. (March 2002). Étude de la polarisation en logique. (A study of polarization in logic.) Thèse de Doctorat. Institut de Mathématiques de Luminy – Université Aix-Marseille II.
Liang, C. and Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logic. Theoretical Computer Science 410 (46) 47474768.
Mac Lane, S. (1971). Categories for the Working Mathematician, Springer.
Manes, E.G. and Arbib, M.A. (1986). Algebraic Approaches to Program Semantics, Springer-Velag.
Melliès, P.-A. (2009). Categorical semantics of linear logic. Published in: ‘Interactive models of computation and program behaviour’. Pierre-Louis Curien, Hugo Herbelin, Jean-Louis Krivine, Paul-André Melliès. Panoramas et Synthèses 27, Société Mathématique de France.
Melliès, P.-A. (2013). Dialogue categories and frobenius monoids. In: Coecke, B., Ong, L., and Panangaden, P. (eds.) Computation, Logic, Games, and Quantum Foundations – The Many Facets of Samson Abramsky, Lecture Notes in Computer Science, vol. 7860, Springer, 197224.
Melliès, P.-A. Dialogue categories and chiralities, manuscript. 2006–2012. Available at:
Miller, D. (2004). An overview of linear logic programming. In: Ehrhard, T., Girard, J.-Y., Ruet, P. and Scott, P. (eds.) Linear Logic and Computer Science, LMS Lecture Note Series, vol. 316, C.U.P., 119150.
Miller, D. Tutorial: Sequent calculus: Overview and recent developments. In: 8th Panhellenic Logic Symposium Ioannina, Greece, July 4–8, 2011 (slides available:
Simpson, A. and Plotkin, G. (2000) Complete axioms for categorical fixed-point operators. In: Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science (LICS), IEEE Computer Society, 30–41.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 1
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 107 *
Loading metrics...

* Views captured on Cambridge Core between 28th September 2017 - 21st March 2018. This data will be updated every 24 hours.