Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-13T06:41:28.875Z Has data issue: false hasContentIssue false
  • English
  • Français

On geometry of interaction for polarized linear logic

Published online by Cambridge University Press:  28 September 2017

MASAHIRO HAMANO  and
PHILIP SCOTT
MASAHIRO HAMANO
Affiliation:
PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan Email: hamano@jaist.ac.jp
PHILIP SCOTT
Affiliation:
Department of Mathematics & Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, K1N 6N5, Canada Email: phil@site.uottawa.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

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:

  1. (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.

  2. (ii) 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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 10 , November 2018 , pp. 1639 - 1694
Copyright
Copyright © Cambridge University Press 2017 

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., Haghverdi, E. and Scott, P.J. (2002). Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12 (5) 1–40.Google Scholar
Andreoli, J.-M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3) 297–347.Google Scholar
Andreoli, J.-M. (2001). Focussing and proof construction. Annals of Pure and Applied Logic 107 (1) 131–163.Google Scholar
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.Google Scholar
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, 467–481.Google Scholar
Chaudhuri, K., Hetzl, S. and Miller, D. A. (2013). Multi-focused proof system isomorphic to expansion proofs. Journal of Logic and Computation 26 (2) 577–603.Google Scholar
Cockett, J.R.B., Hasegawa, M. and Seely, R.A.G. (2006). Coherence of the double involution on *-autonomous categories. TAC 17 17–29.Google Scholar
Cockett, J.R.B. and Seely, R.A.G. (2007). Polarized category theory, modules and game semantics. TAC 18 4–101.Google Scholar
Ehrhard, T. (2012). The Scott model of linear logic is the extensional collapse of its relational model. Theoretical Computer Science 424 (23) 20–45.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1–102.Google Scholar
Girard, J.-Y. (1989). Geometry of interaction I: Interpretation of system F. In: Ferro, R. et al. (eds.) Logic Colloquium '88, North-Holland, 221–260.Google Scholar
Girard, J.-Y. (1991). A new constructive logic: Classical logic. Mathematical Structures in Computer 1 (3) 255–296.Google Scholar
Girard, J.-Y. (1995). Geometry of interaction III: Accommodating the additives. In: Advances in Linear Logic, Lecture Notes in Computer Science, vol. 222, CUP, 329–389.Google Scholar
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, 215–272.Google Scholar
Girard, J.-Y. (2001). Locus solum: From the rules of logic to the logic of rules. Mathematical Structures in Computer Science 11 301–506.Google Scholar
Girard, J.-Y. (2011). The Blind Spot: Lectures in Logic, European Mathematical Society, 550.Google Scholar
Haghverdi, E. (2000). A Categorical Approach to Linear Logic, Geometry of Proofs and Full Completeness. PhD Thesis, University of Ottawa, Canada.Google Scholar
Haghverdi, E. and Scott, P. (2006). A categorical model for the geometry of interaction. Theoretical Computer Science 350 (2–3) 252–274.Google Scholar
Haghverdi, E. and Scott, P.J. (2010). Towards a typed geometry of interaction. Mathematical Structures in Computer Science, 20 473–521.Google Scholar
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.Google Scholar
Hamano, M. and Scott, P. (2007). A categorical semantics for polarized MALL. Annals of Pure & Applied Logic 145 276–313.Google Scholar
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.Google Scholar
Hamano, M. and Takemura, R. (2010). A phase semantics for polarized linear logic and second order conservativity. The Journal of Symbolic Logic 75 (1) 77–102.Google Scholar
Hasegawa, M. (2004). The uniformity principle on traced monoidal categories. Publications of the Research Institute for Mathematical Sciences 40 (3) 991–1014.Google Scholar
Hyland, J.M.E. and Schalk, A. (2003). Glueing and orthogonality for models of linear logic. Theoretical Computer Science 294 183–231.Google Scholar
Joyal, A., Street, R. and Verity, D. (1996). Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society 119 447–468.Google Scholar
Joyal, A. (Nov. 2011) Joyal's CatLab, Distributors and Barrels. Available at: http://ncatlab.org/joyalscatlab/show/Distributors+and+barrels#distributors_6Google Scholar
Lambek, J. and Scott, P.J. (1986). Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics, vol. 7, Cambridge University Press.Google Scholar
Laurent, O. (1999). Polarized proof-nets: Proof-nets for LC (Extended Abstract). In: LNCS 1581 (TLCA '99), 213–217.Google Scholar
Laurent, O. (2001). A token machine for full geometry of interaction (Extended Abstract). In: LNCS 2044 (TLCA '01), 283–297.Google Scholar
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.Google Scholar
Liang, C. and Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logic. Theoretical Computer Science 410 (46) 4747–4768.Google Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician, Springer.Google Scholar
Manes, E.G. and Arbib, M.A. (1986). Algebraic Approaches to Program Semantics, Springer-Velag.Google Scholar
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.Google Scholar
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, 197–224.Google Scholar
Melliès, P.-A. Dialogue categories and chiralities, manuscript. 2006–2012. Available at: http://www.pps.univ-paris-diderot.fr/~mellies/tensorial-logic.htmlGoogle Scholar
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., 119–150.Google Scholar
Miller, D. Tutorial: Sequent calculus: Overview and recent developments. In: 8th Panhellenic Logic Symposium Ioannina, Greece, July 4–8, 2011 (slides available: http://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/pls8.html)Google Scholar
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.Google Scholar