Paper Info
Title
On geometry of interaction for polarized linear logic
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: (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. (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.
Year
DOI
Venue
2015
10.1017/S0960129517000196
MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
Field
DocType
Volume
Discrete mathematics,Algebra,Multiplicative function,Geometry of interaction,Categorical variable,Proof theory,Mathematical proof,Duality (optimization),Traced monoidal category,Linear logic,Mathematics
Journal
28
Issue
ISSN
Citations 
10
0960-1295
0
PageRank 
References 
Authors
0.34
19
2
Name
Order
Citations
PageRank
Masahiro Hamano1397.66
Philip J. Scott212718.30