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Towards a typed Geometry of Interaction

Published online by Cambridge University Press:  21 April 2010

ESFANDIAR HAGHVERDI
Affiliation:
School of Informatics and Computing, Indiana University, Bloomington, IN 47408, U.S.A. Email: ehaghver@indiana.edu
PHILIP SCOTT
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, K1N 6N5, Canada Email: phil@mathstat.uottawa.ca

Abstract

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is related to work of Hyland and Schalk). We develop Girard's original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of cut elimination (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.

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Paper
Copyright
Copyright © Cambridge University Press 2010

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