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Geometry of Interaction and the Dynamics of Proof Reduction: A Tutorial

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New Structures for Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 813))

Abstract

Girard’s Geometry of Interaction (GoI) is a program that aims at giving mathematical models of algorithms independently of any extant languages. In the context of proof theory, where one views algorithms as proofs and computation as cut-elimination, this program translates to providing a mathematical modelling of the dynamics of cut-elimination. The kind of logics we deal with, such as Girard’s linear logic, are resource sensitive and have their proof-theory intimately related to various monoidal (tensor) categories. The GoI interpretation of dynamics aims to develop an algebraic/geometric theory of invariants for information flow in networks of proofs, via feedback.

We shall give an introduction to the categorical approach to GoI, including background material on proof theory, categorical logic, traced and partially traced monoidal *-categories, and orthogonalities.

P. Scott Research partially supported by a Discovery Grant from NSERC, Canada.

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Notes

  1. 1.

    Here \(f_{11}: X\rightarrow Y, f_{12}: U\rightarrow X, f_{21}: X\rightarrow U, f_{22}: U\rightarrow U\).

  2. 2.

    Recent work of Hasegawa and Katsumata [HK09] has shown that the notion of 2-cell in [JSV96] must be changed to invertible monoidal natural transformation.

  3. 3.

    The GoI interpretation of proofs involves manipulation and rearrangment of the interface wires of a proof box. GoI situations, with their reflexive object U and monoidal retracts, give the essential mechanism for modelling the “permuting, splitting, merging, and manipulating” of wires underlying the GoI interpretation of proofs. This is illustrated here for the Cut and Contraction Rules.

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Authors and Affiliations

  1. School of Informatics, Indiana University, Bloomington, IN, USA

    P. Scott

Authors
  1. E. Haghverdi
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  2. P. Scott
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  1. Parks Road, Oxford, OX1 3QD, United Kingdom

    Bob Coecke

Appendices

Appendix 1: Graphical Representation of The Trace Axioms

figure aen

Naturality in X

figure afn

Naturality in Y

figure agn

Dinaturality in U

figure ahn

Vanishing I

figure ain

Vanishing II

figure ajn

Superposing

figure akn

Yanking

Appendix 2: Comparing GoI Notation

Girard

This Paper

\(1\otimes a\)

\(uT(a)v \)

\(p,p^*\)

\(j_1,k_1\)

\(q,q^*\)

\(j_2,k_2\)

\((1\otimes r),(1\otimes r^*)\)

\(uc_1v, uc'_1v\)

\((1\otimes s),(1\otimes s^*)\)

\(uc_2v,uc'_2v\)

\(t,t^*\)

\(ue_{U}(Tv)v, u(Tu)e'_{U}v\)

\(d,d^*\)

\(u d_U, d'_U v\)

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Haghverdi, E., Scott, P. (2010). Geometry of Interaction and the Dynamics of Proof Reduction: A Tutorial. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_5

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