Geometry of Interaction

This is the paper!

Program of Geometry of Interaction

Goal - the model of GoI will be: Thm: it is a model, i.e. is invariant under normalization

System F translates into LL2 (2nd order Linear Logic) F is an extension of lambda calc. with 2nd order LJ <---> lamba calculus (Curry Howard) LJ translates in LL F translates into LL2

Linear Logic

Intuitionistic logic into LL, and system F into LL2 The translation of Intuitionistic logic into LL, and system F into LL2 is given in Girard's original paper

PN2 is LL2 presented as proof nets

PN2 is LL2 presented as proof nets

Integers in LL2

n+1 axiom-links represents the Int n

successor

How to use successor

Primitive recursion

Booleans (coding with additives) but we will rather use the coding in MLL2 (because additives do not work so well in GoI)

We first focus on MLL2

MLL2 MLL + 2nd order QUANTIFIERS

Revision of MLL proof nets

Proof structures

Proof Nets

Correctness guarantees: Graph is image of a proof Normalization terminates

Normalization (local graph reductions)

Properties of normalization

MLL2 proof nets: MLL + quantifiers

X is substituted by B everywhere inside the box Reduction X is substituted by B everywhere inside the box

Let us try out!