1. Geometry of Interaction

2. This is the paper!

3. Program of Geometry of Interaction

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

6. 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

10. Linear Logic

11. 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

13. PN2 is LL2 presented as proof nets

15. PN2 is LL2 presented as proof nets

16. Integers in LL2

18. n+1 axiom-links represents the Int n

20. successor

21. How to use successor

22. Primitive recursion

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

25. We first focus on
MLL2

27. MLL2
MLL + 2nd order QUANTIFIERS

28. Revision of MLL proof nets

29. Proof structures

32. Proof Nets

34. Correctness guarantees:
Graph is image of a proof
Normalization terminates

35. Normalization (local graph reductions)

36. Properties of normalization

37. MLL2 proof nets: MLL + quantifiers

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

41. Let us try out!