1. Geometry of Interaction Models proofs of linear logic as bidirectional computation executed by the flow of data in the proof net

3. This is the paper!

4. Program of Geometry of Interaction

5. In general : the invariant is given by the EXECUTION FORMULA Captures normalization process GoI [Girard 88]

6. Axiom links

9. GoI as computation The interpretation is invariant under normalization If the term is base: calculate the a term in normal form = calculate the invariant

10. The simplified model: permutations Please find details in Girard's notes! Next 3 slides are a summary of: ● Long trip criterion ● Its version as orthogonality condition (polar permutations)

12. Long Trip Criterion: a proof structure is a proof-net if each switching induces a long trip

14. Otherwise: Dynamics (with permutations)

15. The operators algebra model

16. In general : the invariant is given by the EXECUTION FORMULA Captures normalization process GoI [Girard 88]

17. Just think of these graphs described by their adjacency matrix: The interpretation of a proof is an operator, on a finite space. We can just use matrices. In fact (in first approximation) just think of permutation matrices

19. Execution formula Nilpotency of Invariant under normalization Strong normalization!

20. Execution is best understood via examples

24. Interpretation of MLL

25. Identity and cut

26. Interpretation of multiplicatives it is all in our mind... (blocks)

27. Interpretation of multiplicatives

28. problem

31. q q

32. GoI as computation The interpretation is invariant under normalization If the term is base: calculate the term = calculate the invariant

35. ?