by Rudolf Fleischer Fudan University

1. by Rudolf Fleischer Fudan University

2. Not infinitely many problems… …but problems with infinity About trees… …but no data structure

3. Hercules vs. Hydra

4. What is a Hydra?

5. Hydra is a Tree

6. Hydra is a Tree Cut a head: Grow two copies of the subtree one level down the tree (2-Hydra) In step i, grow i copies of the subtree one level down the tree (i-Hydra)

7. Every 2-Hydra Must Die Ordinal tree

8. Every 2-Hydra Must Die Ordinal tree

9. Every 2-Hydra Must Die Ordinal tree

10. Every 2-Hydra Must Die Ordinal tree

11. Every 2-Hydra Must Die Ordinal tree

12. Every 2-Hydra Must Die 0 0 0 0 0 0 node potentials

13. Every 2-Hydra Must Die 0 0 0 0 0 40+40 0 node potentials

14. Every 2-Hydra Must Die 0 0 0 0 0 40+40 0 2·40 40 42·40 node potentials

15. Every 2-Hydra Must Die 0 0 0 0 0 2 0 2 1 42 node potentials

16. Every 2-Hydra Must Die 0 0 0 0 0 2 0 2 1 42 442+42+4+1 442+42+4+1 4 root potential

17. Every 2-Hydra Must Die 0 0 X 0 0 0 2 0 2 1 42 442+42+4+1 442+42+4+1 4 root potential

18. Every 2-Hydra Must Die 0 0 X 0 0 0 1 0 2 1 41 441+42+4+1 441+42+4+1 4 root potential must drop

19. Every 2-Hydra Must Die 0 0 0 0 X 0 0 0 1 1 1 0 2 1 3·41 43·41+42+4+1 43·41+42+4+1 4 root potential must drop

20. Every i-Hydra Must Die 0 0 0 0 0 2 0 2 1 42 442+42+4+1 442+42+4+1 4 root potential

21. Every i-Hydra Must Die 0 0 0 0 0 Ordinals: 0 = {} 1 = {0} 2 = {0,1} 3 = {0,1,2} … ω = {0,1,2,…} ω+1 = {0,1,2,…, ω} … 2 0 2 1 ω2 ωω2+ω2+ω+1 ωω2+ω2+ω+1 ω root potential

22. Every i-Hydra Must Die 0 0 X 0 0 0 1 0 2 1 ω1 ωω1+ω2+ω+1 ωω1+ω2+ω+1 ω root potential must drop (transfinite induction)

23. Mathematics Kirby, Paris ’82: • “i-Hydra must die” cannot be proved in first-order Peano Arithmetic (proof by induction) • “i-Hydra must die” follows from well-orderedness of ordinal numbers (transfinite induction) Luccio, Pagli ’00: • Is there a simple combinatorial proof?

24. Properties (P1) Can only copy subtree that contained the cut head x (P2) Only a subtree that contained x can grow (P3) Subtree can only grow by a copy of a subtree of itself

25. Generalized Hydra

26. Generalized Hydra Arbitrary number of copies at all levels below the head

27. (P1) Cannot Copy Arbitrary Subtrees

28. (P2) Cannot Grow Arbitrary Subtrees

29. (P3) Subtrees Cannot Move Upwards

30. Generalized Hydra Must Die Tree S: • Leaves = subtrees of root of Hydra • Add one level below the leaf where we cut • Split node if subtree gets copied S

31. Generalized Hydra Must Die Tree S: • Leaves = subtrees of root of Hydra • Add one level below the leaf where we cut • Split node if subtree gets copied S

32. Generalized Hydra Must Die • Take smallest height immortal hydra • Immortal Hydra  Infinite path in S(Koenig’s Lemma) • (P1)+(P3): all cuts in a copied subtree could be done in the original subtree • (P2): All other cuts don’t affect this subtree

33. Where Are We Cheating? • Induction on height of Hydra • Koenig’s Lemma not in Peano Arithmetic! (it’s equivalent to weak Axiom of Choice)

34. Buchholz Hydra Can also grow in height

35. Buchholz Hydra Must Die Simple proof?