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by Rudolf Fleischer Fudan University

by Rudolf Fleischer Fudan University. Not infinitely many problems… …but problems with infinity About trees… …but no data structure. Hercules vs. Hydra. What is a Hydra?. Hydra is a Tree. Hydra is a Tree. Cut a head:

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by Rudolf Fleischer Fudan University

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  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?

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