Hydra Games and Tree Ordinals

1
Hydra Games and Tree Ordinals

• 2007 4 Jul.
• Ariya Isihara
• Vrije Universiteit, Amsterdam

2
contents

• 0. contents
• 1. tree ordinals
• 2. hydra games
• 3. relation between them

3
contents

• 0. contents
• 1. tree ordinals
• 2. hydra games
• 3. relation between them

4
tree ordinals
0
0
succ(0)
1
succ(succ(0))
2
succ(succ(succ(0)))
3
w
0123 ...
5
catalogue
0
1
2
3
w
6
catalogue II
w
w
w
w2
w
w3
w
w 1
ww
w4
w w
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semantics
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Dedekind TRS
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Dedekind TRS
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productivity and soundness
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contents

• 0. contents
• 1. tree ordinals
• 2. hydra games
• 3. relation between them

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hydra games

• Kirby and ParisAccessible independence results
  for Peano arithmetic (1982)

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a hydra
root
neck(?)
short head
long head
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chopping heads
Herakles chops the right-most head of the hydra.
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chopping heads
if he chops a short head,then the head will be
removed.
CHOP!
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chopping heads
n copies (Hydra chose 3)
CHOP!
immediate neck
if he chops a long head,then the head will be
removed.but, Hydra will choose a natural number
n and sproutsn-many copies of the immediate neck
of the chopped head.
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chopping heads
copied by Hydra...
CHOP!
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chopping heads
Hydra chose 2.
copied by Hydra...
CHOP!
19
chopping heads
copied by Hydra...
Hydra can choose 0,as you see.
CHOP!
20
chopping heads
copied by Hydra...
Hydra chose 1.
CHOP!
the talker asked Hydra to choose 1 very often so
that he can finish this example within his life.
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chopping heads
Hydra chose 1 again (thank you, Hydra.)
CHOP!
copied by Hydra...
the talker asked Hydra to choose 1 very often so
that he can finish this example within his life.
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chopping heads
Hydra is dead. - we will see that every hydra
will be eventually killed, even if Hydra would
choose 10, 10 !, (10 !) !, ... resp. for every
chop.
(sorry, Hydra.)
Hydra chose 3.
CHOP!
CHOP!
CHOP!
CHOP!
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ordinal assignment
0
1
0
w0
0
w0w1
1w
0
1w1w1
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termination
Theorem(Kirby and Paris). Every hydra will
eventually die.
Proof(sketch) Whenever Herakles chops any head of
a hydra, the assigned ordinal decreases.
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termination (proof)
1. short head chopping
something
something
CHOP!
gt
something 1
something
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termination (proof)
2. long head chopping
CHOP!
gt
wsomething . n
wsomething . w
wsomething1
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contents

• 0. contents
• 1. tree ordinals
• 2. hydra games
• 3. relation between them

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. . .
3
0
2
1
29
. . .
3
0
2
1
30
(No Transcript)
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translation from chopping graphs into tree
ordinals
Theorem. The following diagram commutes
Graph
H
TO
-
-
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contents

• 0. contents
• 1. tree ordinals
• 2. hydra games
• 3. relation between them
• 4. FAQ

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FAQ

• Q. Is this interesting?

A. I'm not sure....The relation may look trivial
for someone, but some of my colleagues say that
it is interesting.
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FAQ

• Q. Can we extend this theory to treat much larger
  ordinals?

A. I have another (messy) extension of Dedekind
TRS which can deal with ordinals up to the Veblen
ordinal. I expect there exists a
hydra-counterpart of that TRS.
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contents

• 0. contents
• 1. tree ordinals
• 2. hydra games
• 3. relation between them
• 4. FAQ
• 5. Appendix

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Appendix toys

• Hydra game simulator
• http//www.few.vu.nl/ariya/app/hydra/hydra.html

Other toys... Extended Dedekind TRS
simulator http//www.few.vu.nl/ariya/app/fg/fg.ht
ml Pebbleflow net simulator http//www.few.vu.nl/
ariya/app/pf/pf.html
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the end

• thank you for being awake.

eventually...
http//www.few.vu.nl/ariya/