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Buchholz hydra

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In mathematics, especially mathematical logic, graph theory and number theory, the Buchholz hydra game is a type of hydra game, which is a single-player game based on the idea of chopping pieces off a mathematical tree. The hydra game can be used to generate a rapidly growing function, , which eventually dominates all recursive functions that are provably total in "", and the termination of all hydra games is not provably total in .[1]

Rules

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The game is played on a hydra, a finite, rooted connected tree , with the following properties:

  • The root of has a special label, usually denoted .
  • Any other node of has a label .
  • All nodes directly above the root of have a label .

If the player decides to remove the top node of , the hydra will then choose an arbitrary , where is a current turn number, and then transform itself into a new hydra as follows. Let represent the parent of , and let represent the part of the hydra which remains after has been removed. The definition of depends on the label of :

  • If the label of is 0 and is the root of , then = .
  • If the label of is 0 but is not the root of , copies of and all its children are made, and edges between them and 's parent are added. This new tree is .
  • If the label of is for some , let be the first node below that has a label . Define as the subtree obtained by starting with and replacing the label of with and with 0. is then obtained by taking and replacing with . In this case, the value of does not matter.
  • If the label of is , is obtained by replacing the label of with .

If is the rightmost head of , is written. A series of moves is called a strategy. A strategy is called a winning strategy if, after a finite amount of moves, the hydra reduces to its root. This always terminates, even though the hydra can get taller by massive amounts.[1]

Hydra theorem

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Buchholz's paper in 1987 showed that the canonical correspondence between a hydra and an infinitary well-founded tree (or the corresponding term in the notation system  associated to Buchholz's function, which does not necessarily belong to the ordinal notation system ), preserves fundamental sequences of choosing the rightmost leaves and the  operation on an infinitary well-founded tree or the  operation on the corresponding term in .[1]

The hydra theorem for Buchholz hydra, stating that there are no losing strategies for any hydra, is unprovable in .[2]

BH(n)

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Suppose a tree consists of just one branch with  nodes, labelled . Call such a tree . It cannot be proven in  that for all , there exists  such that  is a winning strategy. (The latter expression means taking the tree , then transforming it with , then , then , etc. up to .)[2]

Define  as the smallest  such that  as defined above is a winning strategy. By the hydra theorem, this function is well-defined, but its totality cannot be proven in . Hydras grow extremely fast, because the amount of turns required to kill is larger than Graham's number or even the amount of turns to kill a Kirby-Paris hydra; and has an entire Kirby-Paris hydra as its branch. To be precise, its rate of growth is believed to be comparable to  with respect to the unspecified system of fundamental sequences without a proof. Here,  denotes Buchholz's function, and is the Takeuti-Feferman-Buchholz ordinal which measures the strength of .

The first two values of the BH function are virtually degenerate:  and . Similarly to the weak tree function,  is very large, but less so.[citation needed]

The Buchholz hydra eventually surpasses TREE(n) and SCG(n),[citation needed] yet it is likely weaker than loader as well as numbers from finite promise games.

Analysis

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It is possible to make a one-to-one correspondence between some hydras and ordinals. To convert a tree or subtree to an ordinal:

  • Inductively convert all the immediate children of the node to ordinals.
  • Add up those child ordinals. If there were no children, this will be 0.
  • If the label of the node is not +, apply , where is the label of the node, and is Buchholz's function.

The resulting ordinal expression is only useful if it is in normal form. Some examples are:

Conversion
Hydra Ordinal
SVO
LVO
BHO
BO

References

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  1. ^ a b c Buchholz, Wilfried (1987), "An independence result for ", Annals of Pure and Applied Logic, 33 (2): 131–155, doi:10.1016/0168-0072(87)90078-9, MR 0874022
  2. ^ a b Hamano, Masahiro; Okada, Mitsuhiro (1997), "A direct independence proof of Buchholz's Hydra game on finite labeled trees", Archive for Mathematical Logic, 37 (2): 67–89, doi:10.1007/s001530050084, MR 1620664

Further reading

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  • "Hydras", Agnijo's mathematical treasure chest, retrieved 2021-09-04

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