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]
The game is played on a hydra, a finite, rooted connected tree , with the following properties:
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 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]
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]
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.
It is possible to make a one-to-one correspondence between some hydras and ordinals. To convert a tree or subtree to an ordinal:
The resulting ordinal expression is only useful if it is in normal form. Some examples are:
Hydra | Ordinal |
---|---|
SVO | |
LVO | |
BHO | |
BO |
This article incorporates text available under the CC BY-SA 3.0 license.
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]
The game is played on a hydra, a finite, rooted connected tree , with the following properties:
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 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]
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]
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.
It is possible to make a one-to-one correspondence between some hydras and ordinals. To convert a tree or subtree to an ordinal:
The resulting ordinal expression is only useful if it is in normal form. Some examples are:
Hydra | Ordinal |
---|---|
SVO | |
LVO | |
BHO | |
BO |
This article incorporates text available under the CC BY-SA 3.0 license.