Buchholz hydra


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 in.

Rules

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.

Hydra theorem

Buchholz's paper in 1987 showed that the canonical correspondence between a hydra and an infinitary well-founded tree, 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.
The hydra theorem for Buchholz hydra, stating that there are no losing strategies for any hydra, is unprovable in.

BH(n)

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.
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.

Analysis

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: