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 of a mathematical tree. The hydra game can be used to generate a rapidly growing function, ${\displaystyle BH(n)}$, which eventually dominates all recursive functions that are provably total in "${\displaystyle {\textrm {ID}}_{\nu }}$", and the termination of all hydra games is not provably total in ${\displaystyle {\textrm {(}}\Pi _{1}^{1}{\textrm {-CA)+BI}}}$.^[1]

Rules

The game is played on a hydra, a finite, rooted connected tree ${\displaystyle A}$, with the following properties:

• The root of ${\displaystyle A}$ has a special label, usually denoted ${\displaystyle +}$.
• Any other node of ${\displaystyle A}$ has a label ${\displaystyle \nu \leq \omega }$.
• All nodes directly above the root of ${\displaystyle A}$ have a label ${\displaystyle 0}$.

If the player decides to remove the top node ${\displaystyle \sigma }$ of ${\displaystyle A}$, the hydra will then choose an arbitrary ${\displaystyle n\in \mathbb {N} }$, where ${\displaystyle n}$ is a current turn number, and then transform itself into a new hydra ${\displaystyle A(\sigma ,n)}$ as follows. Let ${\displaystyle \tau }$ represent the parent of ${\displaystyle \sigma }$, and let ${\displaystyle A^{-}}$ represent the part of the hydra which remains after ${\displaystyle \sigma }$ has been removed. The definition of ${\displaystyle A(\sigma ,n)}$ depends on the label of ${\displaystyle \sigma }$:

• If the label of ${\displaystyle \sigma }$ is 0 and ${\displaystyle \tau }$ is the root of ${\displaystyle A}$, then ${\displaystyle A(\sigma ,n)}$ = ${\displaystyle A^{-}}$.
• If the label of ${\displaystyle \sigma }$ is 0 but ${\displaystyle \tau }$ is not the root of ${\displaystyle A}$, ${\displaystyle n}$ copies of ${\displaystyle \tau }$ and all its children are made, and edges between them and ${\displaystyle \tau }$'s parent are added. This new tree is ${\displaystyle A(\sigma ,n)}$.
• If the label of ${\displaystyle \sigma }$ is ${\displaystyle u}$ for some ${\displaystyle u\in \mathbb {N} }$, then the first node below ${\displaystyle \sigma }$ is labelled ${\displaystyle v<u}$ as ${\displaystyle \varepsilon }$. ${\displaystyle B}$ is then the subtree obtained by starting with ${\displaystyle A_{\varepsilon }}$ and replacing the label of ${\displaystyle \varepsilon }$ with ${\displaystyle u-1}$ and ${\displaystyle \sigma }$ with 0. ${\displaystyle A(\sigma ,n)}$ is then obtained by taking ${\displaystyle A}$ and replacing ${\displaystyle \sigma }$ with ${\displaystyle B}$. In this case, the value of ${\displaystyle n}$ does not matter.
• If the label of ${\displaystyle \sigma }$ is ${\displaystyle \omega }$, ${\displaystyle A(\sigma ,n)}$ is obtained by replacing the label of ${\displaystyle \sigma }$ with ${\displaystyle n+1}$.

If ${\displaystyle \sigma }$ is the rightmost head of ${\displaystyle A}$, ${\displaystyle A(n)}$ 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. It has been proven that this always terminates, even though the hydra can get taller by massive amounts.

Hydra theorem

Buchholz's paper in 1987^[2] showed that the canonical correspondence between a hydra and an infinitary well-founded tree (or the corresponding term in the notation system ${\displaystyle T}$ associated to Buchholz's function, which does not necessarily belong to the ordinal notation system ${\displaystyle OT\subset T}$), preserves fundamental sequences of choosing the rightmost leaves and the ${\displaystyle (n)}$ operation on an infinitary well-founded tree or the ${\displaystyle [n]}$ operation on the corresponding term in ${\displaystyle T}$.

The hydra theorem for Buchholz hydra, stating that there are no losing strategies for any hydra, is unprovable in ${\displaystyle {\mathsf {\Pi _{1}^{1}-CA+BI}}}$.^[3]

BH(n)

Suppose a tree consists of just one branch with ${\displaystyle x}$ nodes, labelled ${\displaystyle +,0,\omega ,...,\omega }$. Call such a tree ${\displaystyle R^{n}}$. It cannot be proven in ${\displaystyle {\mathsf {\Pi _{1}^{1}-CA+BI}}}$ that for all ${\displaystyle x}$, there exists ${\displaystyle k}$ such that ${\displaystyle R_{x}(1)(2)(3)...(k)}$ is a winning strategy. (The latter expression means taking the tree ${\displaystyle R_{x}}$, then transforming it with ${\displaystyle n=1}$, then ${\displaystyle n=2}$, then ${\displaystyle n=3}$, etc. up to ${\displaystyle n=k}$.)

Define ${\displaystyle BH(x)}$ as the smallest ${\displaystyle k}$ such that ${\displaystyle R_{x}(1)(2)(3)...(k)}$ as defined above is a winning strategy. By the hydra theorem, this function is well-defined, but its totality cannot be proven in ${\displaystyle {\mathsf {\Pi _{1}^{1}-CA+BI}}}$. Hydras grow extremely fast, because the amount of turns required to kill ${\displaystyle R_{x}(1)(2)}$ is larger than Graham's number or even the amount of turns to kill a Kirby-Paris hydra; and ${\displaystyle R_{x}(1)(2)(3)(4)(5)(6)}$ has an entire Kirby-Paris hydra as its branch. To be precise, its rate of growth is believed to be comparable to ${\displaystyle f_{\psi _{0}(\varepsilon _{\Omega _{\omega }+1})}(x)}$ with respect to the unspecified system of fundamental sequences without a proof. Here, ${\displaystyle \psi _{0}}$ denotes Buchholz's function, and ${\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}$ is the Takeuti-Feferman-Buchholz ordinal which measures the strength of ${\displaystyle {\mathsf {\Pi _{1}^{1}-CA+BI}}}$.

The first two values of the BH function are virtually degenerate: ${\displaystyle BH(1)=0}$ and ${\displaystyle BH(2)=1}$. Similarly to the weak tree function, ${\displaystyle BH(3)}$ is very large, but less so.

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

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 ${\displaystyle \psi _{\alpha }}$, where ${\displaystyle \alpha }$ is the label of the node, and ${\displaystyle \psi }$ is Buchholz's function.

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

Conversion

Hydra	Ordinal
${\displaystyle +}$	${\displaystyle 0}$
${\displaystyle +(0)}$	${\displaystyle \psi _{0}(0)=1}$
${\displaystyle +(0)(0)}$	${\displaystyle 2}$
${\displaystyle +(0(0))}$	${\displaystyle \psi _{0}(1)=\omega }$
${\displaystyle +(0(0))(0)}$	${\displaystyle \omega +1}$
${\displaystyle +(0(0))(0(0))}$	${\displaystyle \omega \cdot 2}$
${\displaystyle +(0(0)(0))}$	${\displaystyle \omega ^{2}}$
${\displaystyle +(0(0(0)))}$	${\displaystyle \omega ^{\omega }}$
${\displaystyle +(0(1))}$	${\displaystyle \varepsilon _{0}}$
${\displaystyle +(0(1)(1))}$	${\displaystyle \varepsilon _{1}}$
${\displaystyle +(0(1(0)))}$	${\displaystyle \varepsilon _{\omega }}$
${\displaystyle +(0(1(1)))}$	${\displaystyle \zeta _{0}}$
${\displaystyle +(0(1(1(1))))}$	${\displaystyle \Gamma _{0}}$
${\displaystyle +(0(1(1(1(0)))))}$	SVO
${\displaystyle +(0(1(1(1(1)))))}$	LVO
${\displaystyle +(0(2))}$	BHO
${\displaystyle +(0(\omega ))}$	BO

References

• Buchholz, Wilfried (1987). "An independence result for (Π11 - CA) + BI" (PDF). Annals of Pure and Applied Logic. North-Holland. 33: 131–155. doi:10.1016/0168-0072(87)90078-9. Retrieved 2021-09-03.
• Hamano, Masahiro; Okada, Mitsuhiro (1995). "A relationship among Gentzen's Proof-Reduction, Kirby-Paris' Hydra Game and Buchholz's Hydra Game (Preliminary Report)" (PDF). Research Institute for Mathematical Sciences. 912: 64–81. Retrieved 2021-09-03.
• Hamano, Masahiro; Okada, Mitsuhiro (1997). "A Relationship Among Gentzen's Proof-Reduction, Kirby-Paris' Hydra Game and Buchholz's Hydra Game". Mathematical Logic Quarterly. 43 (1): 103–120. doi:10.1002/malq.19970430113. ISSN 0942-5616. S2CID 26893598.
• Hamano, Masahiro; Okada, Mitsuhiro (1998-03-01). "A direct independence proof of Buchholz's Hydra Game on finite labelled trees". Archive for Mathematical Logic. 37 (2): 67–89. doi:10.1007/s001530050084. ISSN 0933-5846. S2CID 40113368.
• Gordeev, Lev (December 2001). "(Review) Masahiro Hamano and Mitsuhiro Okada. A direct independence proof of Buchholz's Hydra game on finite labelled trees. Archive for mathematical logic, vol. 37 no. 2 (1998), pp. 67–89". Bulletin of Symbolic Logic. 7 (4): 534–535. doi:10.2307/2687805. ISSN 1079-8986. JSTOR 2687805.
• Kirby, Laurie; Paris, Jeff (1982), "Accessible independence results for Peano Arithmetic" (PDF), Bull. London Math. Soc., 14 (4): 285–293, doi:10.1112/blms/14.4.285, retrieved 2021-09-03
• Ketonen, Jussi; Solovay, Robert (1981). "Rapidly growing Ramsey functions". Annals of Mathematics. 113 (2): 267–314. doi:10.2307/2006985. ISSN 0003-486X. JSTOR 2006985. Retrieved 2021-09-03.
• Takeuti, Gaisi (2013). Proof theory (2nd edition (reprint) ed.). Newburyport: Dover Publications. ISBN 978-0-486-32067-0. OCLC 1162507470.
• "Hydras". Agnijo's mathematical treasure chest. Retrieved 2021-09-04.

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