Cooling and heating (combinatorial game theory)

In combinatorial game theory, cooling, heating, and overheating are operations on hot games to make them more amenable to the traditional methods of the theory, which was originally devised for cold games in which the winner is the last player to have a legal move. Overheating was generalised by Berlekamp for the analysis of blockbusting. Chilling (or unheating) and warming are variants used in the analysis of the endgame of go.

Cooling and chilling may be thought of as a tax on the player who moves, who has to pay for the privilege of doing so, while heating, warming and overheating are operations that more or less reverse cooling and chilling.

The cooled game $G_{t}$ ("$G$ cooled by $t$") for a game $G$ and successive integers $t$ is defined by

$G_{t}=\{G_{t}^{L}-t\mid G_{t}^{R}+t\}$ up to and including the first $t$ for which $G_{t}$ is infinitesimally close to some number $x$, thereafter

$G_{t}=x$ for all subsequent $t$.

Chilling is a variant of cooling by $1$ used for go endgames, defined by

$f(G)={\begin{cases}n&{\text{if }}G{\text{ is of the form }}n{\text{ or }}n*,\\\{f(G^{L})-1\mid F(G^{R})+1\}&{\text{otherwise}}.\end{cases}}$

This is proven equivalent to cooling by $1$ when $G$ is an "elementary go position in canonical form".

Warming is defined and proven to be the inverse of chilling.