Cooling and heating (combinatorial game theory)

The cooled game ${\displaystyle G_{t}}$ ("${\displaystyle G}$ cooled by ${\displaystyle t}$") for a game ${\displaystyle G}$ and a (surreal) number ${\displaystyle t}$ is defined by[5]

${\displaystyle G_{t}={\begin{cases}\{G_{t}^{L}-t\mid G_{t}^{R}+t\}&{\text{ for all numbers }}t\leq {\text{ any number }}\tau {\text{ for which }}G_{\tau }{\text{ is infinitesimally close to some number }}m{\text{ , }}\\m&{\text{ for }}t>\tau \end{cases}}}$.

The amount ${\displaystyle t}$ by which ${\displaystyle G}$ is cooled is known as the temperature; the minimum ${\displaystyle \tau }$ for which ${\displaystyle G_{\tau }}$ is infinitesimally close to ${\displaystyle m}$ is known as the temperature ${\displaystyle t(G)}$ of ${\displaystyle G}$; ${\displaystyle G}$ is said to freeze to ${\displaystyle G_{\tau }}$; ${\displaystyle m}$ is the mean value (or simply mean) of ${\displaystyle G}$.

Heating is the inverse of cooling and is defined as the "integral"[6]

${\displaystyle \int ^{t}G={\begin{cases}G&{\text{ if }}G{\text{ is a number, }}\\\{\int ^{t}(G^{L})+t\mid \int ^{t}(G^{R})-t\}&{\text{ otherwise. }}\end{cases}}}$

Norton multiplication is an extension of multiplication to a game ${\displaystyle G}$ and a positive game ${\displaystyle U}$ (the "unit") defined by[7]

${\displaystyle G.U={\begin{cases}G\times U&{\text{ (i.e. the sum of }}G{\text{ copies of }}U{\text{) if }}G{\text{ is a non-negative integer, }}\\-G\times -U&{\text{ if }}G{\text{ is a negative integer, }}\\\{G^{L}.U+(U+I)\mid G^{R}.U-(U+I)\}{\text{ where }}I{\text{ ranges over }}\Delta (U)&{\text{ otherwise. }}\end{cases}}}$

The incentives ${\displaystyle \Delta (U)}$ of a game ${\displaystyle U}$ are defined as ${\displaystyle \{u-U:u\in U^{L}\}\cup \{U-u:u\in U^{R}\}}$.

Overheating is an extension of heating used in Berlekamp's solution of Blockbusting, where ${\displaystyle G}$ overheated from ${\displaystyle s}$ to ${\displaystyle t}$ is defined for arbitrary games ${\displaystyle G,s,t}$ with ${\displaystyle s>0}$ as[8]

${\displaystyle \int _{s}^{t}G={\begin{cases}G.s&{\text{ if }}G{\text{ is an integer, }}\\\{\int _{s}^{t}(G^{L})+t\mid \int _{s}^{t}(G^{R})-t\}&{\text{ otherwise. }}\end{cases}}}$

Winning Ways also defines overheating of a game ${\displaystyle G}$ by a positive game ${\displaystyle X}$, as[9]

${\displaystyle \int _{0}^{t}G=\left\{\int _{0}^{t}(G^{L})+X\mid \int _{0}^{t}(G^{R})-X\right\}}$

Note that in this definition numbers are not treated differently from arbitrary games.

Note that the "lower bound" 0 distinguishes this from the previous definition by Berlekamp

Chilling is a variant of cooling by ${\displaystyle 1}$ used to analyse the Go endgame of Go and is defined by[10]

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

This is equivalent to cooling by ${\displaystyle 1}$ when ${\displaystyle G}$ is an "even elementary Go position in canonical form".[11]

Warming is a special case of overheating, namely ${\displaystyle \int _{1*}^{1}}$, normally written simply as ${\displaystyle \int }$ which inverts chilling when ${\displaystyle G}$ is an "even elementary Go position in canonical form". In this case the previous definition simplifies to the form[12]

${\displaystyle \int G={\begin{cases}G&{\text{ if }}G{\text{ is an even integer, }}\\G*&{\text{ if }}G{\text{ is an odd integer, }}\\\{\int (G^{L})+1\mid \int (G^{R})-1\}&{\text{ otherwise. }}\end{cases}}}$