MuZero Operator

computation

Definition

${\mu }_0$ Operator

The ${\mu }_0$ operator defines a function $f:{\mathbb{N}}^k\to \mathbb{N}$ from a total function $g:{\mathbb{N}}^{k+1}\to \mathbb{N}$.

$$\begin{array}{rl}f& ={\mu }_{0}g\\ f(x_1,\dots ,x_k)& =\underset{y\ge 0}{\min }[g(y,x_1,\dots ,x_k)=0]\end{array}$$

The minimum is taken over the first argument of $g$.

Examples

Example

$$\begin{array}{rl}({\mu }_0{P}_1^1)(n)& =\underset{y\ge 0}{\min }[y=0]\\ & =0\\ ⟹{\mu }_0{P}_1^1& =C_0^0\end{array}$$

Example

$$\begin{array}{rl}({\mu }_{0}S)(n)& =\underset{y\ge 0}{\min }[y+1=0]\end{array}$$

Trivially, there exists no such an $y\in \mathbb{N}$ such that $y+1=0$. Therefore, it’s undefined.

The same holds for all:

$${\mu }_0{C}_k^1\forall k\ge 1$$

${\mu }_{0}E={\mu }_0\left[m,n\mapsto (n+6{)}^{2m+1}\right]$

Define $E$ as:

$$E(m,n)=(n+6{)}^{2m+1}$$

Then:

$$\begin{array}{rl}({\mu }_{0}E)(n)& =\underset{y\ge 0}{\min }\left[E(y,n)=0\right]\\ & =\underset{y\ge 0}{\min }\left[(n+6{)}^{2y+1}=0\right]\end{array}$$

The 1-arity function ${\mu }_{0}E$ is undefined for all $n\in \mathbb{N}$.

$f(x,y)=x^2+y$

$$\begin{array}{rl}f(x,y)& =x^2+y\\ ({\mu }_{0}f)(y)& =\underset{z\ge 0}{\min }\left[z^2+y=0\right]\end{array}$$

Trivially, ${\mu }_{0}f$ is undefined for $y=1$ given that:

$$\cancel{\exists }z\ge 0:z^2=-1$$