We can have some fun on the way to one. \begin{defn}[Busy Beaver Competition] A machine $M$ is an {\em $n$-state entry in the busy beaver competition\/} \index{busy beaver competition} \index{busy beaver competition $n$-state entry} \index{$n$-state entry in busy beaver competition} if: \begin{itemize} \item $M$ has a two-way infinite tape and alphabet $\{1\}$ (see Chapter~\ref{ch:eleven}; \item $M$ has $n+1$ states, but state $n+1$ is used only for halting (so both $M(n+1,0)$ and $M(n+1,1)$ are undefined); \item $M$ eventually halts when given a blank input tape. \end{itemize} $M$'s {\em score\/} \index{score in busy beaver competition} \index{busy beaver competition score in} in the competition is the number of $1$'s on the output tape of its computation from a blank input tape.

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The recursive functions and relations in Problems \ref{p:thirteen6} and \ref{r:rfs} provide most of the necessary tools. \begin{prob} \label{p:fourteen8a} Show that both of the following relations are primitive recursive. \begin{enumerate} \item $\mathsc{TapePos}$, where $\mathsc{TapePos}(n)$ $\iff$ $n$ is the code of a tape position. \index{$\mathsc{TapePos}$} \item $\mathsc{TapePosSeq}$, where $\mathsc{TapePosSeq}(n)$ $\iff$ $n$ is the code of a sequence of tape positions. \index{$\mathsc{TapePosSeq}$} \end{enumerate} \end{prob} \begin{prob} \label{p:fourteen9} Show that each of the following is primitive recursive. \begin{enumerate} \item The $4$-place function $\mathsc{Entry}$\index{$\mathsc{Entry}$} such that \begin{align*} &\mathsc{Entry}(j,w,t,n) \\ &= \begin{cases} \ulcorner (t,i+w-1,\mathbf{a}') \urcorner & \text{if $n = \ulcorner (s,i,\mathbf{a}) \urcorner$, $j \in \{0,1\}$,} \\ & \text{$w \in \{0,2\}$, $i+w-1 \ge 0$, and $t \ge 1$,} \\ & \text{where $a_k' = a_k$ for $k \ne i$ and $a_i' = j$;} \\ 0 & \text{otherwise.} \end{cases} \end{align*} \item For any Turing machine $M$ with alphabet $\{1\}$, the $1$-place function $\mathsc{Step}_M$\index{$\mathsc{Step}_M$} such that $$ \mathsc{Step}_M(n) = \begin{cases} \ulcorner \mathbf{M}(s,i,\mathbf{a}) \urcorner & \text{if $n = \ulcorner (s,i,\mathbf{a}) \urcorner$ and} \\ & \text{$\mathbf{M}(s,i,\mathbf{a})$ is defined;} \\ 0 & \text{otherwise.} \end{cases} $$ \item For any Turing machine $M$ with alphabet $\{1\}$, the $1$-place relation $\mathsc{Comp}_M$\index{$\mathsc{Comp}_M$}, where $$ \mathsc{Comp}_M(n) \iff \text{$n$ is the code of a computation of $M$.} $$ \end{enumerate} \end{prob} The functions and relations above may be primitive recursive, but the last big step in showing that Turing computable functions are recursive requires unbounded minimalization. \begin{prop} \label{p:fourteen9a} For any Turing machine $M$ with alphabet $\{1\}$, the $1$-place (partial) function $\mathsc{Sim}_M$\index{$\mathsc{Sim}_M$} is recursive, where $$ \mathsc{Sim}_M(n) = \ulcorner (t,j,\mathbf{b}) \urcorner $$ if $n = \ulcorner (1,0,\mathbf{a}) \urcorner$ for some input tape $\mathbf{a}$ and $M$ eventually halts in position $(t,j,\mathbf{b})$ on input $\mathbf{a}$. (Note that $\mathsc{Sim}_M(n)$ may be undefined if $n \ne \ulcorner (1,0,\mathbf{a}) \urcorner$ for an input tape $\mathbf{a}$, or if $M$ does not eventually halt on input $\mathbf{a}$.) \end{prop} \begin{lem} \label{l:fourteen9b} Show that the following functions are primitive recursive: \begin{enumerate} \item For any fixed $k \ge 1$, $\mathsc{Code}_k (n_1,\ldots,n_k) = \ulcorner (1,0,01^{n_1}0\ldots 01^{n_k}) \urcorner$. \index{$\mathsc{Code}_k$} \item $\mathsc{Decode}(t) = n$ if $t = \ulcorner (s,i,01^{n+1}) \urcorner$ (and anything you like otherwise). \index{$\mathsc{Decode}$} \end{enumerate} \end{lem} \begin{thm} \label{t:fourteen10} Any $k$-place Turing computable function is recursive. \end{thm} \begin{cor} \label{c:TCiffRec} A function $f: \mathbb{N}^k \to \mathbb{N}$ is Turing computable if and only if it is recursive. \end{cor} Thus Turing machines and recursive functions are essentially equivalent models of computation. \subsection*{An universal Turing machine} One can push the techniques used above little farther to get a recursive function that can simulate {\em any\/} Turing machine , cited: Refugees, Prisoners and Camps: A Functional Analysis of the Phenomenon of Encampment http://sixthskill.com/?books/refugees-prisoners-and-camps-a-functional-analysis-of-the-phenomenon-of-encampment.