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\begin{document}

\centerline{\bf HW 13 CMSC 452}
\centerline{\bf Morally Due TUES May 6 11:00AM} 
\centerline{\bf Dead-Cat Due THU May 8 11:00AM} 


\begin{enumerate}

\item
(40 points) 

{\it Definition} Let $\Sigma=\{a,b\}$. Note that $\$\notin\Sigma$. We will be using
\$ as a separator so we can have a set of ordered pairs. Let $B\subseteq \Sigma^*\$\Sigma^*$.
(For example $aabba\$bba$ could be in $B$ but $aabb$ cannot.) 
We define 

$${\rm JAVIER}(B)= \{ x \colon (\exists y)[|y|=2^{2^{|x|}} \hbox{ AND } x\$y\in B \}.$$

Show that if $B$ is decidable then ${\rm JAVIER}(B)$ is decidable. 

(Here is how the proof begins: Assume you have a program $M$ for $B$. 
Your program for ${\rm JAVIER}(B)$ can use $M$ as a 
subroutine. You may use psuedocode. You DO NOT have to deal with Turing Machines at all.) 

\newpage

\item
(30 points) 
Let $M_0, M_1,\ldots,$ be the set of all Turing Machines (we assume $\Sigma=\{a,b,\$,Y,N,,\#\}$
but this does not matter). You may use Google or ChatGPT or your mothers supercomputer, but you should not need
any of those. We are only saying this since you might ask.

\begin{enumerate}
\item
(15 points) Give a set of the form 

$\{ e \colon M_e \hbox{ BLAH BLAH} \}$ 

that is DECIDABLE and was NOT on the April 29 slides.

\item
(15 points) Give a set of the form 

$\{ e \colon M_e \hbox{ BLAH BLAH BLAH} \}$ 

that is UNDECIDABLE and was NOT on the April 29 slides.
\end{enumerate}

\newpage 

\item
(30 points)
Let $A$ be the set of natural numbers $x$ such that

\begin{itemize}
\item
$x$ is a cube, AND
\item
Either $x\equiv 1 \pmod 5$ OR $x\equiv 2 \mod 8$. 
\end{itemize}

Show a polynomial $p\in \Z[y_1,\ldots,y_n]$ (I am not telling you $n$) such that 

$$A = \{ x \colon (\exists y_1,\ldots,y_n)[x\ge 0 \wedge p(y_1,\ldots,y_n,x)=0]\}.$$

\end{enumerate}

\end{document}