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\begin{document}

\centerline{\bf HW 12 CMSC 452}
\centerline{\bf Morally Due TUES April 29 11:00AM} 
\centerline{\bf Dead-Cat Due THU May    1 11:00AM} 


\begin{enumerate}

\item
(40 points) 
Recall that a Turing Machine had a $\delta$ function

$\delta \colon Q-\{h\} \times \Sigma \into Q \times (\Sigma \cup \{R,L\})$. 


A Leo-Turing-Machine has a $\delta$ function

$\delta \colon Q-\{h\} \times \Sigma \into Q \times \{L,R,S\} \times \Sigma$.

The interpret is that $L$ is Left, $R$ is Right, $S$ is STAY PUT.
The last $\Sigma$ is for writing a character.
So for Leo-Turing-Machines each step is a move AND a write
(unlike our Turing Machines where each step s a move OR a write). 

For example: 

$$\delta(q,a) = (p, R, b)$$

means that if the machine is in state $q$ and the head is looking at an $a$
then the machine will
\begin{itemize}
\item
write a $b$, then 
\item
Move $R$, then 
\item
write a $b$.
\end{itemize}

We want to prove the Cook-Levin Theorem for Leo-Turing machines.

Give the part of the formula that corresponds to 

$\delta(q,a) = (p, R, b).$

We'll help you begin: the formula begins 

$$(z_{i,j,\sigma_1} \wedge z_{i,j+1,(a,q)} \wedge z_{i,j+2,\sigma_2}) \goes$$

YOU FILL IN THE REST. 

\newpage
\item
(40 points) Let 

A {\it Hamiltonian Path} is a path in a graph that hits every vertex exactly once.

A {\it Hamiltonian Cycle } is a cycle in a graph that hits every vertex exactly once.

A {\it Hamiltonian Musical } is a a musical about either {\it Alexander Hamilton} or {\it William Rowan Hamilton}
(I know which one I'd want to see).


\begin{itemize}
\item
$\HAMPATH  =$

$\{ (G,a,b) \colon G \hbox{ has a Ham Path that starts at $a$ and ends at $b$ } \}$.
\item 
$\HAMCYCLE  =$

$ \{ G \colon G \hbox{ has a Hamiltonian Cycle} \}$
\item
$\FHAMCYCLE$ is the function that, on input a graph $G$, either outputs 

NO there is no Ham Cycle, or

YES, there is one, and HERE IT IS: (then outputs a Ham cycle). 

(There may be many Ham cycles, this function outputs one of them.) 
\end{itemize} 

\begin{enumerate}
\item
Show that $\HAMPATH \le \HAMCYCLE$. 
\item
Is the following problem known to be in P? NP-complete? 

$\{ (G,a,b) \colon $

$G \hbox{ has a Ham Path that starts at $a$ and ends at $b$ AND }$

$\hbox{ every vertex in $G$ has degree $\le 2$} \}$.

\smallskip

If you think it is in P then give the algorithm. If you think it is NPC then give a reference (which can be on the web).

\item
Is the following problem known to be in P? NP-complete? 

$\{ (G,a,b) \colon $

$G \hbox{ has a Ham Path that starts at $a$ and ends at $b$ AND }$

$\hbox{ every vertex in $G$ has degree $\le 3$} \}$.

\smallskip

If you think it is in P then give the algorithm. If you think it is NPC then give a reference (which can be on the web).
\end{enumerate}

\newpage

\item
(20 points)
(You are encouraged to go to the web for this one)

Recall that in class we showed that (or will show that, depending on when I do it)
FACTORING is probably NOT NP-complete. 

So are there ANY problems whose inputs are natural numbers that are NP-complete?

FIND ONE on the web and state it. 

\end{enumerate}
\end{document}