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\title{\vspace{-3ex} \bf Assignment 4 \\[1ex]\rm\normalsize CS 311, Fall 2015 \\ Due: November 18, 2014}

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\paragraph{Problem 1}
 A {\em Turing machine with left reset} is similar to an ordinary Turing machine, but the 
 transition function has the form
  \begin{displaymath}
    \delta : Q \times \Gamma \to Q \times \Gamma \times \{R,RESET\}
  \end{displaymath}
If $\delta (q,a) = (r,b,RESET)$, when the machine is in state $q$ reading an $a$, the machine's 
head jumps to the left-hand end of the tape after it writes $b$ on the tape and enters state 
$r$. Note that these machines do not have the usual ability to move the head one symbol left. 
Show that Turing machines with left reset recognize the class of Turing-recognizable languages. [10 points]

  (Hint: Much like previous problems in this class, you'll need to describe some general 
  construction that takes a Turing Machine-with-reset to an ordinary Turing Machine and visa-versa)

\paragraph{Problem 2}
Give the informal descriptions for Turing machines that decide the following languages
\begin{enumerate}[\indent a)]
    \item $\{w \;|\; w \text{ contains twice as many 0s as 1s }\}$ [5 points]
    \item $\{a+b=c \;|\; a,b,c \in \{0,1\}* \text{ and the binary numbers represented by $a$ 
    and $b$ sum to $c$} \}$ [5 points]
\end{enumerate}

\paragraph{Problem 3}
Show that the Turing-decidable languages are closed under
\begin{enumerate}[\indent a)]
    \item union [5 points]
    \item intersection [5 points]
    \item complement [5 points]
    \item set difference [5 points]
\end{enumerate} 

\paragraph{Problem 4}
Show that the Turing-recognizable languages are closed under concatenation

This problem requires providing constructions that take individual Turing machines and combines 
them into a new machine that {\em recognizes} the new language. Remember, this is about 
Turing-recognizable languages not just decideable so that there's a possibility of non-termination. [10 points]

\paragraph{Problem 5} For each of the following Turing machine variants determine if the machine
is more powerful, equivalent, or less powerful than a single-tape Turing machine. If less powerful
describe the class of languages recognized by the machine. Explain your answers.
\begin{enumerate}[\indent a)]
    \item A Turing Machine that can only make moves to the right and never left. [5 points]
    \item A Turing Machine that can move right one space or move left two spaces. [5 points]
    \item A Turing Machine that never writes to a space on the tape that already contains a symbol. [5 points]
\end{enumerate}

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