\documentclass{article}
\usepackage{enumerate,mathtools,amsmath,amssymb,amsthm}

\title{\vspace{-3ex} \bf Assignment 5 \\[1ex]\rm\normalsize CS 311, Fall 2015 \\ Due: December 2, 2014}

\date{}
\author{}

\begin{document}
\maketitle
\paragraph{Problem 1} Use a reduction to show that the language $ALL_{TM}$ is undecidable
\[ALL_{TM} = \{ \langle M \rangle \;|\; \mbox{where $M$ is a TM and $L(M) = \Sigma*$} \}\] [10 points]

\paragraph{Problem 2} A {\em useless state} in a Turing machine
is one that is never entered on any input string.  Consider the 
problem of determining whether a Turing Machine has any useless 
states.  Formulate this problem as a language and show that it 
is undecidable. [10 points]

\paragraph{Problem 3} If $A \leq_m B$ and $B$ is a regular language,
does this imply that $A$ is a regular language? Why or why not? [10 points]

\paragraph{Problem 4} Prove that the language
\[ LOOP_{TM} = \{ \langle M \rangle \;|\; M \text{ is a TM and $M$ loops on all inputs} \} \]
is not recognizable. [10 points]

\paragraph{Problem 5} Prove that the 3-SAT problem discussed in class is an element of $NP$ by
giving a verifier and a NTM decider that run in poly-time. (Only one of these is required for a proof,
but I'd like both for this question.) [10 points]

\end{document}
