\documentclass[11pt]{article}

\usepackage{fullpage,amssymb}
\parindent 0pt
\parskip 1ex
\oddsidemargin -.1in
\textwidth 6.7in
\thispagestyle{empty}

\begin{document}

\centerline{\bf\large COMPSCI 501 Spring 2019 - Homework 3}
\medskip

Released 2/25/2019 \hfill Due in Gradescope on Thu Mar 7, 11:59pm
\bigskip

\textbf{Instructions}.

You may work in groups, but you must write solutions yourself.
List your collaborators. 

All your answers should contain a rigorous argument of correctness.

Please submit a PDF file. You may submit a scanned handwritten document, but a typed submission is preferred. Please assign pages to questions in Gradescope.

\bigskip
\textbf{Q1} (15p). We say some language $L$ \emph{isolates} languages $L_1$ and $L_2$ if $L_1 \subseteq L$ and $L_2 \subseteq \overline{L}$. Suppose $L_1$ and $L_2$ are disjoint co-Turing-recognizable languages. Prove that $L_1$ and $L_2$ are isolated by some \emph{decidable} language. 

\textbf{Q2} (5+5p). 
Prove that a language is decidable if and only if it is enumerable in lexicographic order. \\
Prove that every infinite Turing-recognizable language has an infinite decidable subset.

\textbf{Q3} (5p).  Let $S = \{\langle M\rangle \mid M$ is a DFA that accepts $w^R$ whenever it accepts $w\}$. Show that $S$ is decidable.

\textbf{Q4} (15p). Let $L = \{\langle  M, Q\rangle: M$ is a TM, $Q$ is a state of $M$, and there is some input $w$ whose computation in $M$ reaches the state $Q \}$. Prove or disprove: $L$ is undecidable.

\textbf{Q5} (10p).  Show that determining whether a CFG generates all strings in $1^*$ is decidable.\\ In other words, show that $\{\langle G\rangle \mid G \textrm{ is a CFG over } \{0,1\} \textrm{ and } 1^* \subseteq L(G)\}$ is a decidable language. 

\textbf{Q6} (5p). Goldbach's Conjecture states that every even integer greater than 2 is expressible as the sum of two prime numbers. Show that if HALT were decidable, then we would be able to determine whether Goldbach's Conjecture were true or false.
\end{document}