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\noindent
\begin{center}
\framebox{
\vbox{\vspace{2mm}
\hbox to 6.28in { {\bf CS~21~~~Decidability and Tractability
\hfill Winter 2019} }
\vspace{4mm}
\hbox to 6.28in { {\Large \hfill Midterm \hfill} }
\vspace{2mm}
\hbox to 6.28in { {\it Out: #1 \hfill Due: #2} }
\vspace{2mm}}
}
\end{center}
\vspace*{4mm}
}
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\begin{document}
\midterm{February 6}{\bf {February 13}}
{\bf This is a midterm.} You may consult only the course notes and
the text (Sipser). {\em You may not collaborate.} The full honor code guidelines can be found in the course syllabus.
There are 5 problems
on 2 pages. Please attempt all problems. {\bf Please turn in your solutions via
Gradescope, by 1pm on the due date.} Good luck!
\begin{enumerate}
\item Identify each of the following languages as either (i)
regular, (ii) context-free but not regular, or (iii) not
context-free. For each language, prove that your classification is
correct, using the techniques we have developed in this course.
\begin{enumerate}
\item $L_1 = \{a^ib^jc^k : j > i+k \}$.
\item $L_2 = \{a^ib^jc^k : i = j = k \mbox{ or } i > 1000\}$.
\item $L_3 = \{a^ib^jc^k : i = j = k \mbox{ or } i < 1000\}$.
\end{enumerate}
\item Identify each of the following languages as either decidable
or undecidable, and prove that your classification is correct,
using the techniques we have developed in this course. Recall that for a context free
grammar $G$, we denote by $L(G)$ the language it describes, and
similarly for a regular expression $E$, we denote by $L(E)$ the
language it describes.
\begin{enumerate}
\item $\mbox{\sc cfl-in-reg} = \{(G, E) : \mbox{$G$ is a CFG, $E$
is a regular expression, and $L(G) \subseteq L(E)$}\}$
\item $\mbox{\sc reg-in-cfl} = \{(E, G) : \mbox{$G$ is a CFG, $E$
is a regular expression, and $L(E) \subseteq L(G)$}\}$
\end{enumerate}
Hint: you may wish to use the fact that the intersection of a context free language and a regular language is context-free (Sipser problem 2.18).
\item Two (disjoint) languages $L_1$ and $L_2$ are called {\em
recursively separable} if there is a decidable language $D$ for
which $L_1 \cap D = \emptyset$ and $L_2 \subseteq D$; they are
{\em recursively inseparable} if no such decidable language $D$
exists. Convince yourself that an undecidable language and its
complement are recursively inseparable.
Consider the following languages:
\begin{eqnarray*}
L_1 & = & \{\langle M \rangle : M \mbox{ halts and accepts input
} \langle
M \rangle\} \\
L_2 & = & \{\langle M \rangle : M \mbox{ halts and rejects input
} \langle M \rangle\}
\end{eqnarray*}
Prove that $L_1$ and $L_2$ are recursively inseparable. Hint: your
proof will probably involve supplying a Turing Machine its own
description as input.
\newpage
\item A {\em right-linear} CFG is a context-free grammar in which
every production has the form
\begin{itemize}
\item $A \rightarrow xB$, or
\item $A \rightarrow x$,
\end{itemize}
where $A$ and $B$ are non-terminals, and $x$ can be any string of
terminals. A CFG is {\em linear} if productions of the form $A
\rightarrow Bx$ are allowed in addition to the two types of
productions in a right-linear CFG.
\begin{enumerate}
\item Prove that every language generated by a right-linear CFG is
regular.
\item Prove that every regular language is generated by some
right-linear CFG.
\item Give a linear CFG that generates the non-regular
language
\[L = \{0^n1^n : n \ge 0\}\]
and prove that your grammar indeed generates exactly $L$ (i.e., prove that every string in $L$ is generated by your grammar, and prove that every string generated by your grammar is in $L$).
\end{enumerate}
\item Given a language $L$, define $\mbox{AT-LEAST-100}_L$ as follows:
\begin{eqnarray*}
\mbox{AT-LEAST-100}_L & = & \{\#x_1\#x_2\#\cdots\#x_k\# : k \ge 0 \mbox{ and } |\{i : x_i \in L\}| \ge 100.\}
\end{eqnarray*}
Prove that $\mbox{AT-LEAST-100}_L$ is RE if L is RE. Here the $x_i$
are strings over $L$'s alphabet, and $\#$ is a symbol that is not
in $L$'s alphabet.
\end{enumerate}
\end{document}