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\documentclass{article}
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\usepackage{charter}
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\usepackage{hyperref}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\begin{document}
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\section*{Homework 2}
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\begin{enumerate}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Review the first handout about sets of strings and read
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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the second handout. Assuming the alphabet is $\{a, b\}$,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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decide which of the following equations are true in
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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general for arbitrary languages $A$, $B$ and $C$:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{eqnarray}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(A \cup B) @ C & =^? & A @ C \cup B @ C\nonumber\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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A^* \cup B^* & =^? & (A \cup B)^*\nonumber\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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A^* @ A^* & =^? & A^*\nonumber\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(A \cap B)@ C & =^? & (A@C) \cap (B@C)\nonumber
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{eqnarray}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\noindent In case an equation is true, give an explanation;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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otherwise give a counter-example.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item What is the meaning of a regular expression? Give an
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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inductive definition.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Given the regular expressions $r_1 = \epsilon$ and $r_2
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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= \varnothing$ and $r_3 = a$. How many strings can the
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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regular expressions $r_1^*$, $r_2^*$ and $r_3^*$ each
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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match?
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Give regular expressions for (a) decimal numbers and for
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(b) binary numbers. (Hint: Observe that the empty string
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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is not a number. Also observe that leading 0s are
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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normally not written.)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Decide whether the following two regular expressions are
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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equivalent $(\epsilon + a)^* \equiv^? a^*$ and $(a \cdot
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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b)^* \cdot a \equiv^? a \cdot (b \cdot a)^*$.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Given the regular expression $r = (a \cdot b + b)^*$.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Compute what the derivative of $r$ is with respect to
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$a$, $b$ and $c$. Is $r$ nullable?
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\item Prove that for all regular expressions $r$ we have
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$\textit{nullable}(r) \quad \text{if and only if}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\quad [] \in L(r)$
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Write down clearly in each case what you need to prove and
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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what are the assumptions.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{enumerate}
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\end{document}
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