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\documentclass{article}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../style}
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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 What is the language recognised by the regular expressions
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$(\varnothing^*)^*$.
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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 Review the first handout about sets of strings and read the
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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second handout. Assuming the alphabet is the set $\{a, b\}$, decide
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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which of the following equations are true in general for arbitrary
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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; otherwise
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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 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 regular
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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expressions $r_1^*$, $r_2^*$ and $r_3^*$ each 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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\item Give regular expressions for (a) decimal numbers and for (b)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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binary numbers. (Hint: Observe that the empty string is not a
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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number. Also observe that leading 0s are 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 $(a \cdot b)^* \cdot
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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a \equiv^? a \cdot (b \cdot a)^*$.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Given the regular expression $r = (a \cdot b + b)^*$. Compute
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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what the derivative of $r$ is with respect to $a$, $b$ and $c$. Is
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$r$ 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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Define what is mean by the derivative of a regular expressions
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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with respoect to a character. (Hint: The derivative is defined
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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recursively.)
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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 Assume the set $Der$ is defined as
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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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$Der\,c\,A \dn \{ s \;|\; c\!::\!s \in A\}$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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What is the relation between $Der$ and the notion of derivative of
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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regular expressions?
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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 a regular expression over the alphabet $\{a,b\}$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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recognising all strings that do not contain any substring $bb$ and
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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end in $a$.
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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 Do $(a + b)^* \cdot b^+$ and $(a^* \cdot b^+) + (b^*\cdot b^+)$ define
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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the same language?
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\end{enumerate}
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\end{document}
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%%% Local Variables:
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