122 lines
4.7 KiB
TeX
122 lines
4.7 KiB
TeX
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\documentclass[a4paper,11pt,oneside]{scrartcl}
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\usepackage[utf8]{inputenc}
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\usepackage[a4paper]{geometry}
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\usepackage{hyperref}
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\usepackage{url}
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\usepackage{color}
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\title{Background Questionaire}
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\begin{document}
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\begin{center}
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\usekomafont{sectioning}\usekomafont{part} Background Questionaire
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\end{center}
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\bigskip
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Please try to answer the following questions alone and without using any external aids and hand in the results \emph{anonymously}.
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This is no exam.
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Instead, the results are used to adapt the \emph{Interactive Theorem Proving} course to the background and interests of the audience.
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\section{Functional Programming}
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\begin{enumerate}
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\item Write a functional program \texttt{APPEND} that appends two lists.
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Try to use SML notation.
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\texttt{APPEND} should satisfy the following example behaviours
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\begin{itemize}
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\item \texttt{APPEND [1,2,3] [4,5] = [1,2,3,4,5]}
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\item \texttt{APPEND [] [1] = [1]}
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\end{itemize}
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\item What does \emph{tail-recursion} mean and why is it important for functional programming?
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\item Write a tail-recursive version of \texttt{APPEND}.
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\end{enumerate}
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\section{Induction Proofs}
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\begin{enumerate}
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\item Prove that \texttt{APPEND} is associative (hint: via induction). You can use the following
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properties of \texttt{APPEND} and \texttt{CONS}. For convenience use the notation \texttt{l1 ++ l2} for \texttt{APPEND l1 l2} and the notation \texttt{x::xs} for \texttt{CONS x xs}. Be very detailed and formal.
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\begin{itemize}
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\item \texttt{$\forall$ l.\ [] ++ xs = l}
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\item \texttt{$\forall$ l x xs.\ (x ::\ xs) ++ l = x ::\ (xs ++ l)}
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\end{itemize}
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\item Which induction principles do you know? Which one did you use for proving the associativity of \texttt{APPEND}?
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\end{enumerate}
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\section{Logic}
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\begin{enumerate}
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\item Explain briefly what's wrong with the following reasoning:
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``No cat has two tails. A cat has one more tail than no cat. Therefore, a cat has three tails.''
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\item What's wrong with the following classical example of a false chain syllogism?
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``Qui bene bibit, bene dormit; qui bene dormit, non peccat; qui non peccat, salvatur; ergo qui bene bibit, salvatur. (Ergo, bibamus!)'' (``Who drinks well, sleeps well. Who sleeps, does not sin. Who does not sin, will go to heaven. Therefore, who drinks well, will go to heaven. (So, let's drink!)'')
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\item Formalise the following riddle\footnote{found in the German Wikipedia entry for resolution} using first order logic.
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It is well known that in ancient times (a) all \emph{Spartans} were \emph{brave} and (b) all \emph{Athenians} were \emph{wise}.
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Spartans and Athenians always fought with each other. So there was (c) no dual citizenship.
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Once upon a time, 3 Greek philosophers met: Diogenes, Platon and Euklid. In contrast to their
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famous namesakes, not much is known about them. We know however, that
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(d) they all came from Sparta or Athens and did not like each other much.
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Being philosophers they however never told a lie; even while insulting each other.
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A few fragments of their squabbles have come to us through the centuries:
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\begin{enumerate}
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\item[(e)] Euklid: ``If Platon is from Sparta, then Diogenes is a coward.''
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\item[(f)] Platon: ``Diogenes is a coward, provided Euklid is from Sparta.''
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\item[(g)] Platon: ``If Diogenes is from Athens, then Euklid is a coward.''
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\item[(h)] Diogenes: ``If Platon is from Athens, then Euklid is a moron.''
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\end{enumerate}
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Can you reconstruct where each of them came from?
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Prove that Plato is from Sparta using \emph{resolution}.
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\item Explain very briefly what \emph{Skolemisation} is?
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\item Let the function \textit{myst} for all \textit{R} of type $\alpha \to \alpha \to \textit{bool}$ be given by
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\[
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\textit{myst}(\textit{R}) = \lambda a\, b.\ \forall Q. \left(
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\begin{array}{cr}
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\forall x.\ Q\, x\, x\ & \wedge \\
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\forall x\,y.\ R\,x\,y\ \Longrightarrow\ Q\, x\, y\ & \wedge \\
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\forall x\,y\,z.\ (Q\,x\,y\ \wedge\ Q\,y\,z)\ \Longrightarrow\ Q\,x\,z
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\end{array}
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\right) \Longrightarrow Q\,a\,b
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\]
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Which concept does the function \textit{myst} define? If you don't see it, describe as much
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as possible. Which type does it have? What is represented by this type? Explain the formula in English using as high level concepts as possible.
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\end{enumerate}
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\section{General}
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Please write down anything else that you believe is relevant for
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selecting and prioritising topics of the Interactive Theorem Proving
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course. Do you have any comments or suggestions? Why are you
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attending the course? Do you have a concrete application in mind?
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Have you used interactive theorem provers before? Which ones? Do you
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have experience with other formal method tools like model checkers,
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SAT solvers, SMT solvers, first order provers \ldots? Which ones?
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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