146 lines
5.3 KiB
TeX
146 lines
5.3 KiB
TeX
\documentclass[a4paper,10pt,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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\usepackage{amsfonts}
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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
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any external aids. If you have trouble, just skip a question instead
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of guessing and thinking very hard. Try not to write lengthy answers,
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often bullet-points are enough. This is no exam, please hand in the
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results \emph{anonymously}. The results are used only to adapt the
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\emph{Interactive Theorem Proving} course to the background and
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interests of the audience.
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\section{Functional Programming}
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\begin{enumerate}
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\item Consider the following functional program on lists.
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\begin{verbatim}
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fun SNOC x [] = [x]
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| SNOC x (y::ys) = y::(SNOC x ys)
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\end{verbatim}
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Please answer the following questions:
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\begin{itemize}
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\item What is the result of \texttt{SNOC 5 [7,3,2]}?
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\item Describe informally (and very briefly) what the function \texttt{SNOC} does.
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\end{itemize}
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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 is \emph{tail-recursion} 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 the following method to calculate the sum of the first $n$ natural numbers is correct (notice $0 \notin \mathbb{N}$, i.\,e.\ $\mathbb{N} = \{1, 2, 3, \ldots\}$):
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\[\forall n \in \mathbb{N}.\ \sum_{1 \leq i \leq n} i = \frac{n * (n+1)}{2}\]
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\item Prove that \texttt{$\forall$x l.\ LENGTH (SNOC x l) = LENGTH l + 1} holds for the function \texttt{SNOC} given above. You can use arithmetic facts and the
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following properties of \texttt{LENGTH}:
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\begin{itemize}
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\item \texttt{LENGTH [] = 0}
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\item \texttt{$\forall$x xs.\ LENGTH (x ::\ xs) = LENGTH xs + 1}
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\end{itemize}
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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}. Use the notation \texttt{l1 ++ l2} for \texttt{APPEND l1 l2}.
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\begin{itemize}
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\item \texttt{$\forall$l.\ [] ++ l = 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 ones did you use above?
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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 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.
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During their meeting, the 3 philosophers started to argue and finally insulted each other.
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Being philosophers they were very careful not to tell a lie, though.
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A few fragments of what they said 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 from which town each of these philosophers came?
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\begin{enumerate}
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\item Formalise the relevant parts of the text above in first order logic. Model \emph{is coward} as \emph{is not brave} and \emph{is moron} as \emph{is not wise}.
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\item Using the proof method of \emph{resolution} show that Platon is from Sparta. If you
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don't know the resolution method, try to show it using some other method.
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\item Which town did Euklid come from?
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\end{enumerate}
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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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\begin{enumerate}
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\item Which type does \textit{myst} have?
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\item What concept does the type $\alpha \to \alpha \to \textit{bool}$ represent?
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\item Translate the formula in English using as high level concepts as possible.
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\item What concept does the function \textit{myst} define?
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\end{enumerate}
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\end{enumerate}
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\section{General}
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Do you have any comments or suggestions? Is there something that you
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believe is relevant for selecting and prioritising topics of the
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Interactive Theorem Proving course. I'm happy if you don't write
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anything here.
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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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