From 24301f945086b836d08c29af2cdffc8fc4a7be44 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sat, 18 May 2013 11:20:00 +0200 Subject: [PATCH] added draft for proof of correctnes for pogo algorithm --- documents/proof-of-correctness-pogo/Makefile | 8 + .../proof-of-correctness-pogo.pdf | Bin 0 -> 181662 bytes .../proof-of-correctness-pogo.tex | 159 ++++++++++++++++++ 3 files changed, 167 insertions(+) create mode 100644 documents/proof-of-correctness-pogo/Makefile create mode 100644 documents/proof-of-correctness-pogo/proof-of-correctness-pogo.pdf create mode 100644 documents/proof-of-correctness-pogo/proof-of-correctness-pogo.tex diff --git a/documents/proof-of-correctness-pogo/Makefile b/documents/proof-of-correctness-pogo/Makefile new file mode 100644 index 0000000..4bfa692 --- /dev/null +++ b/documents/proof-of-correctness-pogo/Makefile @@ -0,0 +1,8 @@ +SOURCE = proof-of-correctness-pogo +make: + pdflatex $(SOURCE).tex -output-format=pdf + pdflatex $(SOURCE).tex -output-format=pdf + make clean + +clean: + rm -rf $(TARGET) *.class *.html *.log *.aux *.out diff --git a/documents/proof-of-correctness-pogo/proof-of-correctness-pogo.pdf b/documents/proof-of-correctness-pogo/proof-of-correctness-pogo.pdf new file mode 100644 index 0000000000000000000000000000000000000000..981c5957e2cec3351f27ccf6a5a4712e608a5ace GIT binary patch literal 181662 zcma&NV~j6M6sP%b+qV05_ifv@ZQHhO+qP}r#_uZD)p`EIr%*~ zWb(qIGz_#X&}4HuS^F`+d` zgskvj?3IBfQ@tEEmn>HucTeq>Tc=PGEj=9ZPKNrsaNG7ySM~1sWbjq10>;B>kj#FO zzABU?O+VOPt^CD2==J{7RHHWFo@rNkTI`8RV!QB()Mk-8oqG9h+BU_Or1E^a?aJ|J z*5mot_!Y7g*CBep{2@H=FIH_QW32rmeEv2&R!8~D^iBFX^Kfy1lzlt zo%gj`g4!$&l`)CRr+O#beIr7_)pj?e-#@mNR!ZMHhV@u=PAyyVk#1Jny96hpWG?p# zb3CY>l*Hi4m2tT`F|5U%#Bkoyh~vVdxDjG2#C25)>{V$=IHFWzX~Ys)Q;9Aji@~uZ z3utTN@Pa+KhcArOo5=Cja^E~uhTRr2X*Zv;d3|v3zF5k{;R$s5C&b~oJ${6HWJvU~ zV66;ecj%*nuiH%-lV>3fl24l{3InLsnpTr5Zh>ganu$mel$7_nfRBqw&boOUDd)qq z_eS-pHiaqrTj*Qx8<8o|(FhLZBSvgaDQ1?0IR~8lC?;b6_$kB*7qTz4^~9a%I^V7b 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\gets solution + W$ + \State $x \gets x + i$ + \EndIf + \Else + \If{$y > 0$} + \State $solution \gets solution + N$ + \State $y \gets y - i$ + \Else + \State $solution \gets solution + S$ + \State $y \gets y + i$ + \EndIf + \EndIf + \EndFor + \\ + \State \Return $solution$ + \EndFunction + \end{algorithmic} + \caption{Algorithm to solve the pogo problem} + \label{alg:solvePogo} +\end{algorithm} + +\section{Correctness} +\subsection{calculateSteps} +Let $x,y \in \mathbb{Z}$ and $s := \Call{calculateSteps}{x, y}$. + +Let $s_{\min}$ be the minimum amount of necessary steps to get from $(0,0)$ +to $(x,y)$ when you move $i$ units in your $i$'th step. + +\textbf{Theorem: } $s = s_{\min}$ + +It's enough to proof $s \geq s_{\min}$ and $s \leq s_{\min}$. + +\begin{myindentpar}{1cm} +\textbf{Theorem: } $s \leq s_{\min}$ (we don't make too many steps) + +\textbf{Proof: } +\begin{myindentpar}{1cm} +We have to get from $(0,0)$ to $(x, y)$. As we may only move in +taxicab geometry we have to use the taxicab distance measure $d_1$: +\[d_1 \left (p, q \right ) := \sum_{i=1}^2 |p_i -q_i|\] + +So in our scenario: +\[d_1 \left ((0,0), (x,y) \right ) = |x| + |y|\] + +This means we have to move at least $|x| + |y|$ units to get +from $(0,0)$ to $(x, y)$. As we move $i$ units in the $i$'th step, +we have to solve the following equations for $s_{\min1}$: +\begin{align} + \sum_{i=1}^{s_{\min1}} i &\geq |x| + |y| &&\text{ and } &|x| + |y| &> \sum_{i=1}^{s_{\min1} - 1} i\\ + \frac{s_{\min1}^2 + s_{\min1}}{2} &\geq |x| + |y| && & &> \sum_{i=1}^{s_{\min1} - 1} i & +\end{align} + +This is what algorithm \ref{alg:calculateSteps} check with condition 1. +As the algorithm increases $s$ only by one in each loop, it makes +sure that $\sum_{i=1}^{s_{\min1} - 1} i$ is bigger than $|x| + |y|$. + +TODO: Proof necessarity of condition two + +TODO: I guess I should initialize $s$ with 0 (should only make a difference when (x,y) = (0,0)) +\end{myindentpar} + +\textbf{Theorem: } $s \geq s_{\min}$ (we make enough steps) + +\textbf{Proof: } +\begin{myindentpar}{1cm} +TODO +\end{myindentpar} +\end{myindentpar} + +\subsection{solvePogo} +\textbf{Theorem: } \Call{solvePogo}{$x,y$} returns a valid, minimal sequence of steps to get from $(0, 0)$ to $(x,y)$ + +\textbf{Proof: } +\begin{myindentpar}{1cm} +TODO +\end{myindentpar} + +\end{document}