first draft of the final presentation

final-pres/finalpres.tex

@@ -206,7 +206,8 @@
   % \caption{different fixed-point formats}
 \end{figure}
 Representable numbers: \\
-Biggest (range): \(2^{\msb}\) \hspace{\stretch{1}} Smallest (accuracy): \(2^{\lsb}\)
+Biggest (range): \(2^{\msb+1} - 2^{\lsb}\) \hspace{\stretch{1}} Smallest
+(accuracy): \(2^{\lsb}\)
 \end{frame}
 
 \section{Bounding Signals to Avoid Overflow (MSB)}
@@ -243,7 +244,7 @@ Biggest (range): \(2^{\msb}\) \hspace{\stretch{1}} Smallest (accuracy): \(2^{\ls
 \end{itemize}
 \end{frame}
 
-\begin{frame}{Recursive Signals}
+\begin{frame}{Recursive Signals Problem}
   Int phaser : \lstinline{process = \%(10)~ +(1);}
   \begin{figure}
     \centering
@@ -255,14 +256,14 @@ Biggest (range): \(2^{\msb}\) \hspace{\stretch{1}} Smallest (accuracy): \(2^{\ls
   previous sample \(s(t-1)\).
 \end{frame}
 
-\begin{frame}{Fixpoint search}
+\begin{frame}{Abstract Analysis on signals}
   Adapt \emph{Abstract analysis} to signals:
 
   for each time \(t\), find an
   interval \(S(t)\) containing \(s(t)\)
   \[\small
   \recSig{t}{s(t) = 0}{s(t) = f(s(t-1))}  \leadsto
-  \recSig{t}{S(t) = \bot}{S(t) = f(S(t-1)) \bigcup S(t)}
+  \recSig{t}{S(t) = [0, 0]}{S(t) = f(S(t-1)) \bigcup S(t)}
   \]
   
   \(\bar{S} = \bigcup_{t\in\Rel} S(t)\) upperbounds the signal \(s(t)\) at each time
@@ -277,11 +278,122 @@ Biggest (range): \(2^{\msb}\) \hspace{\stretch{1}} Smallest (accuracy): \(2^{\ls
 \end{itemize}
 \end{frame}
 
+\begin{frame}{Example}
+  On example \only<1-3>{\lstinline{process = \%(10)~ +(1);}
+    \(f:z \mapsto z + 1 \mod 10 \)}
+  \only<4>{\lstinline{process = \_~ +(1);}
+    \(f:z \mapsto z + 1\)}
+  \begin{figure}
+    \centering
+    \only<1-3>{\includegraphics[width = 0.5\textwidth]{../code/rec-block.png}}
+    \only<4>{\includegraphics[width = 0.5\textwidth]{../code/ramp-block.png}}
+  \end{figure}
+  \only<1>{Step 0 : \([0, 0]\)}
+  \only<2>{Step 1 : \(f([0, 0]) \cup [0,0] = [1,1] \cup [0,0] = [0,1]\)}
+  \only<3>{Step 10 : \(f([0, 10]) \cup [0,10] = [0, 10] \cup [0, 10] = [0, 10]\) Done !}
+  \only<4>{Step 18752 : \(f([0, 18752]) \cup [0,18752] = [0, 18753] \cup [0, 18752] = [0, 18753]\)\\
+    used solution: \emph{widening}, set to \(\IntR\)} then
+  \(f(\IntR) \cup \IntR = \IntR \)
+\end{frame}
+
+\begin{frame}{Limitations}
+  \faustexfig{A smoother}{\label{fig:smooth}}
+  {0.4}{../code/smooth-block.png}
+  {0.5}{../code/smooth-sig-old.png}
+  {\(y(t) = x(t) + \frac{1}{2} y(t-1)\)}{../code/smooth.dsp}
+  With \(f: X \mapsto [-1, 1] + 1/2 * X\)\\
+  \only<1>{Step 0 : \([0, 0]\)}
+  \only<2>{Step 1 : \(f([0, 0]) \cup [0,0] = [-1,1] \cup [0,0] = [-1,1]\)}
+  \only<3>{Step 2 : \(f([-1, 1]) \cup [-1, 1] = [-1.5, 1.5] \cup [-1, 1] = [-1.5, 1.5]\)}
+  \only<4>{Step 3 : \(f([-1.5, 1.5]) \cup [-1.5, 1.5] = [-1.75, 1.75] \)}
+  % \only<5>{Step \(t\) : \(f([- 2 + 2^{-t-1}, 2 - 2^{-t-1}]) \cup
+  %   [- 2 + 2^{-t-1}, 2 - 2^{-t-1}]  = [- 2 + 2^{-t}, 2 - 2^{-t}] \)}
+\end{frame}
 
 \section{Designing Accuracy for Small but Correct Computations (LSB)}
 
-\section{Conclusion and Future Work}
+\begin{frame}{Error Model}
+  Use a rounding operator \(\rnd\) for computed arithmetic primitives.\\
+  \lstinline{(a + b) * c} implements \emph{exactly} \(\rnd(\rnd(a + b) \times c)\)\\
+  Absolute error \(\abserr_f = (a + b) \times c - \rnd(\rnd(a + b) \times c)\)
+  \begin{eqnarray*}
+    \footnotesize
+    \abserr_f &=& (a + b)\times c - \rnd(\rnd(a + b)\times c) \pause\\
+              &=& (a + b)\times c - \rnd(a + b)\times c
+                  + \rnd(a + b)\times c -\rnd(\rnd(a + b)\times c)\pause\\
+              &=& [(a + b) - \rnd(a + b)]\times c +
+                  \left[\rnd(a + b)\times c -\rnd(\rnd(a + b)\times c)
+                  \right] \pause\\
+              &=& \abserr^+(a, b)\times c +
+                  \abserr^\times (\rnd(a + b), c) \pause\\
+    |\abserr_f| &=& |\abserr^+(a, b)\times c +
+                    \abserr^\times (\rnd(a + b), c) |\\
+              &\leq& |\abserr^+(a, b)\times c| +
+                     |\abserr^\times (\rnd(a + b), c)|\\
+              &\leq& \maxerr^+ \times c + \maxerr^\times
+  \end{eqnarray*}
+  Upperbounding \(\abserr\) for any \(a,b,c\): \emph{worst-case absolute error}
+\end{frame}
+
+\begin{frame}{Local Rules}
+  Local rule: devising the LSB of a processor depending only on its type
+  \(+, \times\dots\) and the LSB of its inputs/outputs.
+  
+  \faustexfig{Weighted sum}{\label{fig:pond}}%
+  {0.4}{../code/pond-block.png}%
+  {0.5}{../code/pond-sig-old.png}%
+  {}{../code/pond.dsp}%
+\end{frame}
+
+\begin{frame}{Tentatives of local rules}
+  \todo{détailler, ou pas?}
+  \begin{itemize}
+  \item From the Input to the Outputs, scale to best
+  \item From the Input to the Outputs, scale to worst
+  \item From the Outputs to the Inputs, satisfying requirements
+  \end{itemize}
+\end{frame}
+
+\begin{frame}{Global rules, one step back}
+  \begin{itemize}
+  \item Detect patterns on the signal graph and contract them into one node
+  \item Apply local rules on the contracted graph
+  \item Compute the LSB of the elements of each pattern according to its global
+    inputs and outputs
+  \end{itemize}
 
+  \begin{minipage}{1.0\linewidth}
+    \centering
+  \includegraphics[width=.7\textwidth]{../code/pond-sig-old.png}
+\end{minipage}
+\end{frame}
+
+\section{Conclusion and Perspectives}
+
+\begin{frame}{Conclusion}
+  \begin{itemize}
+  \item Most Significant Bit:
+    \begin{itemize}
+    \item Strong theoretical background
+    \item Needs an efficient interval library
+    \item Undecidable, needs sometimes annotations
+    \end{itemize}
+  \item Least Significant Bit:
+    \begin{itemize}
+    \item No efficient local rule
+    \item Needs a step-back approach to understand the code in its globality
+    \end{itemize}
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Perspectives}
+  \begin{itemize}
+  \item Find a suboptimal backup pragmatic rule to derive MSB and LSB in every case
+  \item Improve the interval library
+  \item Interface Faust with external tools
+  \item Use statistical error instead of worst-case error
+  \end{itemize}
+\end{frame}
 
 \end{document}