\subsection{Faculteit}
Als we de analyse op een compleet programma toepassen komt er een \textit{Piet}
-programma uit dat behoorlijk fors is maar netjes zijn werk doet. Dit programma
-berekent de faculteit van $x$ en stopt dat uiteindelijk in $y$.
+programma uit dat behoorlijk fors is maar netjes zijn werk doet. Het programma
+dat we gaan vertalen berekent de faculteit van $x$ en stopt dat uiteindelijk in
+$y$.
\begin{lstlisting}[title=Faculteit in \textit{While}]
x:=5;
y:=1;
)
\end{lstlisting}
-In \textit{Piet} ziet dit er als volgt uit...
+In \textit{Piet'} ziet dit er als volgt uit...
\begin{lstlisting}[title=Faculteit in \textit{Piet'}]
-push 5 // x is vanaf nu variabele 1
-push 1 // y is vanaf nu variabele 2
+push 10 // x:=5
+push 1 // y:=1
MARKERING A:
-Un(1) // variabele x wordt klaargezet
-push 1 // de waarde 1 wordt klaargezet
-sub // subtractie om straks gelijkheid te kunnen bepalen
-push 1 // de waarde 1 word gepushed om 0 te maken
-dup // de waarde 1 staat er nu twee keer op
-sub // 0 staat op de stack om groterdan relatie te bepalen
-push 2 // klaarmaken om de bovenste twee te swappen
-push 1 // idem
-roll // swappen
-gre // als x>0 dan ligt 1 op de stack anders 0
-not // als x=0 dan ligt 1 op de stack anders 0
-pointer // als x=0 dan draait de DP niet en gaat het programma naar pad A
- // als x$\neq$ dan draait de DP en gaat het programma naar pad B
+ // Un(2)
+ // $\neg$(x=1) $\equiv$ $\neg$(x-1)>0)
+
+
+ // x-1
+ // Un(2)
+push 2 //
+push 1 //
+roll //
+dup //
+push 3 //
+push 1 //
+roll //
+push 1 //
+sub //
+not // $\neg(x-1)$
+
+push 1 // zet $0$ op de stack
+push 1
+sub
+
+gre // $\neg(x-1)>0$ oftewel x=1
+
+pointer // als x=0 dan draait de DP niet en gaat het programma naar pad B
+ // als x$\neq$ dan draait de DP en gaat het programma naar pad A
PAD A:
-skip //een oneindig aantal witte blokken, y is nu x!
+skip // een oneindig aantal witte blokken, y is nu x!
+ // evt een outchar om $y$ naar standardout te printen
PAD B:
-Bin(1, 2) // x en y worden klaargezet om y opnieuw te assignen
-mul // x*y ligt boven op de stack
-Ass(2) // x*y wordt geassigned aan y
-Un(1) // x wordt klaargezet
-push 1 // 1 wordt klaargezet
-sub // x-1 ligt op de stack
-Ass(1) // x-1 wordt geassigned aan x
- // nu gaat het programma weer via een wit pad naar markering A
+ // y:=y*x
+ // Bin(1, 2)
+dup // Un(1)
+push 3 // Un(2+1)
+push 2 //
+roll //
+dup //
+push 4 //
+push 1 //
+roll //
+mul // x*y
+push 2 // Ass(1)
+push 1
+roll
+pop
+
+ // x:=x-1
+push 2 // Un(2)
+push 1
+roll
+dup
+push 3
+push 1
+roll
+
+push 1 //
+sub // x-1
+push 3 // Ass(2)
+push 2
+roll
+pop
+push 2
+push 1
+roll
+ // nu gaat het programma weer via een wit pad naar markering A
\end{lstlisting}
+
+Wat in \textit{Piet} er uit ziet als in Figuur~\ref{fig:img3}
+\begin{figure}[H]
+ \caption{Faculteit vanuit \textit{While} naar \textit{Piet}}
+ \label{fig:img3}
+ \centering
+ \fbox{\includegraphics[natheight=320px, natwidth=256px, height=320px,
+ width=256px]{img3.png}}
+\end{figure}
+
+\begin{figure}[H]
+ \caption{Faculteit vanuit \textit{While} naar \textit{Piet}(geannoteerd)}
+ \label{fig:img5}
+ \centering
+ \fbox{\includegraphics[natheight=320px, natwidth=256px, height=160px,
+ width=128px]{img5.png}}
+\end{figure}
+De code blokken komen als volgt overeen met stukken\textit{While} code.\\
+\begin{tabular}{ll}
+ A & \lstinline{x:=5; y:=1}\\
+ B & \lstinline{$\neg$(x=1)}\\
+ C & \lstinline{y:=y*x}\\
+ D & \lstinline{x:=x-1}\\
+ E & het stukje achter de while loop, hierin printen we de uitkomst zodat we er ook nog wat aan hebben, \\
+ & dit gebeurt in het originele \textit{While}
+ programma niet. Het pyramidevormige stuk zorgt voor terminatie.\\
+ F & \lstinline{while$\neg$(x=1) do(y:=y*x; x:=x-1)}
+\end{tabular}
+
+\begin{landscape}
+De afleidingsrij voor 1! ziet er als volgt uit:\\
+$
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2;push \: 1;push \: 1, [\:], [\:], [\:]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2;push \: 1, [\:], [\:], [1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2, [\:], [\:], [1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1, [\:], [\:], [2,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll, [\:], [\:], [1,2,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup, [\:], [\:], [1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1, [\:], [\:], [3,1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll, [\:], [\:], [1,3,1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;, [\:], [\:], [1,1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [0,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1, [\:], [\:], [1,1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract, [\:], [\:], [1,1,1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater, [\:], [\:], [0,1,1,1]\rangle \Rightarrow\\
+\langle pointer; not, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
+\langle pointer [\:], [\:], [0,1,1]\rangle \Rightarrow\\
+\langle [\:], [\:], [1,1]\rangle\\
+\\
+$
+Eerst worden x en y op de stack gestopt zodat de stack [y,x] = [1,1].
+Deze stack is gelijk aan de stack in de termineerende configuratie: 1 = 1! dus y = x!.
+Als bewijs dat de wiskundige operaties op de stack goed zijn uitgevoerd staat in
+de appendix onder het kopje voorbeelden een stukje code. In het programma
+eenfaculteit worden alle wiskundige operaties die bij de commando's horen
+achter elkaar uitgevoerd op de stack, waarbij het resultaat [1,1] is.\\
+Bij de voorbeelden staan ook nog de programma's tweefaculteit en
+vijffaculteit die respectievelijk 2! en 5! uitrekenen.
+\end{landscape}
+
+\newpage
+
+\begin{landscape}
+De afleidingsrij voor 2! ziet er als volgt uit:\\
+$
+\langle R_1; R_2; R_3; greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2;push \: 1;push \: 2, [\:], [\:], [\:]\rangle \Rightarrow
+\langle R_1; R_2; R_3; greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2;push \: 1, [\:], [\:], [2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll, [\:], [\:], [1,2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup, [\:], [\:], [2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3, [\:], [\:], [2,2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1, [\:], [\:], [3,2,2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll, [\:], [\:], [1,3,2,2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;, [\:], [\:], [1,2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1, [\:], [\:], [0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1, [\:], [\:], [1,0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract, [\:], [\:], [1,1,0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater, [\:], [\:], [0,0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup; pointer; not, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup; pointer, [\:], [\:], [0,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2, [\:], [\:], [3,1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll, [\:], [\:], [2,3,1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup, [\:], [\:], [2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4, [\:], [\:], [2,2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1, [\:], [\:], [4,2,2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll, [\:], [\:], [1,4,2,2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul, [\:], [\:], [2,1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1, [\:], [\:], [2,2,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll, [\:], [\:], [1,2,2,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop, [\:], [\:], [1,2,2]\rangle \Rightarrow\\
+\langle R_1;R_2, [\:], [\:], [2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll; push \: 1; push \: 2, [\:], [\:], [2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll; push \: 1, [\:], [\:], [2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll, [\:], [\:], [1,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup, [\:], [\:], [2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3, [\:], [\:], [2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1, [\:], [\:], [3,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll, [\:], [\:], [1,3,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1, [\:], [\:], [2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub, [\:], [\:], [1,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3, [\:], [\:], [1,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2, [\:], [\:], [3,1,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll, [\:], [\:], [2,3,1,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; roll, [\:], [\:], [1,2,1,2]\rangle \Rightarrow\\
+\langle R_1, [\:], [\:], [2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2, [\:], [\:], [2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1, [\:], [\:], [2,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll, [\:], [\:], [1,2,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1, [\:], [\:], [3,1,1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll, [\:], [\:], [1,3,1,1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1, [\:], [\:], [1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;, [\:], [\:], [1,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [0,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1, [\:], [\:], [1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1, [\:], [\:], [1,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract, [\:], [\:], [1,1,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater, [\:], [\:], [0,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not, [\:], [\:], [1,2,1]\rangle \Rightarrow\\
+\langle pointer [\:], [\:], [0,2,1]\rangle \Rightarrow\\
+\langle [\:], [\:], [2,1]\rangle\\
+$
+
+In deze afleidingsrij zijn de volgende afkortingen gebruikt:\\
+$
+R_1: pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2\\
+R_2: roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll; push \: 1; push \: 2\\
+R_3: pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup; pointer; not\\
+$
+
+Eerst worden x en y op de stack gestopt zodat de stack [y,x] = [1,2].
+De stack in de termineerende configuratie is [y',x'] = [2,1] en 2 = 2! dus x! = y' .
+Als bewijs dat de wiskundige operaties op de stack goed zijn uitgevoerd staat in
+de appendix onder het kopje voorbeelden een stukje code. In het programma
+tweefaculteit worden alle wiskundige operaties die bij de commando's horen
+achter elkaar uitgevoerd op de stack, waarbij het resultaat [2,1] is.\\
+\end{landscape}
\ No newline at end of file
repositories / sec1415.git / blobdiff