--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides2.thy Wed Jul 17 10:33:19 2013 +0100
@@ -0,0 +1,524 @@
+(*<*)
+theory Slides2
+imports "../thys/UTM" "../thys/Abacus_Defs"
+begin
+declare [[show_question_marks = false]]
+
+hide_const (open) s
+
+abbreviation
+ "update2 p a \<equiv> update a p"
+
+
+notation (latex output)
+ Cons ("_::_" [48,47] 48) and
+ Oc ("1") and
+ Bk ("0") and
+ exponent ("_\<^bsup>_\<^esup>") and
+ inv_begin0 ("I\<^isub>0") and
+ inv_begin1 ("I\<^isub>1") and
+ inv_begin2 ("I\<^isub>2") and
+ inv_begin3 ("I\<^isub>3") and
+ inv_begin4 ("I\<^isub>4") and
+ inv_begin ("I\<^bsub>cbegin\<^esub>") and
+ inv_loop1 ("J\<^isub>1") and
+ inv_loop0 ("J\<^isub>0") and
+ inv_end1 ("K\<^isub>1") and
+ inv_end0 ("K\<^isub>0")
+
+
+
+lemma inv_begin_print:
+ shows "s = 0 \<Longrightarrow> inv_begin n (s, tp) = inv_begin0 n tp" and
+ "s = 1 \<Longrightarrow> inv_begin n (s, tp) = inv_begin1 n tp" and
+ "s = 2 \<Longrightarrow> inv_begin n (s, tp) = inv_begin2 n tp" and
+ "s = 3 \<Longrightarrow> inv_begin n (s, tp) = inv_begin3 n tp" and
+ "s = 4 \<Longrightarrow> inv_begin n (s, tp) = inv_begin4 n tp" and
+ "s \<notin> {0,1,2,3,4} \<Longrightarrow> inv_begin n (s, l, r) = False"
+apply(case_tac [!] tp)
+by (auto)
+
+
+lemma inv1:
+ shows "0 < (n::nat) \<Longrightarrow> Turing_Hoare.assert_imp (inv_begin0 n) (inv_loop1 n)"
+unfolding Turing_Hoare.assert_imp_def
+unfolding inv_loop1.simps inv_begin0.simps
+apply(auto)
+apply(rule_tac x="1" in exI)
+apply(auto simp add: replicate.simps)
+done
+
+lemma inv2:
+ shows "0 < n \<Longrightarrow> inv_loop0 n = inv_end1 n"
+apply(rule ext)
+apply(case_tac x)
+apply(simp add: inv_end1.simps)
+done
+
+
+lemma measure_begin_print:
+ shows "s = 2 \<Longrightarrow> measure_begin_step (s, l, r) = length r" and
+ "s = 3 \<Longrightarrow> measure_begin_step (s, l, r) = (if r = [] \<or> r = [Bk] then 1 else 0)" and
+ "s = 4 \<Longrightarrow> measure_begin_step (s, l, r) = length l" and
+ "s \<notin> {2,3,4} \<Longrightarrow> measure_begin_step (s, l, r) = 0"
+by (simp_all)
+
+lemma inv_begin01:
+ assumes "n > 1"
+ shows "inv_begin0 n (l, r) = (n > 1 \<and> (l, r) = (Oc \<up> (n - 2), [Oc, Oc, Bk, Oc]))"
+using assms by auto
+
+lemma inv_begin02:
+ assumes "n = 1"
+ shows "inv_begin0 n (l, r) = (n = 1 \<and> (l, r) = ([], [Bk, Oc, Bk, Oc]))"
+using assms by auto
+
+(*>*)
+
+
+
+text_raw {*
+ \renewcommand{\slidecaption}{ITP, 24 July 2013}
+ \newcommand{\bl}[1]{#1}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}
+ \frametitle{%
+ \begin{tabular}{@ {}c@ {}}
+ \\[-3mm]
+ \Large Reasoning about Turing Machines\\[-1mm]
+ \Large and Low-Level Code\\[-3mm]
+ \end{tabular}}
+
+ \begin{center}
+ Christian Urban\\[1mm]
+ %%\small King's College London
+ \end{center}\bigskip
+
+ \begin{center}
+ in cooperation with Jian Xu and Xingyuan Zhang
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{A Trend in Verification}%
+
+ \begin{itemize}
+ \item in the past:\\
+ \begin{quote}
+ model a problem mathematically and proof properties about the
+ model
+ \end{quote}\bigskip
+
+ \item needs elegance, is still very hard\bigskip\pause
+
+ \item does not help with ensuring the correctness of running programs
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{A Trend in Verification}%
+
+ \begin{itemize}
+ \item make the specification executable (e.g.~Compcert)\bigskip\pause
+
+ you would expect the trend would be to for example model C, implement
+ your programs in C and verify the programs written in C (e.g.~seL4)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{A Trend in Verification}%
+
+ \begin{itemize}
+ \item but actually people start to verify machine code directly
+ (e.g.~bignum arithmetic implemented in x86-64 -- 700 instructions)
+
+ \item CPU models exists, but the strategy is to use a small subset
+ which you use in your programs
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Why Turing Machines}%
+
+ \begin{itemize}
+ \item at the beginning it was just a nice student project
+ about computability
+
+ \begin{center}
+ \includegraphics[scale=0.3]{boolos.jpg}
+ \end{center}
+
+ \item found an inconsistency in the definition of halting computations
+ (Chap.~3 vs Chap.~8)
+ \pause
+
+ \item \small Norrish formalised computability via lambda-calculus (and nominal);
+ Asperti and Riccioti formalised TMs but didn't get proper UTM
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Turing Machines}%
+
+ \begin{itemize}
+ \item tapes contain 0 or 1 only
+
+ \begin{center}
+\begin{tikzpicture}[scale=1]
+ \draw[very thick] (-3.0,0) -- ( 3.0,0);
+ \draw[very thick] (-3.0,0.5) -- ( 3.0,0.5);
+ \draw[very thick] (-0.25,0) -- (-0.25,0.5);
+ \draw[very thick] ( 0.25,0) -- ( 0.25,0.5);
+ \draw[very thick] (-0.75,0) -- (-0.75,0.5);
+ \draw[very thick] ( 0.75,0) -- ( 0.75,0.5);
+ \draw[very thick] (-1.25,0) -- (-1.25,0.5);
+ \draw[very thick] ( 1.25,0) -- ( 1.25,0.5);
+ \draw[very thick] (-1.75,0) -- (-1.75,0.5);
+ \draw[very thick] ( 1.75,0) -- ( 1.75,0.5);
+ \draw[rounded corners=1mm] (-0.35,-0.1) rectangle (0.35,0.6);
+ \draw[fill] (1.35,0.1) rectangle (1.65,0.4);
+ \draw[fill] (0.85,0.1) rectangle (1.15,0.4);
+ \draw[fill] (-0.35,0.1) rectangle (-0.65,0.4);
+ \draw[fill] (-1.65,0.1) rectangle (-1.35,0.4);
+ \draw (-0.25,0.8) -- (-0.25,-0.8);
+ \draw[<->] (-1.25,-0.7) -- (0.75,-0.7);
+ \node [anchor=base] at (-0.85,-0.5) {\small left list\;\;\;\;\mbox{}};
+ \node [anchor=base] at (0.40,-0.5) {\small \mbox{}\;\;\;\;right list};
+ \node [anchor=base] at (0.1,0.7) {\small \;\;head};
+ \node [anchor=base] at (-2.2,0.2) {\ldots};
+ \node [anchor=base] at ( 2.3,0.2) {\ldots};
+ \end{tikzpicture}
+ \end{center}
+
+ \item steps function
+
+ \begin{quote}
+ What does the tape look like after the TM has executed n steps?
+ \end{quote}\pause
+
+ designate the 0-state as halting state and remain there forever
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Copy Turing Machines}%
+
+ \begin{itemize}
+ \item TM that copies a number on the input tape
+
+
+ \begin{center}
+\begin{tikzpicture}[scale=0.7]
+ \node [anchor=base] at (2.2,0.1) {\small$\Rightarrow$};
+ \node [anchor=base] at (5.6,0.1) {\small$\Rightarrow$};
+ \node [anchor=base] at (10.5,0.1) {\small$\Rightarrow$};
+ \node [anchor=base] at (2.2,-0.8) {\small$\overbrace{\text{cbegin}}^{}$};
+ \node [anchor=base] at (5.6,-0.8) {\small$\overbrace{\text{cloop}}^{}$};
+ \node [anchor=base] at (10.5,-0.8) {\small$\overbrace{\text{cend}}^{}$};
+
+
+ \begin{scope}[shift={(0.5,0)}]
+ \draw[very thick] (-0.25,0) -- ( 1.25,0);
+ \draw[very thick] (-0.25,0.5) -- ( 1.25,0.5);
+ \draw[very thick] (-0.25,0) -- (-0.25,0.5);
+ \draw[very thick] ( 0.25,0) -- ( 0.25,0.5);
+ \draw[very thick] ( 0.75,0) -- ( 0.75,0.5);
+ \draw[very thick] ( 1.25,0) -- ( 1.25,0.5);
+ \draw[rounded corners=1mm] (-0.35,-0.1) rectangle (0.35,0.6);
+ \draw[fill] (-0.15,0.1) rectangle (0.15,0.4);
+ \draw[fill] ( 0.35,0.1) rectangle (0.65,0.4);
+ \draw[fill] ( 0.85,0.1) rectangle (1.15,0.4);
+ \end{scope}
+
+ \begin{scope}[shift={(2.9,0)}]
+ \draw[very thick] (-0.25,0) -- ( 2.25,0);
+ \draw[very thick] (-0.25,0.5) -- ( 2.25,0.5);
+ \draw[very thick] (-0.25,0) -- (-0.25,0.5);
+ \draw[very thick] ( 0.25,0) -- ( 0.25,0.5);
+ \draw[very thick] ( 0.75,0) -- ( 0.75,0.5);
+ \draw[very thick] ( 1.25,0) -- ( 1.25,0.5);
+ \draw[very thick] ( 1.75,0) -- ( 1.75,0.5);
+ \draw[very thick] ( 2.25,0) -- ( 2.25,0.5);
+ \draw[rounded corners=1mm] (0.15,-0.1) rectangle (0.85,0.6);
+ \draw[fill] (-0.15,0.1) rectangle (0.15,0.4);
+ \draw[fill] ( 0.35,0.1) rectangle (0.65,0.4);
+ \draw[fill] ( 0.85,0.1) rectangle (1.15,0.4);
+ \draw[fill] ( 1.85,0.1) rectangle (2.15,0.4);
+ \end{scope}
+
+ \begin{scope}[shift={(6.8,0)}]
+ \draw[very thick] (-0.75,0) -- ( 3.25,0);
+ \draw[very thick] (-0.75,0.5) -- ( 3.25,0.5);
+ \draw[very thick] (-0.75,0) -- (-0.75,0.5);
+ \draw[very thick] (-0.25,0) -- (-0.25,0.5);
+ \draw[very thick] ( 0.25,0) -- ( 0.25,0.5);
+ \draw[very thick] ( 0.75,0) -- ( 0.75,0.5);
+ \draw[very thick] ( 1.25,0) -- ( 1.25,0.5);
+ \draw[very thick] ( 1.75,0) -- ( 1.75,0.5);
+ \draw[very thick] ( 2.25,0) -- ( 2.25,0.5);
+ \draw[very thick] ( 2.75,0) -- ( 2.75,0.5);
+ \draw[very thick] ( 3.25,0) -- ( 3.25,0.5);
+ \draw[rounded corners=1mm] (-0.35,-0.1) rectangle (0.35,0.6);
+ \draw[fill] (-0.15,0.1) rectangle (0.15,0.4);
+ \draw[fill] ( 2.35,0.1) rectangle (2.65,0.4);
+ \draw[fill] ( 2.85,0.1) rectangle (3.15,0.4);
+ \draw[fill] ( 1.85,0.1) rectangle (2.15,0.4);
+ \end{scope}
+
+ \begin{scope}[shift={(11.7,0)}]
+ \draw[very thick] (-0.75,0) -- ( 3.25,0);
+ \draw[very thick] (-0.75,0.5) -- ( 3.25,0.5);
+ \draw[very thick] (-0.75,0) -- (-0.75,0.5);
+ \draw[very thick] (-0.25,0) -- (-0.25,0.5);
+ \draw[very thick] ( 0.25,0) -- ( 0.25,0.5);
+ \draw[very thick] ( 0.75,0) -- ( 0.75,0.5);
+ \draw[very thick] ( 1.25,0) -- ( 1.25,0.5);
+ \draw[very thick] ( 1.75,0) -- ( 1.75,0.5);
+ \draw[very thick] ( 2.25,0) -- ( 2.25,0.5);
+ \draw[very thick] ( 2.75,0) -- ( 2.75,0.5);
+ \draw[very thick] ( 3.25,0) -- ( 3.25,0.5);
+ \draw[rounded corners=1mm] (-0.35,-0.1) rectangle (0.35,0.6);
+ \draw[fill] (-0.15,0.1) rectangle (0.15,0.4);
+ \draw[fill] ( 2.35,0.1) rectangle (2.65,0.4);
+ \draw[fill] ( 2.85,0.1) rectangle (3.15,0.4);
+ \draw[fill] ( 1.85,0.1) rectangle (2.15,0.4);
+ \draw[fill] ( 0.35,0.1) rectangle (0.65,0.4);
+ \draw[fill] ( 0.85,0.1) rectangle (1.15,0.4);
+ \end{scope}
+ \end{tikzpicture}
+ \end{center}\bigskip
+
+ \footnotesize
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-8mm}}c@ {\hspace{2mm}}c@ {\hspace{2mm}}c}
+ \begin{tabular}[t]{@ {}l@ {}}
+ @{text cbegin} @{text "\<equiv>"}\\
+ \hspace{2mm}@{text "["}@{text "(W\<^bsub>0\<^esub>, 0), (R, 2), (R, 3),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(R, 2), (W\<^bsub>1\<^esub>, 3), (L, 4),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(L, 4), (L, 0)"}@{text "]"}
+ \end{tabular}
+ &
+ \begin{tabular}[t]{@ {}l@ {}}
+ @{text cloop} @{text "\<equiv>"}\\
+ \hspace{2mm}@{text "["}@{text "(R, 0), (R, 2), (R, 3),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(W\<^bsub>0\<^esub>, 2), (R, 3), (R, 4),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(W\<^bsub>1\<^esub>, 5), (R, 4), (L, 6),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(L, 5), (L, 6), (L, 1)"}@{text "]"}
+ \end{tabular}
+ &
+ \begin{tabular}[t]{@ {}l@ {}}
+ @{text cend} @{text "\<equiv>"}\\
+ \hspace{2mm}@{text "["}@{text "(L, 0), (R, 2), (W\<^bsub>1\<^esub>, 3),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(L, 4), (R, 2), (R, 2),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(L, 5), (W\<^bsub>0\<^esub>, 4), (R, 0),"}\\
+ \hspace{2mm}\phantom{@{text "["}}@{text "(L, 5)"}@{text "]"}
+ \end{tabular}
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ \definecolor{mygrey}{rgb}{.80,.80,.80}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Hoare Logic for TMs}%
+
+ \begin{itemize}
+ \item Hoare-triples and Hoare-pairs:
+
+ \small
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-6mm}}c@ {\hspace{-3mm}}c@ {}}
+ \begin{tabular}[t]{@ {}l@ {}}
+ \colorbox{mygrey}{@{thm (lhs) Hoare_halt_def}} @{text "\<equiv>"}\\[1mm]
+ \hspace{5mm}@{text "\<forall>"}@{term "tp"}.\\
+ \hspace{7mm}if @{term "P tp"} holds then\\
+ \hspace{7mm}@{text "\<exists>"}@{term n}. such that\\
+ \hspace{7mm}@{text "is_final (steps (1, tp) p n)"} \hspace{1mm}@{text "\<and>"}\\
+ \hspace{7mm}@{text "Q holds_for (steps (1, tp) p n)"}
+ \end{tabular} &
+ \begin{tabular}[t]{@ {}l@ {}}
+ \colorbox{mygrey}{@{thm (lhs) Hoare_unhalt_def}} @{text "\<equiv>"}\\[1mm]
+ \hspace{5mm}@{text "\<forall>"}@{term "tp"}.\\
+ \hspace{7mm}if @{term "P tp"} holds then\\
+ \hspace{7mm}@{text "\<forall>"}@{term n}. @{text "\<not> is_final (steps (1, tp) p n)"}
+ \end{tabular}
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ \definecolor{mygrey}{rgb}{.80,.80,.80}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Hoare Reasoning}%
+
+ \begin{itemize}
+ \item reasoning is still quite difficult---invariants
+
+ \footnotesize
+ \begin{center}
+\begin{tabular}{@ {\hspace{-4mm}}lcl@ {\hspace{-0.9cm}}l@ {}}
+ \hline
+ @{thm (lhs) inv_begin1.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_begin1.simps} & (starting state)\\
+ @{thm (lhs) inv_begin2.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_begin2.simps}\\
+ @{thm (lhs) inv_begin3.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_begin3.simps}\\
+ @{thm (lhs) inv_begin4.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_begin4.simps}\\
+ @{thm (lhs) inv_begin0.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_begin01} @{text "\<or>"}& (halting state)\\
+ & & @{thm (rhs) inv_begin02}\smallskip \\
+ \hline
+ @{thm (lhs) inv_loop1.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_loop1_loop.simps} @{text "\<or>"}\\
+ & & @{thm (rhs) inv_loop1_exit.simps} & (starting state)\\
+ @{thm (lhs) inv_loop0.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_loop0.simps}& (halting state)\smallskip\\
+ \hline
+ @{thm (lhs) inv_end1.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_end1.simps} & (starting state)\\
+ @{thm (lhs) inv_end0.simps} & @{text "\<equiv>"} & @{thm (rhs) inv_end0.simps} & (halting state)\smallskip\\
+ \hline
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Register Machines}%
+
+ \begin{itemize}
+ \item instructions
+
+ \begin{center}
+ \begin{tabular}{rcl@ {\hspace{10mm}}l}
+ @{text "I"} & $::=$ & @{term "Inc R\<iota>"} & increment register @{text "R"} by one\\
+ & $\mid$ & @{term "Dec R\<iota> L\<iota>"} & if content of @{text R} is non-zero,\\
+ & & & then decrement it by one\\
+ & & & otherwise jump to instruction @{text L}\\
+ & $\mid$ & @{term "Goto L\<iota>"} & jump to instruction @{text L}
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Recursive Functions}%
+
+ \begin{itemize}
+ \item addition, multiplication, \ldots
+ \item logical operations, quantifiers\ldots
+ \item coding of numbers (Cantor encoding)
+ \item UF\pause\bigskip
+
+ \item Recursive Functions $\Rightarrow$ Register Machines
+ \item Register Machines $\Rightarrow$ Turing Machines
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Sizes}%
+
+ \begin{itemize}
+ \item UF (size: 140843)
+ \item Register Machine (size: 2 Mio instructions)
+ \item UTM (size: 38 Mio states)
+ \end{itemize}\bigskip\bigskip
+
+ \small
+ old version: RM (12 Mio) UTM (112 Mio)
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Separation Algebra}%
+
+ \begin{itemize}
+ \item introduced a separation algebra framework for register machines and TMs
+ \item we can semi-automate the reasoning for our small TMs
+ \item we can assemble bigger programs out of smaller components\bigskip
+
+ \item looks awfully like ``real'' assembly code\pause
+ \item Conclusion: we have a playing ground for reasoning about low-level
+ code; we work on automation
+ \end{itemize}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+
+(*<*)
+end
+(*>*)
\ No newline at end of file