(*<*)
theory Slides1
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 ("0") and
  Bk ("1") and
  exponent ("_\<^bsup>_\<^esup>") and
  inv_begin0 ("I\<^sub>0") and
  inv_begin1 ("I\<^sub>1") and
  inv_begin2 ("I\<^sub>2") and
  inv_begin3 ("I\<^sub>3") and
  inv_begin4 ("I\<^sub>4") and 
  inv_begin ("I\<^bsub>cbegin\<^esub>") and
  inv_loop1 ("J\<^sub>1") and
  inv_loop0 ("J\<^sub>0") and
  inv_end1 ("K\<^sub>1") and
  inv_end0 ("K\<^sub>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}{SMAL, 16 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
(*>*)