ladder-calculus/beamer/lc-beamer/example.tex

\documentclass{beamer}


\usepackage[utf8]{inputenc}
\usepackage{formal-grammar}
\usepackage[dvipsnames]{xcolor}
\usepackage{mathpartir}
\usepackage{stmaryrd}
%\usepackage{minted}
%\usemintedstyle{tango}


\usepackage{listings}% http://ctan.org/pkg/listings
\lstset{
  basicstyle=\ttfamily,
  mathescape
}


\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
\newcommand{\exprterminal}[1]{\textcolor{Sepia}{#1}}



\title{A functional core calculus with ladder-types}
%\date[ISPN ’80]{27th International Symposium of Prime Numbers}


\author[Euclid]{Michael Sippel \texttt{michael.sippel@mailbox.tu-dresden.de}\inst{1}}
\institute{\inst{1} Technische Universität Dresden}

\usetheme{ccc}

\begin{document}

\begin{frame}
\titlepage
\end{frame}



\begin{frame}[t, fragile]
  \frametitle{Type Systems}
  \framesubtitle{Safety <> Flexibility}

\centering
  \includegraphics[width=0.9\textwidth]{intro.pdf}

	\pause
	\begin{block}{type information in comments}
	\begin{lstlisting}
	/* this is an angle in degrees */
	double hue = 156.4;
	\end{lstlisting}
\end{block}

\end{frame}

\begin{frame}[t, fragile]{Example - Image Processing}
\begin{lstlisting}
image-read :: FilePath -> [Color]
image-write :: FilePath -> [Color] -> ()

image-mix :: $\mathbb{R}$ -> [Color] -> [Color] -> [Color]
image-saturate :: $\mathbb{R}$ -> [Color] -> [Color]

let im1 = image-read "in1.png"
let im2 = image-read "in2.png"
let im3 = image-mix 0.5
		im1
		(image-saturate 0.8 im2)		

image-write "out.png" im3
\end{lstlisting}
\end{frame}


\begin{frame}{Performance Checklist}
	\framesubtitle{(singlethreaded)}
  \begin{enumerate}
  \item<1-> algorithmic
    \begin{itemize}
     \item choose appropriate data structures\\
			(reduce complexity in frequent cases)

     \item identify special cases
    \end{itemize}

 \item<2-> implementation detail:
   \begin{itemize}
	\item optimize memory bottleneck
      \begin{itemize}
       \item improve cache \& bus efficiency with compact representations
  	   \item<3-> Synchronize locality in access order and memory layout.
       (elements accessed in succession shall be close in memory,
	     AoS vs SoA)

	   \item<4-> \(\rightarrow\) De-/Encode Overhead
	         vs. Fetch-Overhead,\\
       		account for machine specific cache-sizes
	\end{itemize}
	\item

  \item<6-> use SIMD instructions
  \item<7-> interfacing with kernels running on accelerator devices
 \end{itemize}
\end{enumerate}
\end{frame}



\begin{frame}[t, fragile]{Example - Image Processing}
\begin{lstlisting}
image-read :: FilePath -> [Color]
image-write :: FilePath -> [Color] -> ()

image-mix :: $\mathbb{R}$ -> [Color] -> [Color] -> [Color]
image-saturate :: $\mathbb{R}$ -> [Color] -> [Color]

let im1 = image-read "in1.png"
let im2 = image-read "in2.png"
let im3 = image-mix 0.5
		im1
		(image-saturate 0.8 im2)		

image-write "out.png" im3
\end{lstlisting}
\end{frame}

\begin{frame}{Example - Image Processing}
\framesubtitle{More Types}

	\begin{itemize}
		\item \texttt{ColorRGBu8}
		\item \texttt{ColorRGBf32}
		\item \texttt{ColorHSVu8}
		\item \texttt{ColorHSVf32}
	\end{itemize}
\end{frame}


\begin{frame}[t, fragile]{Example - Image Processing}
\framesubtitle{More Types}

\begin{lstlisting}
image-read :: FilePath -> [ColorRGBu8]
image-write :: FilePath -> [ColorRGBu8] -> ()

image-mix :: f32 -> [ColorRGBf32] -> [ColorRGBf32] -> [ColorRGBf32]
image-saturate:: f32->[ColorHSVf32]->[ColorHSVf32]

let im1 = image-read "in1.png"
let im2 = image-read "in2.png"
let im3 = image-mix 0.5
		im1
		(image-saturate 0.8 im2)		

image-write "out.png" im3
\end{lstlisting}
\end{frame}


\begin{frame}[t, fragile]{Example - Image Processing}

\framesubtitle{AoS vs SoA}

\begin{verbatim}
[cacheline][cacheline][cacheline]
\end{verbatim}

\begin{verbatim}
[RGBRGBRG][BRGBRGBR][GBRGBRGB]
\end{verbatim}

\begin{verbatim}
[HSVHSVHS][VHSVHSVH][SVHSVHSV]
\end{verbatim}

\begin{verbatim}
[HHHHHHHH][SSSSSSSS][VVVVVVVV]
\end{verbatim}

\end{frame}

\begin{frame}{Example - Image Processing}
  \framesubtitle{OpenGL Image Formats}

  \begin{itemize}
    \item \texttt{GL\_R3\_G3\_B2}
    \item \texttt{GL\_RGB10\_A2}
	\item \texttt{GL\_R11F\_G11F\_B10F}
	\item \texttt{GL\_SRGB8\_ALPHA8}
	\item \ldots
  \end{itemize}
\end{frame}


\begin{frame}{Sea of Types}
\centering
\includegraphics[width=0.5\textwidth]{sea-of-types/target/sea-of-types-0-bytes.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[width=0.5\textwidth]{sea-of-types/target/sea-of-types-1-ieee754.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.5\textheight]{sea-of-types/target/sea-of-types-2-real.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.5\textheight]{sea-of-types/target/sea-of-types-3-real.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-4-degrees.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-5-turns.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-6-radians.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-7-angle.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-8-hue.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[height=0.8\textheight]{sea-of-types/target/sea-of-types-9-u8.png}
\end{frame}

\begin{frame}{Sea of Types}
\centering
\includegraphics[width=\textwidth]{sea-of-types/target/sea-of-types-10-u8.png}
\end{frame}




\begin{frame}{Solutions (I): Traits / Typeclasses / Interfaces}
\centering
	\includegraphics[width=0.5\textwidth]{solution-traits.png}
	\vskip1cm
  \begin{itemize}
	\item \texttt{impl Angle for Degrees \{ \dots \}}
	\item \texttt{impl Angle for Turns \{ \dots \}}
	\item \texttt{impl Angle for Radians \{ \dots \}}
  \end{itemize}
\end{frame}


\begin{frame}{Solutions (II): Wrapper structs / newtype}
\centering
	\includegraphics[width=0.5\textwidth]{solution-wrappers.png}

  \vskip1cm
  \begin{itemize}
	\item \texttt{struct Degrees\{ value: f32 \}}
	\item \texttt{struct Turns\{ value: f32 \}}
	\item \texttt{struct Radians\{ value: f32 \}}
  \end{itemize}
\end{frame}


\begin{frame}{Solutions (II): Wrapper structs / newtype}
\centering
	\includegraphics[width=0.5\textwidth]{solution-wrappers.png}

  \vskip1cm
  \begin{itemize}
	\item \texttt{struct Degrees<T: \(\mathbb{R}\)>\{ value: T \}}
	\item \texttt{struct Turns<T: \(\mathbb{R}\)>\{ value: T \}}
	\item \texttt{struct Radians<T: \(\mathbb{R}\)>\{ value: T \}}
  \end{itemize}
\end{frame}

\begin{frame}{Coercions: (In-)Coherence with Transitivity}
  \begin{itemize}
    \item<1-> Common example: coerce  \texttt{Int -> Float}
    (subsumption \(\mathbb{N} \subset \mathbb{R}\))

      \begin{itemize}
        \item \texttt{Int -> Float}
        \item \texttt{Int -> String}
        \item \texttt{Float -> String}
      \end{itemize}

    \item<2-> transitivity creates two extensionally \emph{different} coercion paths:
    \begin{itemize}
      \item \texttt{Int -> String}  (3 -> "3")
      \item \texttt{Int -> Float -> String} (3 -> "3.0")
    \end{itemize}

    \item<3-> Subsumptive interpretation is misleading!
    \item<3-> \texttt{Int -> Float} is not as unambiguous as it might seem.
    \begin{itemize}
      \item value ranges (e.g. map int to unit interval [0,1])
      \item multiple quantization functions (linear, exponential, \dots)
    \end{itemize}
    
  \end{itemize}
\end{frame}

\begin{frame}{Ladder Types}
  \begin{block}{Intuition}
    Encode "represented-as" relation into type terms
  \end{block}

  \begin{block}{}
    \( \tau_1 \sim \tau_2 \)  reads "\(\tau_1\) represented as \(\tau_2\)"
  \end{block}
  
  \begin{example}
	\small
	\(
    Color \sim sRGB
    \sim HSV \sim \{ Angle \sim Turns \sim \mathbb{R}_{[0,1]}, \mathbb{R}_{[0,1]}, \mathbb{R}_{[0,1]} \}
	\sim \langle \text{Vec3} \quad \mathbb{R}_{[0,1]}
      \sim \langle \text{QuantizeLinear} \quad 256 \rangle
      \sim \mathbb{Z}_{256} \sim \text{machine.UInt8} \sim \text{Byte}
      \rangle 
    \sim \text{[Byte; 3]}
  \)\end{example}
\end{frame}

\begin{frame}{Morphisms}
  \begin{block}{Intuition}
	Transform between semantically equivalent representations of the same abstract concept
  \end{block}

  \begin{block}{Type}
    \( \tau \sim \tau_1 \rightarrow_\text{morph} \tau \sim \tau_2 \)
  \end{block}

  \begin{example}\(
	 Angle \sim Degrees \sim \mathbb{R}
	\rightarrow_\text{morph}
	 Angle \sim Radians \sim \mathbb{R}
  \)\end{example}
\end{frame}

\begin{frame}{Morphisms}
  \begin{block}{Morphism Condition}
	Assume\\
	\(\metavariable{m_\sigma} : \typeterminal{\metavariable{\sigma} \sim \metavariable{\sigma_1}  \rightarrow_\text{morph}  \metavariable{\sigma} \sim \metavariable{\sigma_2}}\)\\
	\(\metavariable{m_\tau} : \typeterminal{\metavariable{\tau} \sim \metavariable{\tau_1}  \rightarrow_\text{morph}  \metavariable{\tau} \sim \metavariable{\tau_2}}\)\\
	\(\metavariable{f_1} : \typeterminal{\metavariable{\sigma} \sim \metavariable{\sigma_1}  \rightarrow  \metavariable{\tau} \sim \metavariable{\tau_1} }\)\\
	\(\metavariable{f_2} : \typeterminal{\metavariable{\sigma} \sim \metavariable{\sigma_2}  \rightarrow  \metavariable{\tau} \sim \metavariable{\tau_2} }\)\\
	\vskip1cm
	then it holds that:\\
	\(\forall x:\typeterminal{\metavariable{\sigma}\sim\metavariable{\sigma_1}},\quad
	\metavariable{m_\tau}(\metavariable{f_1} x) = \metavariable{f_2} (\metavariable{m_\sigma} x) \)

	\vskip-4.5cm
	\hfill\includegraphics[width=0.3\textwidth]{morphisms.png}
  \end{block}

\end{frame}



\begin{frame}{Extending System F}
\small
\begin{grammar}

\firstcase{ T }{
	\metavariable{\sigma}
}{Base Type}

\otherform{
	\metavariable{\alpha}
}{Type Variable}

\otherform{
	\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \nonterm{T}
}{Universal Type}

\otherform{
	\typeterminal{<} \nonterm{T} \quad \nonterm{T} \typeterminal{>}
}{Specialized Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\rightarrow} \quad \nonterm{T}
}{Function Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\rightarrow_\text{morph}} \quad \nonterm{T}
}{Morphism Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\sim} \quad \nonterm{T}
}{Ladder Type}

$$\\$$

\firstcase{ E
% T_\selexpr
}
	{ \metavariable{x}
} {Variable}

\otherform{
	\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad 
	\nonterm{ E }
	\quad \exprterminal{in} \quad
	\nonterm{ E }
}{Variable Binding}

\otherform{
	$$ \exprterminal{\Lambda} \metavariable{\alpha}
	\quad \exprterminal{\mapsto} \quad $$
	\nonterm{ E }
}{Type Abstraction}

\otherform{
	$$ \exprterminal{\lambda} \metavariable{x} $$
	\exprterminal{:} \nonterm{ T }
    \quad $$\exprterminal{\mapsto}$$ \quad
	\nonterm{ E }
}{Value Abstraction}

\otherform{
	$$ \exprterminal{\lambda} \metavariable{x} $$
	\exprterminal{:} \nonterm{ T }
    \quad $$\exprterminal{\mapsto_\text{morph}}$$ \quad
	\nonterm{ E }
}{Value Morphism}

\otherform{
	\nonterm{ E }
	\quad
	\nonterm{ T }
}{Type Application}

\otherform{
	\nonterm{ E }
	\quad
    \nonterm{ E }
}{Value Application}

\otherform{
    \nonterm{ E }
	\quad
	\exprterminal{as}
	\quad
    \nonterm{ T }
}{Ascension}

\otherform{
    \nonterm{ E }
	\quad
	\exprterminal{des}
	\quad
    \nonterm{ T }
}{Descension}

\end{grammar}
\end{frame}




\begin{frame}{Typing}
\begin{definition}[Typing Relation]
\begin{mathpar}
	\inferrule[T-App]{
		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
		\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\
		\Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
	}{
		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
	}
\end{mathpar}
\end{definition}
\end{frame}


\begin{frame}{Morphism Graph}

\begin{definition}[Morphism Paths]
\label{def:pathrules}
\begin{mathpar}

	\inferrule[M-Sub]{
		\metavariable{\tau} \leq \metavariable{\tau'}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}

	\inferrule[M-Single]{
		(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}

	\inferrule[M-Chain]{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
		\Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
	}
	
	\inferrule[M-MapSeq]{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}{
		\Gamma \vdash
		\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
		\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
	}
\end{mathpar}
\end{definition}  
\end{frame}


\begin{frame}{Morphism Graph}
\begin{example}[Morphism Graph]
\begin{mathpar}
\small
\text{Assume  } 
\Gamma := \{\\
	\exprterminal{\text{degrees-to-turns}} : \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R} \quad \rightarrow_\text{morph} \quad \text{Angle}\sim\text{Turns}\sim\mathbb{R}},\\
	\exprterminal{\text{turns-to-radians}} : \typeterminal{\text{Angle}\sim\text{Turns}\sim\mathbb{R} \quad  \rightarrow_\text{morph} \quad  \text{Angle}\sim\text{Radians}\sim\mathbb{R}},\\
\}.
\end{mathpar}
\end{example}
\end{frame}



\begin{frame}{Morphism Graph}
\begin{example}[Morphism Graph]
Then ..
\small
\begin{itemize}
\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\mathbb{R}}\) (by \textsc{M-Sub})
\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\text{Angle}\sim\text{Radians}\sim\mathbb{R}}\) (by \textsc{M-Chain})
\item \(\Gamma \vdash \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle} \leadsto \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}\) (by \textsc{M-MapSeq})
\end{itemize}
\end{example}
\end{frame}



\begin{frame}{Translation}
\begin{definition}[Morphism Translation]
\begin{mathpar}
\small

\Big{\llbracket}
	\inferrule[M-Sub]{
		\metavariable{\tau} \leq \metavariable{\tau'}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}
\Big{\rrbracket} = \exprterminal{\lambda x:\metavariable{\tau} \mapsto x}
\and

\Big{\llbracket}
	\inferrule[M-Single]{
		(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}
\Big{\rrbracket} = \metavariable{h}
\and

\end{mathpar}
\end{definition}
\end{frame}

\begin{frame}{Translation}
\begin{definition}[Morphism Translation (2)]
\begin{mathpar}
\small
\Big{\llbracket}
	\inferrule[M-Chain]{
		C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
		C_2 :: \Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
	}
\Big{\rrbracket} = \exprterminal{\lambda \text{x}:\metavariable{\tau} \mapsto}
		\Big{\llbracket} C_2 \Big{\rrbracket}
		\exprterminal{(}\Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{x})}
\and


\Big{\llbracket}
	\inferrule[M-MapSeq]{
		C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}{
		\Gamma \vdash
		\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
		\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
	}
\Big{\rrbracket} = \exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq }\metavariable{\tau}\rangle} \mapsto}
		\exprterminal{( \text{map}} \Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{xs})}

\end{mathpar}
\end{definition}
\end{frame}


\begin{frame}{Translation}
\begin{example}[Morphism Translation]
\begin{mathpar}
\small
\Big{\llbracket}
	\inferrule[M-Chain]{
		\ldots
	}{
		\Gamma \vdash \typeterminal{\langle\text{Seq}\quad\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle}\\
		\leadsto \typeterminal{\langle\text{Seq}\quad\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}
	}
\Big{\rrbracket} =
\exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq}\quad \text{Angle}\sim\text{Degrees}\sim\mathbb{R} \rangle}}
\newline
\exprterminal{\mapsto (map \text{ }(\lambda \text{x}:\metavariable{\tau} \mapsto \text{turns-to-radians } (\text{degrees-to-turns } x)) \text{ } xs)	}
\end{mathpar}
\end{example}
\end{frame}


\begin{frame}{Translation}
\begin{definition}[Expression Translation]
Translates a type-derivation tree into a fully expanded expression

\begin{mathpar}
\Big{\llbracket}
	\inferrule[T-App]{
		D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
		D_2 :: \Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\\\
		C :: \Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
	}{
		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
	}
\Big{\rrbracket} =
	\Big{\llbracket}D_1\Big{\rrbracket}
	\exprterminal{(}
	\Big{\llbracket}C\Big{\rrbracket}
	\Big{\llbracket}D_2\Big{\rrbracket}
	\exprterminal{)}
\end{mathpar}
\end{definition}
\end{frame}


\begin{frame}{Soundness}

\begin{lemma}[Morphism generation has correct type]
\begin{mathpar}
C :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
\Rightarrow \Gamma \vdash \llbracket C \rrbracket : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}
\end{mathpar}
\end{lemma}

\begin{lemma}[Expression Translation preserves typing]
\begin{mathpar}
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
\Rightarrow \Gamma \vdash \llbracket D \rrbracket : \metavariable{\tau}
\end{mathpar}
\end{lemma}
\end{frame}




\begin{frame}{Soundness}
\begin{theorem}[Preservation]
\begin{mathpar} \forall \Gamma, \metavariable{e}, \metavariable{\tau},\\
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
\Rightarrow \llbracket D \rrbracket \rightarrow_{eval} \metavariable{e'} \\
\Rightarrow \Gamma \vdash \metavariable{e'} : \metavariable{\tau}
\end{mathpar}
\end{theorem}
\end{frame}




\begin{frame}{Soundness}
\begin{theorem}[Progress]
\begin{mathpar} \forall \Gamma, \metavariable{e}, \metavariable{\tau},\\
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
\Rightarrow \llbracket D \rrbracket \text{ is a value} \\ \vee
\exists \metavariable{e'} . \llbracket D \rrbracket \rightarrow_{eval} \metavariable{e'} \\
\end{mathpar}
\end{theorem}
\end{frame}


\begin{frame}[t, fragile]{Summary}

\begin{tabular*}{\textwidth}{p{0.5\textwidth}|p{0.5\textwidth}}

	\textbf{Problem}
	\small
	\begin{itemize}
	\item fixed memory representation derived from type
		  (loss of low level control)
				
	\item weak type interpretations
		(loss of safety)
	\end{itemize}

	&

	\textbf{Goal}
	\small
	\begin{itemize}
	\item handle data transformations in type-safe way
	\item allow low-level optimizations
	 with strong, unambiguous types
	\end{itemize}

	\\

	\hline

	\textbf{Ladder-Types}
	\small
	\begin{itemize}
	\item connect low-level representation \&
		high-level concept via structure of type terms
	\end{itemize}

	&

	\textbf{Result}
	\small
	\begin{itemize}
	\item extended SystemF with ladder types \& morphisms
	\item formalized in Coq
	\item proved basic lemmas about correctness of translation
	\end{itemize}

\end{tabular*}
\end{frame}



%%%
%%% BACKUP
%%% 

\begin{frame}{Subtype Relation}
\includegraphics[width=\textwidth]{sot-int.png}
\end{frame}


\begin{frame}{Subtype Relation}
\includegraphics[width=\textwidth]{sot-json.png}
\end{frame}


\end{document}