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--- /dev/null +++ b/talk/cmu-fall-2018/cmu-talk.tex @@ -0,0 +1,1479 @@ +\documentclass{beamer} +\usetheme{boxes} + +\newcommand{\notepr}[1]{} + +%% \newif\ifshort +%% \newif\iflong + +%% \shorttrue +%% %% \longtrue + +\usepackage{rotating} +\usepackage{listings} +\usepackage{xcolor} +\lstset{ + basicstyle=\ttfamily, + escapeinside=~~ +} + +\usepackage{microtype} +\usepackage{latexsym,amsmath,amssymb,amsthm} +\usepackage{mathpartir} +\usepackage{tikz} +\usepackage{tikz-cd} +\input{talk-defs} +\newcommand{\ltdyn}{\dynr} +\newcommand{\uarrow}{\mathrel{\rotatebox[origin=c]{-30}{$\leftarrowtail$}}} +\newcommand{\darrow}{\mathrel{\rotatebox[origin=c]{30}{$\twoheadleftarrow$}}} +\renewcommand{\upcast}[2]{\langle{#2}\uarrow{#1}\rangle} +\renewcommand{\dncast}[2]{\langle{#1}\darrow{#2}\rangle} +\newcommand{\gtdyn}{\sqsupseteq} +\AtBeginSection[] +{ + \begin{frame}<beamer> + %% \frametitle{Outline for section \thesection} + \tableofcontents[currentsection] + \end{frame} +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% WARNING: I changed these commands +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\renewcommand{\u}{\underline} + +\newcommand{\pipe}{\,\,|\,\,} +\newcommand{\dynv}{{?}} +\newcommand{\dync}{\u {\text{?`}}} +\newcommand{\result}{\text{result}} +\newcommand{\Set}{\text{Set}} +\newcommand{\relto}{\to} +\newcommand{\M}{\mathcal{M}} +\newcommand{\sq}{\square} +\newcommand{\lett}{\kw{let}} +\newcommand{\letXbeYinZ}[2]{\lett#2 = #1;} +\newcommand{\bindXtoYinZ}[2]{\kw{bind}#2 \leftarrow #1;} +\newcommand{\too}{\kw{to}} +\newcommand{\case}{\kw{case}} +\newcommand{\kw}[1]{\texttt{#1}\,\,} +\newcommand{\absurd}{\kw{absurd}} +\newcommand{\caseofXthenYelseZ}[3]{\case #1 \{ #2 \pipe #3 \}} +\newcommand{\caseofX}[1]{\case #1} +\newcommand{\thenY}{\{} +\newcommand{\elseZ}[1]{\pipe #1 \}} +\newcommand{\pmpairWtoXYinZ}[4]{\kw{split} #1\,\kw{to} (#2,#3). #4} +\newcommand{\pmpairWtoinZ}[2]{\kw{split} #1\,\kw{to} (). #2} +\newcommand{\pmmuXtoYinZ}[3]{\kw{unroll} #1 \,\kw{to} \roll #2. #3} +\newcommand{\ret}{\kw{ret}} +\newcommand{\thunk}{\kw{thunk}} +\newcommand{\force}{\kw{force}} +\newcommand{\abort}{\kw {abort}} +\newcommand{\with}{\mathbin{\&}} +\DeclareMathOperator*{\With}{\&} + + +\title{A Type Theoretic Approach to Gradual Typing} +\author{Max S. New} +\institute{Northeastern University} +\date{\today} + +\begin{document} + +\begin{frame} + \titlepage + \begin{small} + Joint work with\\ + Amal Ahmed (Northeastern),\\ + Dan Licata (Wesleyan) + \end{small} +\end{frame} + +\begin{frame} + \frametitle{Outline} + \tableofcontents +\end{frame} + +\section{Intro to Gradual Typing} + +\begin{frame}[fragile]{Gradual Typing is about Program Evolution} +\begin{columns} +\begin{column}{0.5\textwidth} +\begin{onlyenv}<1> +\begin{footnotesize} +\begin{lstlisting} +view(st) -> { ... } + +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { + st = initial_state; + while true { + evt = input(); + state = update(evt,st); + display(view(st)); + } +} +\end{lstlisting} +\end{footnotesize} +\end{onlyenv} +\begin{onlyenv}<2-> +\begin{footnotesize} +\begin{lstlisting} +view(st) -> { ... } + +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { .. } +\end{lstlisting} +\end{footnotesize} +\end{onlyenv} +\end{column} +\begin{column}{0.1\textwidth} + \begin{onlyenv}<3-> + $\Longrightarrow$ +\end{onlyenv} +\end{column} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<3-> +\begin{lstlisting} +view(st) -> { ... } + +~\colorbox{green}{type State = (Boolean, ?)}~ +~\colorbox{green}{update : (?, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} +\end{footnotesize}} \end{column} +\end{columns} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing is about Program Evolution} +\begin{columns} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<1-> +\begin{lstlisting} +view(st) -> { ... } + +~\colorbox{green}{type State = (Boolean, ?)}~ +~\colorbox{green}{update : (?, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} +\end{footnotesize}} +\end{column} +\begin{column}{0.1\textwidth} +\begin{onlyenv}<2-> + $\Longrightarrow$ +\end{onlyenv} +\end{column} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<2-> +\begin{lstlisting} +view(st) -> { ... } +~\colorbox{green}{type State = (Boolean, Num)}~ +~\colorbox{green}{type Evt = NewData Num }~ + ~\colorbox{green}{| Cancel}~ +~\colorbox{green}{update : (Evt, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Up to date +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} +\end{footnotesize}} +\end{column} +\end{columns} +\begin{onlyenv}<3-> + Code runs at every point, even though only \texttt{update} is typed. +\end{onlyenv} +\end{frame} + +\begin{frame}{Gradual Typing Goals} + Gradually Typed languages let dynamically typed and statically typed + code interoperate within a single language. + \begin{enumerate} + \item Can write scripts, experimental code in the dynamically typed + fragment. + \item Can later add types to enforce interfaces, add documentation, + prevent future bugs. + \item But don't have to add types to the \emph{entire} program to + run it! + \end{enumerate} +\end{frame} + +\begin{frame}{What do the Types Mean?} + How do the types affect the semantics of the program? + Two answers. + \begin{enumerate} + \item Optional Typing: they don't, just run the type erased + programs. Dynamic typing is already safe anyway. + \item Gradual Typing: a type system should give us a type + \emph{soundness} theorem, need to check types at runtime in order + to satisfy that. + \end{enumerate} +\end{frame} + +\begin{frame}{Cast Insertion} + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A \to B \and u : A' \and + A \sim A'} + {\Gamma \vdash t u : B} + \end{mathpar} + elaborates to a term with an explicit cast + \[ t (\obcast{A'}{A}u) \] +\end{frame} + +\begin{frame}{Semantics of Casts} + \begin{center} + \includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio]{Wadler-Findler-operational-semantics} + \end{center} +\end{frame} + +\begin{frame}{Semantics of Casts} + \begin{align*} + \obcast{A}{A}v &\mapsto v\\ + \obcast{A\to B}{A'\times B'}v & \mapsto \err\\ + \obcast{A\to B}{A' \to B'}v &\mapsto \lambda x. \obcast{B}{B'}(v (\obcast{A'}{A}x))\\ + \cdots + \end{align*} +\end{frame} + +\begin{frame}{} + \begin{itemize} + \item Many different semantics of casts + \begin{itemize} + \item For simple types alone: lazy, eager, up-down, partially + eager, transient + \end{itemize} + \item Some seem especially natural: functorial interpretation. + \end{itemize} + What are the underlying principles? + Which of them give us sound typing? +\end{frame} + +\begin{frame}{A Type Theoretic Approach} + \begin{enumerate} + \item Simple axiomatization of the desired properties of types and + casts. + \item \emph{Derive} semantics of casts from these properties. + \item We prove certain cast semantics are the only ones that satisfy + our desired properties. + \end{enumerate} + Caveat: applies directly to the cast calculus, not the surface + gradual language. See Abstracting Gradual Typing for a good + explanation of the surface type checking. +\end{frame} + +\begin{frame}{Properties of Gradual Typing} + \begin{enumerate} + \item \pause Dynamic Typing: Have a dynamic type into which all types can + be encoded. + \item \pause Type Soundness: Types should give associated + reasoning principles: $\beta, \eta$ equality. + \item \pause \emph{Graduality}: Transition from dynamic to static + typing should only add type checking, not otherwise change + behavior. + \end{enumerate} +\end{frame} + +\begin{frame}[fragile]{Graduality} +\begin{columns} +\begin{column}{0.5\textwidth} +\begin{onlyenv}<1-> +\begin{footnotesize} +\begin{lstlisting} +view(st) -> { ... } + +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { .. } +\end{lstlisting} +\end{footnotesize} +\end{onlyenv} +\end{column} +\begin{column}{0.1\textwidth} + \begin{onlyenv}<1> + $\Longrightarrow$ + \end{onlyenv} + \begin{onlyenv}<2-> + $\gtdyn$ +\end{onlyenv} +\end{column} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<1-> +\begin{lstlisting} +view(st) -> { ... } + +~\colorbox{green}{type State = (Boolean, ?)}~ +~\colorbox{green}{update : (?, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} + \end{footnotesize}} +\end{column} +\end{columns} + \begin{onlyenv}<2-> + Adding types \emph{refines} program behavior + \end{onlyenv} +\end{frame} + +\begin{frame}{Graduality} + Syntactically, + \[ t \ltdyn t' \] + if $t$ and $t'$ have the same type erasure, and every type in $t$ is + ``less dynamic'' than the corresponding type in $t'$ + + \pause Operationally, if $t, t' : \texttt{Bool}$, then + \begin{enumerate} + \item $t$ runs to a type error (and $t'$ can do anything) + \item $t$ and $t'$ diverge + \item $t \mapsto \texttt{true}$ and $t' \mapsto \texttt{true}$ + \item $t \mapsto \texttt{false}$ and $t' \mapsto \texttt{false}$ + \end{enumerate} + + \pause + \begin{itemize} + \item Originally called the ``gradual guarantee'' in SNAPL '15 + (Siek-Vitousek-Cimini-Boyland) + \item ``Graduality from Embedding-projection Pairs'' ICFP '18 with + Amal Ahmed for more on the operational semantics POV + \end{itemize} +\end{frame} + +%% \begin{frame}{Evaluation in Gradual Typing} +%% Say we have a function $\lambda x. t : \texttt{bool} \to \texttt{string}$ and +%% we apply it to a dynamically typed value $v : \dyn$. +%% \pause +%% If $v$ is a boolean, fine +%% \[ (\lambda x. t)(\texttt{true}) \mapsto t[\texttt{true}/x] +%% \] +%% \pause +%% Otherwise, we raise an error +%% \[ (\lambda x. t)(5) \mapsto \texttt{type error: expected ...} +%% \] +%% \end{frame} + +%% \begin{frame}{Evaluation in Gradual Typing} +%% What if it's is a higher order function $\lambda x.t : +%% (\texttt{number} \to \texttt{string})\to \texttt{num}$? +%% \pause +%% If $v$ is not a function, we error +%% \[ (\lambda x.t)(\texttt{true}) \mapsto \texttt{type error:...}\] +%% But if $v$ is a function? How can we verify that it takes numbers +%% to strings? +%% \begin{enumerate} +%% \item\pause Run the type checker at runtime :( +%% \item\pause Findler-Felleisen wrapping implementation: +%% \[ (\lambda x.t)(\lambda y. u) \mapsto t[\lambda y. \texttt{string?}(u)/x]\] +%% \item \pause Some combination: Annotate every value with best-known type +%% information, use wrapping if the type information is not precise +%% enough to tell. +%% \end{enumerate} +%% \end{frame} + +%% \begin{frame}{Type Soundness} +%% \begin{enumerate} +%% \item For simple types, usually defined as a safety property: if $t +%% : A$ then $t$ diverges or $t$ runs to one of the allowed errors or +%% $t$ runs to a value that has type $A$. +%% \item \pause But there are also stronger, \emph{relational} soundness +%% theorems +%% \begin{enumerate} +%% \item Function extensionality: $f_1 = f_2 : A \to B$ if for every +%% $x : A$, $f_1(x)$ has the same behavior as $f_2(x)$. +%% \item Universal Polymorphism: $v : \forall \alpha. A$, then +%% $v[B_1]$ and $v[B_2]$ ``do the same thing''. +%% \item Local State/Existential Polymorphism: If I don't expose my +%% representation, I can change it without breaking the program. +%% \end{enumerate} +%% \end{enumerate} +%% \end{frame} + +\section{CBN Gradual Type Theory} + +\begin{frame}{Call-by-name Gradual Type Theory} + Start simple: $\lambda$ calculus with only negative types $\to, + \times, 1$. Since effects are involved this is call-by-name + evaluation. \\ + + ``Call-by-name Gradual Type Theory'', FSCD 2018 with Dan Licata. +\end{frame} + +\subsection{Syntax of GTT} + +\begin{frame}{Syntax of Gradual Type Theory} + 4 sorts in Gradual Type Theory + \begin{enumerate} + \item Types $A,B,C$ + \item Terms $x_1 : A_1,\ldots, x_n:A_n \vdash t : A$ + \item \pause Type Dynamism: $A \ltdyn B$ + \item Term Dynamism: $x_1 \ltdyn y_1 : A_1 \ltdyn B_1,\ldots \vdash t \ltdyn u : A \ltdyn B$ + \end{enumerate} +\end{frame} + +\begin{frame}{Types and Terms} + Add dynamic type, casts, error to typed lambda calculus. + \begin{mathpar} + \begin{array}{rrl} + \text{types} & A,B,C \bnfdef & \colorbox{lightgray}{$\dyn$} \bnfalt A \to B \bnfalt A \times B \bnfalt 1\\ + \text{terms} & s,t,u \bnfdef& x \bnfalt \lambda (x : A). t \bnfalt t u\\&& + \bnfalt (t, u) \bnfalt \pi_i t \bnfalt ()\\&& + \bnfalt \colorbox{lightgray}{$\err_{A}$} \bnfalt \colorbox{lightgray}{$\upcast A B t$} \bnfalt \colorbox{lightgray}{$\dncast A B t$} + \end{array} + \end{mathpar} +\end{frame} +%% \begin{frame}{Terms} +%% Terms are all intrinsically typed, we have variables and +%% substitution as usual. +%% %% and we can also add function symbols as +%% %% primitives. + +%% \begin{mathpar} +%% \inferrule +%% {x:A \in \Gamma} +%% {\Gamma \vdash x : A} + +%% \inferrule +%% {\Gamma \vdash u : B\and +%% \Gamma, x: B \vdash t : A} +%% {\Gamma \vdash t[u/x] : A} + +%% \inferrule +%% {\Gamma \vdash t_1 : A_1,\ldots, \Gamma \vdash t_n : A_n \and f : (A_1,\ldots,A_n) \to B} +%% {\Gamma \vdash f(t_1,\ldots,t_n) : B} +%% \end{mathpar} +%% \end{frame} + +\begin{frame}{Type Dynamism} + \[ A \ltdyn B \] + read: $A$ is ``less dynamic''/''more precise'' than $B$. \\ + \begin{itemize} + \item Syntactic intuition: $B$ has more $\dyn$s in it. + \item Semantic intuition: $A$ supports fewer interactions than $B$. + \end{itemize} +\end{frame} + +\begin{frame}{Type Dynamism Axioms} + \begin{mathpar} + \inferrule{~}{A \dynr A}\and + \inferrule{A \dynr B \and B \dynr C}{A \dynr C}\and + \inferrule{~}{A \ltdyn \dyn}\\ + \inferrule{A \ltdyn A' \and B \ltdyn B'}{A \to B \ltdyn A' \to B'}\and + \inferrule{A \ltdyn A' \and B \ltdyn B'}{A \times B \ltdyn A' \times B'} + \end{mathpar} + NB: Covariant function rule +\end{frame} + +%% \begin{frame}{Term Dynamism INTUITION} +%% Models the refinement relation of gradual guarantee. Here's an example +%% \emph{model} for intuition.\\ +%% \pause +%% For every $A \ltdyn A'$, if +%% \begin{mathpar} +%% \cdot \vdash t \ltdyn t' : A \ltdyn A' +%% \end{mathpar} +%% Then either +%% \begin{enumerate} +%% \item $t$ runs to a type error. +%% \item $t,t'$ both diverge +%% \item $t \Downarrow v$ and $t' \Downarrow v'$ and $v \ltdyn v'$. +%% \end{enumerate} + +%% \notepr{ +%% The $A \ltdyn A'$ restriction ensures we have a sensible +%% way to interpret that. +%% } +%% \end{frame} + +\begin{frame}{Term Dynamism} + \begin{mathpar} + x_1 \ltdyn x_1' : A_1 \ltdyn A_1' ,\ldots \vdash t \ltdyn t' : B \ltdyn B' + \end{mathpar} + read: ``if all the $x_i$s are less than $x_i'$s then $t$ is less than $t'$''.\\ + \pause + Note that it is only well formed if all of the $A_i \ltdyn A_i'$ and + $B \ltdyn B'$. and + \begin{mathpar} + x_1:A_1,\ldots\vdash t : B\\ + x_1':A_1',\ldots\vdash t' : B'\\ + \end{mathpar} + %% We sometimes abbreviate the context ordering as $\Phi \vdash t + %% \ltdyn t' : B \ltdyn B'$. + \pause + This very explicit notation gets very big, so I'll simplify in the + talk. Abbreviate the above as: + \[ + \Phi \vdash t \ltdyn t' : B \ltdyn B' + \] +\end{frame} + +\begin{frame}{Type Error} + The type error is the \emph{least} dynamic term, modeling + graduality. + + \[ \Phi \vdash \err_A \ltdyn t : A \ltdyn A' \] +\end{frame} + +\begin{frame}{Term Dynamism Axioms} + Identity and substitution are monotone. + %% For open terms, identity and substitution properties that should + %% evoke a logical relation interpretation. Substitution says that + %% every term is monotone in its free variables. + \begin{mathpar} + \inferrule + {(x \ltdyn x' : A \ltdyn A') \in \Phi} + {\Phi \vdash x \ltdyn x' : A \ltdyn A'} + + \inferrule + {x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' \and + u \ltdyn u' : A \ltdyn A'} + {t[u/x] \ltdyn t'[u'/x']} + \end{mathpar} + %% \pause Note that these are the same as ordinary identity and + %% substitution but ``monotonically''. +\end{frame} + +\begin{frame}{Term Dynamism} + Like type dynamism, term dynamism has reflexivity and transitivity + rules: + %% note + %% how well-formedness relies on the reflexivity and transitivity of + %% type dynamism + + \begin{mathpar} + \inferrule + {\vdash t : B} + {t \ltdyn t : B \ltdyn B} + + \inferrule + {t \ltdyn t' : B \ltdyn B'\\ + t' \ltdyn t'' : B' \ltdyn B''} + {t \ltdyn t'' : B \ltdyn B''} + \end{mathpar} +\end{frame} + +%% \begin{frame}[fragile]{Term Dynamism Squares} +%% These can be nicely visualized as squares: +%% For $\Phi : \Gamma \ltdyn \Gamma'$, we can think of +%% \[ \Phi \vdash t \ltdyn t' : A \ltdyn A' \] +%% as +%% \[ +%% \begin{tikzcd} +%% \Gamma \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "t" ] +%% & +%% \Gamma' \arrow[d , "t'"] +%% \\ +%% A \arrow[ dashed , r, "\preciser" ] +%% & +%% A' +%% \end{tikzcd} +%% \] + +%% \end{frame} + +\begin{frame}[fragile]{Graphical View of Dynamism} + Can help to visualize these as \emph{squares}. + \[ + x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' : B \ltdyn B' + \] + + becomes + \begin{tikzcd} + x:A \arrow[dd] \arrow[rr, "t"] & & B \arrow[dd] \\ + & \sqsubseteq & \\ + x':A' \arrow[rr, "t'"] & & B' + \end{tikzcd} +\end{frame} + +\begin{frame}[fragile]{Graphical View of Dynamism} + Substitution principle/monotonicity corresponds to horizontal + composition of squares. + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + A \arrow[r,"u"]\arrow[d] & B \arrow[r,"t"]\arrow[d] & C\arrow[d]\\ + A' \arrow[r,"u'"] & B'\arrow[r,"t'"]& C + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + A \arrow[d] \arrow[r, "id"] & A \arrow[d] \\ + B \arrow[r, "id"] & B + \end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule{x \ltdyn x' \vdash t \ltdyn t'\\ u \ltdyn u'} + {t[u/x] \ltdyn t'[u'/x]} + \end{mathpar} + \end{column} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule + {} + {x\ltdyn x' : A \ltdyn A' \vdash x \ltdyn x' : A \ltdyn A'} + \end{mathpar} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{Graphical View of Dynamism} + Vertical composition of squares is \emph{transitivity and reflexivity}. + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} +\Gamma \arrow[d] \arrow[r,"t"] & A \arrow[d] \\ +\Gamma' \arrow[d] \arrow[r,"t'"] & A' \arrow[d] \\ +\Gamma'' \arrow[r,"t''"] & A'' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \begin{tikzcd} +A \arrow[d, "id"] \arrow[r, "t"] & B \arrow[d, "id"] \\ +A \arrow[r, "t"] & B +\end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule{\Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A'\\ + \Gamma' \ltdyn \Gamma'' \vdash t' \ltdyn t'' : A' \ltdyn A''} + {\Gamma \ltdyn \Gamma'' \vdash t \ltdyn t'' : A \ltdyn A''} + \end{mathpar} + \end{column} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A} + {\Gamma \ltdyn \Gamma \vdash t \ltdyn t : A \ltdyn A} + \end{mathpar} + \end{column} + \end{columns} + \pause Aside: Squares can be thought of as 2-cells in a double + category, which was the inspiration for the whole approach. +\end{frame} + +\begin{frame}{Function Type} + First, we give have the usual term structure for the function type + \begin{mathpar} + \inferrule + {\Gamma, x : A \vdash t : B} + {\Gamma \vdash \lambda x:A. t : A \to B} + + \inferrule + {\Gamma \vdash t : A \to B\and \Gamma \vdash u : A} + {\Gamma \vdash t u : B} + \end{mathpar} + \pause + And we add the $\beta, \eta$ laws as ``equi-dynamism'' axioms: + + \begin{mathpar} + \inferrule + {} + {(\lambda x:A. t) u \equidyn t[u/x]} + + \inferrule + {t : A \to B} + {t \equidyn (\lambda x:A. t x)} + \end{mathpar} +\end{frame} + +\begin{frame}{Function Type} + Next to model graduality, we add \emph{congruence} rules. + \begin{mathpar} + \inferrule + {A \ltdyn A' \and B \ltdyn B'} + {A \to B \ltdyn A' \to B'}\\ + \inferrule + {x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' : B \ltdyn B'} + {\lambda x:A. t \ltdyn \lambda x':A'.t' } + + \inferrule + {t \ltdyn t' : A \to B \ltdyn A' \to B'\and + u \ltdyn u' : A \ltdyn A' %% \and B \ltdyn B' + } + {t\, u \ltdyn t'\, u' : B \ltdyn B'} + \end{mathpar} +\end{frame} + +\begin{frame}{Product Type} + Product and (unit types) get a similar treatment. + \begin{mathpar} + \inferrule + {\Gamma \vdash t_1 : A_1\\ \Gamma \vdash t_2 : A_2} + {\Gamma \vdash (t_1,t_2) : A_1 \times A_2} + + \inferrule + {\Gamma \vdash t : A_1 \times A_2} + {\Gamma \vdash \pi_i t : A_i}\\ + + \inferrule + {} + {\pi_i(t_1,t_2) \equidyn t_i} + + \inferrule + {t : A_1 \times A_2} + {t \equidyn (\pi_1 t, \pi_2 t)}\\ + + \inferrule{t_1 \ltdyn t_1' : A_1 \ltdyn A_1' \and t_2 \ltdyn t_2' : A_2 \ltdyn A_2'}{(t_1,t_1') \ltdyn (t_2,t_2') : A_1 \times A_2 \ltdyn A_1' \times A_2'} + + \inferrule{t \ltdyn t' : A_1 \times A_2 \ltdyn A_1' \times A_2' \and A_i \ltdyn A_i'}{\pi_i t \ltdyn \pi_i t' : A_i \ltdyn A_i'} + \end{mathpar} +\end{frame} + +\begin{frame}{Casts} + How to define the casts axiomatically? + \begin{itemize} + \item We \emph{characterize} them by a property with respect to $\ltdyn$. + \item We can do this when our casts are \emph{upcasts} or + \emph{downcasts}. + \end{itemize} +\end{frame} + +\begin{frame}{Casts} + When $A \ltdyn B$, we get two functions: the \emph{upcast} $\upcast + A B$ and the \emph{downcast} $\dncast A B$. + + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A \and A \ltdyn B} + {\Gamma \vdash \upcast A B t : B} + + \inferrule + {\Gamma \vdash u : B \and A \ltdyn B} + {\Gamma \vdash \dncast A B u : A} + \end{mathpar} +\end{frame} + +\begin{frame}{Oblique Casts} + Not every cast in gradual typing is an upcast or a downcast, for + instance the cast + + \[ \obcast {\texttt{Bool} \times \dyn} {\dyn \times \texttt{Bool}} \] + + But in almost every gradual system, these are equivalent to casting + through the dynamic type: + + \[ \obcast{A}{B}t = \dncast B \dyn \upcast A \dyn t\] +\end{frame} + +\begin{frame}{Casts} + How should casts behave? + %% Intuition: the casted term should act as much like the original term + %% as possible while still satisfying the new type. + \pause + + So for a downcast, the casted term should refine the original: + %% ``Elimination Rule'' + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\dncast A B u \ltdyn u : A \ltdyn B} + \end{mathpar} + + \pause But this would be satisfied even by the type error $\err_A$. + \pause + We want to make sure that the casted term has \emph{as much} + behavior as possible: i.e., it's the \emph{biggest} term below + $u$. + \begin{mathpar} + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash t \ltdyn \dncast A B u : A} + \end{mathpar} + \pause Order-theoretic summary: $\dncast A B u$ is the greatest + lower bound of $u$ in $A$. +\end{frame} + +\begin{frame}{Casts} + Upcasts are the exact dual + + For an upcast, the casted term should be \emph{more} dynamic than + the original: + \begin{mathpar} + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \upcast A B t : A \ltdyn B} + \end{mathpar} + + But it has no more behaviors than dictated by the original term, + it's the \emph{least} term above $t$. + + \begin{mathpar} + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash \upcast A B t \ltdyn u : B} + \end{mathpar} + \pause Order-theoretic summary: $\upcast A B u$ is the least upper + bound of $t$ in $B$. +\end{frame} + +\begin{frame}{Casts} + \begin{enumerate} + \item Note that this is a \emph{precise} specification because lubs + and glbs are unique (up to $\equidyn$). + \item Category theoretic terminology: this is a \emph{universal + property} for upcasts and downcasts. + \item Since they are exact \emph{duals} most theorems about one + dualize for the other. + \end{enumerate} +\end{frame} + +\subsection{Theorems in GTT} + +\begin{frame}{Consequences of GTT} + \begin{itemize} + \item Now we have a simple ``logic'' of term dynamism, what can we + \emph{prove} in the theory? + \item Theorems in the theory apply to any model, i.e. a simple lazy + gradually typed language. + \end{itemize} +\end{frame} + +\begin{frame}{Derivation of Function Casts} + We can prove that the ``function contract'' is the unique + implementation of casts between function types. I.e. if $A \ltdyn + A'$ and $B \ltdyn B'$ and $t : A' \to B'$ then + + \[ + \dncast {A \to B} {A' \to B'} t \equidyn \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + and + \[ + \upcast {A \to B} {A' \to B'} t \equidyn \lambda x':A'. \upcast{B}{B'}(t (\dncast A {A'} x')) + \] +\end{frame} + +\begin{frame}{Derivation of Function Casts} + One direction: + To prove + \[ \dncast {A \to B} {A' \to B'} t \ltdyn \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + By $\eta$, it is sufficient to show + \[ + x : A \vdash (\dncast {A \to B} {A' \to B'} t) x \ltdyn \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + Then it follows by congruence because: + \begin{enumerate} + \item $(\dncast {A \to B} {A' \to B'} t)\ltdyn t$ + \item $x \ltdyn \upcast A {A'} x$ + \item if $u \ltdyn u'$ then $u \ltdyn \dncast {B} {B'} u'$ + \end{enumerate} + \pause + Other 3 cases are similar/dual. + \pause + All just $\beta,\eta$ and congruence. Applies to any type defined + this way: $\times$, $1$. +\end{frame} + +\begin{frame}{Casts are Galois Connections} + \notepr{ First, we can show that every pair of upcast and downcast forms a + Galois connection, which} + + Two ``round-trip'' properties: + \pause + Casting down and back up has ``more errors'' + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\upcast A B \dncast A B u \ltdyn u : B} + \end{mathpar} + \pause + And casting up and back down has ``fewer errors'' + \begin{mathpar} + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \dncast A B \upcast A B t : A} + \end{mathpar} + \pause In practice, we strengthen this to an + equality, so we can add the ``retract axiom'' as well to the system + (though none of the theorems I'll show depend on it): + \begin{mathpar} + \inferrule*[right=RETRACT-AXIOM] + {t : A \and A \ltdyn B} + {\dncast A B \upcast A B t \ltdyn t : A} + \end{mathpar} + but only need to add it as an axiom for ``base casts'' +\end{frame} + +\begin{frame}{Identity and Composition} + Reflexivity gives the identity cast and transitivity gives the + composition of casts + \begin{enumerate} + \item $\dncast A A t \equidyn t$ + \item If $A \ltdyn B \ltdyn C$, then $\dncast A C t \equidyn \dncast A B \dncast B C t$ + \end{enumerate} + \pause + These are part of the operational semantics for instance: + \begin{align*} + \dncast{\dyn}{\dyn} v &\mapsto v\\ + \dncast{A \to B}{\dyn} v &\mapsto \dncast {A \to B}{\dyn \to \dyn}\dncast {\dyn \to \dyn} \dyn v + \end{align*} +\end{frame} + +%% \begin{frame}{Identity Cast} +%% The upcast/downcast from $A$ to $A$ is the identity function, +%% i.e. $\upcast A A t \equidyn t$. Each direction is an instance of +%% the specification of upcasts: + +%% \begin{mathpar} +%% \inferrule*[right=UR] +%% {~} +%% {t \ltdyn \upcast{A}{A}{t} : A}\and + +%% \inferrule*[right=UL] +%% {t \ltdyn t : A \ltdyn A} +%% {\upcast{A}{A}{t} \ltdyn t : A} +%% \end{mathpar} + +%% Downcast proof is precise dual. +%% \end{frame} + +%% \begin{frame}{Casts (de)compose} +%% If $A \ltdyn B \ltdyn C$ then a cast between $A$ and $B$ factors +%% through $C$: + +%% \begin{mathpar} +%% \inferrule +%% {A \ltdyn B \ltdyn C} +%% {\dncast A B \dncast B C t \equidyn \dncast A C t} +%% \end{mathpar} + +%% Example: casts between dynamic type and function types factor +%% through ``most dynamic'' function type: +%% \begin{mathpar} +%% \dncast {A \to B}{\dyn \to \dyn}\dncast \dyn {\dyn \to \dyn} t \equidyn \dncast{A \to B}{\dyn} t +%% \end{mathpar} +%% This is often a rule of the operational semantics. +%% \end{frame} + +\begin{frame}{Non-theorem: Disjointness of Constructors} + The only standard operational rules we cannot prove are the ones like + \[ + \obcast {A \times B} {A' \to B'} t \mapsto \err + \] + that say that the type constructors $\times, \to, 1$ are all + disjoint. + + Independent of the axioms: have models where it's true and + false. This is the actual ``design space''. + + %% In fact, written as equidynamism instead of stepping, these are + %% \emph{independent} of the other axioms: we can construct adequate + %% models where it's true and ones where it's false(!) + + %% The reason is that our definition of function types, product types + %% et cetera only defines them \emph{up to isomorphism}, not up to cast + %% equivalence. And we know very little about the dynamic type, just + %% that everything embeds in it. +\end{frame} + +%% \begin{frame}{Term Dynamism Reduces to Globular Dynamism} +%% The presence of the casts means that we can reduce all type dynamism +%% to ordering \emph{at the same type}, which is more typical for +%% logical relations. +%% \begin{mathpar} +%% {x \ltdyn y : A \ltdyn B \vdash t \ltdyn u : C \ltdyn D}\\ +%% \iff\\ +%% {x \ltdyn x : A \ltdyn A \vdash t \ltdyn \dncast C D u[\upcast A B x/y] : C \ltdyn C}\\ +%% \iff\\ +%% {x \ltdyn x : A \ltdyn A \vdash \upcast C D t \ltdyn u[\upcast A B x/y] : D \ltdyn D}\\ +%% \iff\\ +%% {y \ltdyn y : B \ltdyn B \vdash \upcast C D t[\dncast A B y/x] \ltdyn u : D \ltdyn D}\\ +%% \iff\\ +%% {y \ltdyn y : B \ltdyn B \vdash t[\dncast A B y/x] \ltdyn \dncast C D u : C \ltdyn C}\\ +%% \end{mathpar} +%% \end{frame} + +\subsection{Models of GTT} + +\begin{frame}{Models of GTT} + \begin{enumerate} + \item In FSCD Paper, + \begin{enumerate} + \item Sound and complete category theoretic semantics using double + categories. + \item Adequate domain theoretic model where domains have domain + structure and error ordering modeling $\ltdyn$. + \end{enumerate} + \end{enumerate} +\end{frame} + +\begin{frame}{Summary} + \begin{enumerate} + \item Can derive that the functorial interpretation of casts is the + only one that satisfies graduality and $\beta, \eta$. + \item In practice, other cast semantics break $\eta$, breaking + type-based reasoning! + \end{enumerate} +\end{frame} + +\section{CBPV Gradual Type Theory} + +\subsection{Overview of Call-by-push-value} + +\begin{frame}{Call-by-push-value GTT} + Want to extend theorems to CBV, but still want a simple axiomatic + presentation of the type theory that works with positives, negatives + and effects. + + Answer is a polarized type theory: Call-by-push-value.\\ + + Conditionally accepted to POPL 2018: ``Gradual Type Theory'' with + Dan Licata and Amal Ahmed. +\end{frame} + +\begin{frame}{Types, Judgments of CBPV} + \begin{mathpar} + \begin{array}{rl} + \text{value types}\,\,A ::= & \colorbox{lightgray}{$\dynv$} \mid U \u B \mid 0 \mid A_1 + A_2 \mid 1 \mid A_1 \times A_2\\ + \text{computation types}\,\,\u B ::= & \colorbox{lightgray}{$\dync$} \mid \u F A \mid \top \mid \u B_1 \with \u B_2 \mid A \to \u B\\ + \text{types}\,\, T ::= & A \mid \u B\\ + \Gamma ::= & x:A,\ldots\\ + \Delta ::= & \bullet : \u B\\ + \end{array} + \end{mathpar} + Three judgments + \begin{enumerate} + \item Computations $\Gamma \vdash M : \u B$: effectful terms + \item Values $\Gamma \vdash V : A$: pure function that returns an $A$ + \item Stacks $\Gamma \pipe \bullet:B \vdash S : B'$, strict function + that forces the hole $\bullet : B$ whenever it is forced. + \end{enumerate} +\end{frame} + +\begin{frame}{Functions} + \begin{mathpar} + \inferrule + {\Gamma, x : A \vdash M : B} + {\Gamma \vdash \lambda x. M : A \to B} + + \inferrule + {\Gamma \vdash M : A \to B \and + \Gamma \vdash V : A} + {\Gamma \vdash M V : B} + + (\lambda x. M) V \equiv M[V/x]\\ + (M : A \to B) \equiv \lambda x. M x\\ + + \pause + \inferrule + {\Gamma, x: A \pipe \Delta \vdash S : B } + {\Gamma \pipe \Delta \vdash \lambda x. S : A \to B} + + \inferrule + {\Gamma \pipe \Delta \vdash S : A \to B\and + \Gamma \vdash V : A} + {\Gamma \pipe \Delta \vdash S V : B} + \end{mathpar} +\end{frame} + +\begin{frame}{Strict Pairs} + \begin{mathpar} + \inferrule + {\Gamma \vdash V_i : A_i} + {\Gamma \vdash (V_1,V_2) : A_1\times A_2} + + \inferrule + {\Gamma \vdash V : A_1\times A_2\and + \Gamma, x_1:A_1,x_2:A_2 \vdash E : T} + {\Gamma \vdash \pmpairWtoXYinZ V {x_1} {x_2} E : T} + + \pmpairWtoXYinZ {(V_1,V_2)} {x_1} {x_2} E \equiv E[V_1/x_1][V_2/x_2]\\ + x : A_1\times A_2 \vdash E \equiv \pmpairWtoXYinZ x {x_1}{x_2} E[(x_1,x_2)/x] + \end{mathpar} +\end{frame} + +\begin{frame}{Shifts} + A computation of type $\u F A$ is something that performs effects + and returns $A$s. + \begin{mathpar} + \inferrule + {\Gamma \vdash V : A} + {\Gamma \vdash \ret V : \u F A} + + \inferrule + {\Gamma \vdash M : \u F A\and + \Gamma, x : A \vdash N : \u B} + {\Gamma \vdash \bindXtoYinZ M x N : \u B} + + \inferrule + {\Gamma\pipe\Delta \vdash S : \u F A\and + \Gamma, x : A \vdash N : \u B} + {\Gamma \vdash \bindXtoYinZ S x N : \u B} + + \bindXtoYinZ {\ret V} x N \equiv N[V/x]\\ + \bullet : \u F A \vdash S \equiv \bindXtoYinZ \bullet x S[\ret x/\bullet] + \end{mathpar} +\end{frame} + +\begin{frame}{Shifts} + A value of type $U \u B$ is a thunk that when forced behaves as $\u B$. + \begin{mathpar} + \inferrule + {\Gamma \vdash M : \u B} + {\Gamma \vdash \thunk M : U \u B} + + \inferrule + {\Gamma \vdash V : U \u B} + {\Gamma \vdash \force V : \u B}\\ + + \force \thunk M \equiv M\\ + V : U \u B \equiv \thunk \force V + \end{mathpar} +\end{frame} + +\begin{frame}{Embedding CBV, CBN} + CBV types become value types: + \begin{align*} + (A \mathop{+_{cbv}} A') &= A + A'\\ + (A \mathop{\to_{cbv}} A') &= U(A \to \u F A') + \end{align*} + And CBV computations: + \[ \Gamma \mathrel{\vdash^{cbv}} M : A \Rightarrow \Gamma \vdash M : \u F A \] + CBN types become computation types: + \begin{align*} + (\u B \mathop{+_{cbn}} \u B') &= \u F (U\u B + U \u B')\\ + (\u B \mathop{\to_{cbn}} \u B') &= U\u B \to \u B' + \end{align*} + And CBN computations: + \[ + x_1:\u B_1,\ldots \vdash M : \u B \Rightarrow x_1:U\u B_1,\ldots \vdash M : \u B + \] +\end{frame} + +\subsection{Casts and Dynamic Types} + +\begin{frame}{CBPV Gradual Type Theory} + Goal: gradualize CBPV. Should reproduce our CBN work and the + existing CBV semantics. + + \begin{enumerate} + \item Add dynamism ordering $\ltdyn$. + \item Define casts by universal property. + \item Add dynamic type(s?) + \end{enumerate} +\end{frame} + +\begin{frame}{Dynamism Ordering} + Easy: new ordering judgment for types and values/stacks/computations. + Everything is monotone. + + \begin{mathpar} + \inferrule + {A \ltdyn A'} + {\u F A \ltdyn \u F A'} + + \inferrule + {A_1 \ltdyn A_1' \and A_2 \ltdyn A_2'} + {A_1 \times A_2 \ltdyn A_1' \times A_2'}\\ + + \inferrule + {\u B \ltdyn \u B'} + {U \u B \ltdyn U \u B'} + + \inferrule + {A \ltdyn A' \and \u B \ltdyn \u B'} + {A \to \u B \ltdyn A' \to \u B'} + \end{mathpar} +\end{frame} + +\begin{frame}{Errors} + \begin{mathpar} + \Gamma \vdash \err : \u B\\ + + \inferrule + {} + {\Gamma \vdash \err \ltdyn M : \u B} + + \inferrule + {} + {\Gamma \vdash S[\err/\bullet] \equidyn \err : \u B} + \end{mathpar} +\end{frame} + +\begin{frame}{Which casts?} + If we embed from cbv, cbn. we get ``Kleisli'' and ``CoKleisli'' + casts. + + From CBV: + \begin{mathpar} + \inferrule + {\Gamma \vdash V : A \and A \ltdyn A'} + {\Gamma \vdash \upcast{A}{A'}V : \u F A'} + + \inferrule + {\Gamma \vdash V : A' \and A \ltdyn A'} + {\Gamma \vdash \dncast{A}{A'}V : \u F A} + \end{mathpar} + + From CBN: + \begin{mathpar} + \inferrule + {\Gamma \vdash V : U \u B \and \u B \ltdyn \u B'} + {\Gamma \vdash \upcast{\u B}{\u B'}V : \u B'} + + \inferrule + {\Gamma \vdash V : U \u B' \and \u B \ltdyn \u B'} + {\Gamma \vdash \dncast{\u B}{\u B'}V : \u B} + \end{mathpar} +\end{frame} + +\begin{frame}{Embeddings are Values, Projections are Stacks} + Observation from CBV: in practice the upcasts are ``pure'': the + basic ones are sum type injections, others are functorial liftings + of pure functions. + + CBPV lets us axiomatize this directly, along with the \emph{dual} + that downcasts \emph{stacks} between computation types: + \begin{mathpar} + \inferrule + {\Gamma \vdash V : A} + {\Gamma \vdash \upcast{A}{A'}V : A'} + + %% \inferrule + %% {\Gamma \vdash M : \u B} + %% {\Gamma \vdash \dncast{\u B}{\u B'}M : \u B'} + + \inferrule + {\Gamma\pipe \Delta \vdash S : \u B} + {\Gamma\pipe \Delta \vdash \dncast{\u B}{\u B'}S : \u B'} + \end{mathpar} + Should be useful for compilation/optimization. +\end{frame} + +\begin{frame}{Embeddings are Values, Projections are Stacks} + ``Missing'' casts come from the shifts: + \begin{mathpar} + \inferrule + {\Gamma \vdash V : A \and A \ltdyn A'} + {\Gamma \vdash \upcast{A}{A'}V : A'} + + %% \inferrule + %% {\Gamma \vdash M : \u B} + %% {\Gamma \vdash \dncast{\u B}{\u B'}M : \u B'} + + \inferrule + {\Gamma\pipe \Delta \vdash S : \u B \and \u B \ltdyn \u B'} + {\Gamma\pipe \Delta \vdash \dncast{\u B}{\u B'}S : \u B'} + + \inferrule + {\Gamma \vdash M : \u F A \and \inferrule{A \ltdyn A'}{\u F A \ltdyn \u F A'}} + {\Gamma \vdash \dncast{\u F A}{\u F A'}M : \u F A} + + \inferrule + {\Gamma \vdash V : U \u B \and + \inferrule{\u B \ltdyn \u B'}{U \u B \ltdyn U \u B'}} + {\Gamma \vdash \upcast{U \u B}{U \u B'}V : U\u B'} + \end{mathpar} +\end{frame} + +\begin{frame}{Cast Universal Properties} + Just like for CBN: + \begin{mathpar} + \inferrule + {V : A \and A \ltdyn A'} + {V \ltdyn \upcast A {A'} V : A \ltdyn A'} + + \inferrule + {V \ltdyn V' : A \ltdyn A'} + {\upcast A {A'} V \ltdyn V' : A'} + + \inferrule + {M : \u B \and \u B \ltdyn \u B'} + {\dncast{\u B}{\u B'}M \ltdyn M} + + \inferrule + {M \ltdyn M' : \u B \ltdyn \u B'} + {M \ltdyn \dncast {\u B}{\u B'} M' : \u B} + \end{mathpar} +\end{frame} + +\begin{frame}{Strict Product Cast} + Derivable: + \begin{mathpar} + \upcast{A_1 \times A_2}{A_1'\times A_2'}V \equidyn\\ + \pmpairWtoXYinZ V {x_1}{x_2} (\upcast{A_1}{A_1'}x_1, \upcast{A_2}{A_2'}x_2)\\ + \end{mathpar} + + \begin{mathpar} + + \dncast{\u F (A_1' \times A_2')}{\u F (A_1 \times A_2)}{M}\\ + \begin{array}{rcl} +& + \equidyn & + \bindXtoYinZ{M}{p'} {\pmpairWtoXYinZ{p'}{x_1'}{x_2'}{}}\\ + & & \bindXtoYinZ{\dncast{\u F A_1}{\u F A_1'}{\ret x_1'}}{x_1}\\ + & & \bindXtoYinZ{\dncast{\u F A_2}{\u F A_2'}{\ret x_2'}}{x_2} {\ret (x_1,x_2) }\\ + \end{array} + \end{mathpar} + Note: evaluation order of the two casts doesn't matter. +\end{frame} + +\begin{frame}{CBV and CBN Uniqueness Principles} + \begin{mathpar} + \dncast{A \to \u B}{A' \to \u B'}M \equidyn + \lambda x. \dncast{\u B}{\u B'}(M (\upcast{A}{A'}x)) + \end{mathpar} + \begin{mathpar} + \upcast{U (A \to \u B)}{U (A' \to \u B')}{f} \equidyn + \begin{array}{ll} + \thunk (\lambda x'. & \bindXtoYinZ{\dncast{\u F A}{\u F A'}{(\ret x')}}{x}{} \\ + & { \force{(\upcast{U \u B}{U \u B'}{(\thunk{(\force{f}\,x)})})}}) + \end{array} + \end{mathpar} +\end{frame} + +\begin{frame}{Two Dynamic Types} + Add a dynamic value type $\dynv$ and a dynamic computation type + $\dync$. + + \begin{mathpar} + \inferrule + {} + {A \ltdyn \dynv} + + \inferrule + {} + {\u B \ltdyn \dync} + \end{mathpar} + + So we always have casts: + \begin{mathpar} + \inferrule{V : A}{\upcast{A}{\dynv}V : \dynv}\and + \inferrule{M : \dync}{\dncast{\u B}{\dync}M : \u B} + \end{mathpar} +\end{frame} + +%% \begin{frame}{Translation to CBPV} +%% A model of GTT in CBPV +%% \begin{enumerate} +%% \item A gradual value type $A$ denotes a pair of a type $[A]$ and an +%% interpretation of the upcast $\upcast{[A]}{[\dynv]}$ and the +%% downcast $\dncast{\u F [A]}\u F{\dynv}$ that form an ep pair. +%% \item A gradual computation type $\u B$ is a pair of type $[\u B]$ +%% and interpretations of $\dncast{\u B}{\dync}$ and $\upcast{U \u +%% B}{U\u \dync}$ that form an ep pair. +%% \item $A \ltdyn A'$ is modeled by an ep pair $\upcast{[A]}{[A']}$ +%% and $\dncast{\u F [A]}{\u F [A']}$ that factorizes the casts for +%% $A, A'$. +%% \end{enumerate} +%% \end{frame} + +\begin{frame}{Implementing Two Dynamic Types} + Can use CBPV as a metalanguage to define models of CBPV GTT. + \pause + \begin{mathpar} + \dynv \cong (1+1) + U\dync + (\dynv \times \dynv)\\ + \dync \cong (\dynv \to \u F \dync) + \end{mathpar} + + $\dynv$: S-expressions.\\ + + $\dync$: supports call-by-value single argument functions as the + only computation type, similar to ML. +\end{frame} + +\begin{frame}{Implementing Two Dynamic Types} + Can use CBPV as a metalanguage to define models of CBPV GTT. + \begin{mathpar} + \dynv \cong (1+1) + U\dync + (\dynv \times \dynv)\\ + \dync \cong (\dynv \to \dync) \with \u F \dynv + \end{mathpar} + + $\dynv$: S-expressions.\\ + + $\dync$: Supports Scheme's multi-argument and variable-arity + functions. +\end{frame} + +\section{Conclusion} + +\begin{frame}{Summary} + \begin{enumerate} + \item Deconstructed gradual typing to basic axioms + \item By axiomatizing term precision/dynamism, we can give casts a + specification. + \item Graduality + $\beta\eta$ = functor interpretation of casts, so + all other cast semantics sacrifice one or the other. + \item Polarized Languages make effectful equations as easy as + non-effectful. + \end{enumerate} +\end{frame} + +\begin{frame}{Conclusion} + \begin{itemize} + \item Type theoretic approach justifies the ``most canonical'' + version of gradual typing. Can other semantics be similarly + justified? + \item Plays nicely with good type theories: what to gradualize next? + Polarized session types? + \item Papers: FSCD '18, ICFP'18, POPL'19 + \end{itemize} + + \pause + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\dncast A B u \ltdyn u : A \ltdyn B} + + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash t \ltdyn \dncast A B u : A}\\ + + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \upcast A B t : A \ltdyn B} + + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash \upcast A B t \ltdyn u : B} + \end{mathpar} + +\end{frame} +\end{document} + +%% Local Variables: +%% compile-command: "pdflatex talk.tex" +%% End: diff --git a/talk/cmu-fall-2018/fscd-talk.tex b/talk/cmu-fall-2018/fscd-talk.tex new file mode 100644 index 0000000..224583f --- /dev/null +++ b/talk/cmu-fall-2018/fscd-talk.tex @@ -0,0 +1,797 @@ +\documentclass{beamer} +\usetheme{boxes} + +\newcommand{\notepr}[1]{} + +\usepackage{rotating} +\usepackage{listings} +\usepackage{xcolor} +%% \lstset{ +%% basicstyle=\ttfamily, +%% escapeinside=~~ +%% } + +\usepackage{microtype} +\usepackage{latexsym,amsmath,amssymb,amsthm} +\usepackage{mathpartir} +\usepackage{tikz} +\usepackage{tikz-cd} +\input{talk-defs} +\newcommand{\ltdyn}{\dynr} +\newcommand{\uarrow}{\mathrel{\rotatebox[origin=c]{-30}{$\leftarrowtail$}}} +\newcommand{\darrow}{\mathrel{\rotatebox[origin=c]{30}{$\twoheadleftarrow$}}} +\renewcommand{\upcast}[2]{\langle{#2}\uarrow{#1}\rangle} +\renewcommand{\dncast}[2]{\langle{#1}\darrow{#2}\rangle} +\newcommand{\gtdyn}{\sqsupseteq} + +\newcommand{\converge}{\Downarrow} +\newcommand{\diverge}{\Uparrow} + +\title{Call-by-Name Gradual Type Theory} +\author{Max S. New\inst{1} \and Daniel R. Licata\inst{2}} +\institute{\inst{1}Northeastern University \and \inst{2}{Wesleyan University}} +\date[]{FSCD 2018} + +\begin{document} +\begin{frame} + \titlepage +\end{frame} + +\section{What is Gradual Typing?} + +\begin{frame}{Gradual Typing} + Gradually Typed languages support a \emph{mix} of static and dynamic + typing, while maintaining benefits of each. + \begin{enumerate} + \item Dynamically typed parts of the language have minimal syntactic + discipline. + \item Statically typed parts of the language have sound type-based + reasoning. + \item Allow for safe/sound interaction by inserting dynamic type + checks at the boundary. + \end{enumerate} +\end{frame} + +\begin{frame}{Gradual Typing} + Applications + \begin{enumerate} + \item Allow static code to interact with legacy/untrusted code. + \item Gradually strengthen the types in an existing codebase + (Scripts to Programs) + \begin{enumerate} + \item \pause Dynamic $\to$ Partially Static $\to$ Static + \item \pause ADT $\to$ GADT $\to$ Dependent Types + \end{enumerate} + \end{enumerate} + We already have to do write this kind of glue code, gradual typing + is about doing it compositionally and automatically. +\end{frame} + +\begin{frame}{The Problem with Gradual Typing} + \begin{itemize} + \item Many implementations: Typed Racket, Thorn, Safe Typescript, + Reticulated Python + \item Useful type systems have strong type-based reasoning + principles, and categorical semantics! + \begin{enumerate} + \item Extensionality principles ($\eta$ laws), Representation + independence, Parametricity. + \item Need to respect these if we want to add gradual typing to + existing \emph{typed} languages. + \end{enumerate} + \item \pause But the state of the art$\ldots$ + \begin{enumerate} + \item Many incompatible semantics of casts have been proposed: + \begin{itemize} + \item For simple types: lazy, eager, up-down, partially eager, + transient + \item for state: monotonic state, proxied state + \item Are these really ``sound''? What does that mean exactly? + \end{itemize} + \item Blame Safety Theorem: + \begin{enumerate} + \item Gradually typed programs \emph{do} go wrong, but + \item \pause If a type error occurs, the well-typed + part of the program is never \emph{blamed}. + \end{enumerate} + \end{enumerate} + \end{itemize} +\end{frame} + +\begin{frame}{Semantic Gradual Typing} + Wanted: semantic approach to gradual typing. + \begin{enumerate} + \item Which semantics enable which type-based reasoning principles? + \item Help guide design of gradual typing with different programming + features. + \end{enumerate} +\end{frame} + +\section{Gradual CCCs and GTT} +\begin{frame}{First Step: Gradual CCCs and Gradual Type Theory} + \begin{footnotesize} + \begin{center} + \begin{tabular}{|c|c|} + \hline + \textbf{Type Theory} & \textbf{Category} \\ + \hline + \hline + Negative Typed Lambda Calculus & Cartesian Closed Categories\\ + \hline + Negative \textbf{Gradual} Lambda Calculus? & Gradual CCC?\\ + \hline + \end{tabular} + \end{center} + \end{footnotesize} + \begin{itemize} + \item Model the runtime semantics of gradual typing, not the surface + syntax. + \item Type Theory: syntax for proving results + \item Category Theory: Develop concrete models based on domains and + logical relations. + \end{itemize} +\end{frame} + +\begin{frame}{A Guide: Domain Theory} + \begin{itemize} + \item Gradual Typing is very similar to domain theory with universal + types + \begin{enumerate} + \item Dynamic type $\sim$ Universal object + \item Type casts $\sim$ Embedding-projection pairs + \end{enumerate} + \item But the usual notion of ep pair is not an \emph{adequate} + model of gradual typing: + \begin{enumerate} + \item Models type errors as $\bot$, same as diverging computation. + \item Come back to this later + \end{enumerate} + \item Analogy exploited in work on semantics of contracts in mid + 2000s by Findler, Blume and Felleisen (some unpublished), but + abandoned. + \end{itemize} +\end{frame} + +\begin{frame}{Desiderata} + What should GCCC/GTT be? + \begin{itemize} + \item Have a fully dynamically typed sub-language that is + computationally complete for the language. + \begin{itemize} + \item \pause There should be a \emph{universal type} $\dyn$ such that + every type is a \emph{retract} of it. + \end{itemize} + \item \pause Have static types, with type-based reasoning principles. + \begin{itemize} + \item \pause The category should be a CCC (satsifies $\beta, \eta$) + \end{itemize} + \item \pause Have a smooth transition from dynamic to static typing. + \begin{itemize} + \item \pause ??? + \end{itemize} + \end{itemize} +\end{frame} + +\begin{frame}{Graduality} + How do we formalize the idea of a ``smooth transition''? + \begin{itemize} + \item Siek-Vitousek-Cimini-Boyland, SNAPL 2015, propose the + \emph{gradual guarantee}. + \item \pause Intuition: changing the precision of the types of the + program should only add or remove checking and not otherwise + change behavior. + \item \pause More Formally: refining the \emph{types} of a gradual + program should result in a refinement of \emph{behavior}. + \item \pause Analogous to parametricity: changing the type can only result + in certain changes to behavior. + \end{itemize} +\end{frame} + +% Refining a type means... +\begin{frame}{Type Dynamism Ordering} + $A \ltdyn B$ means $A$ is ``less dynamic''/ ``more precise'' than $B$. + \pause + \begin{mathpar} + \begin{array}{rrl} + \text{types} & A,B,C \bnfdef & \dyn \bnfalt A \to B \bnfalt A \times B \bnfalt 1 + \end{array}\\ + \pause + \inferrule + {} + {A \ltdyn \dyn} + + \inferrule + {} + {A \ltdyn A} + + \inferrule + {A \ltdyn B \ltdyn C} + {A \ltdyn C}\\ + + \inferrule + {A \ltdyn A' \and B \ltdyn B'} + {A \to B \ltdyn A' \to B'} + + \inferrule + {A \ltdyn A' \and B \ltdyn B'} + {A \times B \ltdyn A' \times B'} + \end{mathpar} + \pause + Why covariant function space? Consider + \[ N \to N \sqsubseteq \dyn \to \dyn \] +\end{frame} + +% Now what does it mean to refine the behavior of a program... +\begin{frame}{Operational Definition of Graduality} + Extend type ordering to terms by congruence: + \begin{itemize} + \item Define $t \ltdyn u$ to hold if $t$ is formed by replacing all + type annotatons in $u$ with a less dynamic type. + \end{itemize} + \begin{theorem}[Gradual Guarantee] + If $t \ltdyn u$ and $\cdot \vdash t : A$ and $\cdot \vdash u : B$, + then either + \begin{enumerate} + \item $t \converge \err$ + \item $t \diverge, u \diverge$ + \item $t \converge v, u \converge v'$ and $v \ltdyn v'$ + \end{enumerate} + \end{theorem} + \pause Defines an ordering on effects, but where error is the least + element, not divergence. +\end{frame} + +\begin{frame}{Graduality Violations} + Rules out: + \begin{itemize} + \item Making program more static turning a type error into a + success(!) + \item Or changing the successful result + \item Or causing a terminating program to diverge + \end{itemize} + + Nota Bene: + \begin{itemize} + \item Rules out unrestricted \emph{catching} of type errors + \emph{within} the gradual language. + \item So if the language is sequential, type errors must crash. + \end{itemize} +\end{frame} + +\begin{frame}{Extensional Term Dynamism} + To model this theorem, we want an \emph{axiomatic ,extensional} + version of the term ordering. I.e., we include the rules: + + \begin{mathpar} + \inferrule* + {~} + {\err \ltdyn t} + + \inferrule*[Right=$\beta\to$] + {~} + {(\lambda x.t) u \equidyn t[u/x]} + + \inferrule*[Right=$\eta\to$] + {t : A \to B} + {t \equidyn \lambda x. t x} + + \inferrule*[Right=$\beta\times$] + {~} + {\pi_i (t_1, t_2) \equidyn t_i} + + \inferrule*[Right=$\eta\times$] + {t : A \times B} + {t \equidyn (\pi_1 t, \pi_2 t)} + + \inferrule*[Right=$\eta 1$] + {t : 1} + {t \equidyn ()} + \end{mathpar} +\end{frame} + +\begin{frame}{Term Dynamism Ordering} + \alt<2>{\emph{Heterogeneously typed congruence rules}}{Also add \emph{congruence rules}} + \begin{mathpar} + \alt<2>{\inferrule {t \ltdyn t' : A \to B \ltdyn A' \to B' \and u + \ltdyn u' : A \ltdyn A'} {t\, u \ltdyn t' \, u' : B \ltdyn + B'}} {\inferrule {t \ltdyn t' \and u \ltdyn u'} {t\, u \ltdyn + u \ltdyn u'}} + + \alt<2>{\inferrule + {\Gamma,x:A\ltdyn \Gamma',x':B \vdash t \ltdyn u : B \to B'} + {\Gamma \ltdyn \Gamma' \vdash \lambda x : A. t \ltdyn \lambda y : B . u : A \to B \ltdyn B \to B'}} + {\inferrule + {x:A\ltdyn y:B \vdash t \ltdyn u \and A \ltdyn B} + {\lambda x : A. t \ltdyn \lambda y : B . u}} + + \alt<2>{ + {\inferrule{t_1 \ltdyn u_1 : A_1 \ltdyn B_1 \and t_2\ltdyn u_2 : A_2 \ltdyn B_2}{(t_1,t_2)\ltdyn (u_1,u_2) : A_1 \times A_2 \ltdyn B_1 \times B_2}} + } + {\inferrule{t_1 \ltdyn u_1 \and t_2\ltdyn u_2}{(t_1,t_2)\ltdyn (u_1,u_2)}} + + \alt<2> + {\inferrule{t \ltdyn u : A_1 \times A_2 \ltdyn B_1 \times B_2}{\pi_i t \ltdyn \pi_i u : A_i \ltdyn B_i}} + {\inferrule{t \ltdyn u}{\pi_i t \ltdyn \pi_i u}\\} + + \alt<2>{ + \inferrule + {\Gamma \vdash t : A} + {\Gamma \ltdyn \Gamma \vdash t \ltdyn t}} + {\inferrule{~}{t \ltdyn t}} + + \alt<2>{ + \inferrule{\Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A'\\ + \Gamma' \ltdyn \Gamma'' \vdash t' \ltdyn t'' : A' \ltdyn A''} + {\Gamma \ltdyn \Gamma'' \vdash t \ltdyn t'' : A \ltdyn A''} + }{\inferrule{t \ltdyn t' \ltdyn t''}{t \ltdyn t''}} + \end{mathpar} +\end{frame} + +\begin{frame}{Formalizing Graduality} + How to formalize this in our categories? + \begin{enumerate} + \item Types and terms are ordered, type constructors and term + constructors are monotone\ldots\pause Use internal categories! + \item \pause Want categories internal to preorders, i.e., thin categories. + \item \pause Categories internal to categories are well-studied and called + \emph{double categories}. + \end{enumerate} +\end{frame} + +%\subsection{Double Categories} +\begin{frame}{Double Categories} + A double category is a kind of 2-category with two different kinds + of 1-arrows between the same objects.\\ + Examples: + \begin{itemize} + \item Functions and Relations + \item EP Pairs and Continuous functions + \item Functors and Profunctors + \item Adjunctions and Functors + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{Double Categories} + A double category consists of + \begin{itemize} + \item A set of objects $O$ + \item A set of \emph{vertical arrows} $p,q,r$, with category structure. + \begin{tikzcd} + A \arrow[d, "p"]\\ B + \end{tikzcd} + \item A set of \emph{horizontal arrows}, with category structure. + \[ + \begin{tikzcd} + A \arrow[r, "f"] & B + \end{tikzcd} + \] + \item A set of \emph{squares} which are ``framed'' by horizontal, vertical arrows: + \[\begin{tikzcd} + A \arrow[dd,"p"] \arrow[rr, "f"] & & B \arrow[dd,"r"] \\ + & \alpha & \\ + C \arrow[rr, "g"] & & D + \end{tikzcd}\] + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{Double Categories} + Squares can be composed in 2 ways\\ + \begin{columns} + \begin{column}{0.45\textwidth} + First, \emph{horizontally}\\ + \begin{tikzcd} + A \arrow[r]\arrow[d] & B \arrow[r]\arrow[d] & C\arrow[d]\\ + A' \arrow[r] & B'\arrow[r]& C + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + With for every vertical arrow, a horizontal identity: + \begin{tikzcd} +A \arrow[d, "p"] \arrow[r, "id"] & A \arrow[d, "p"] \\ +B \arrow[r, "id"] & B +\end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + Second, \emph{vertically}\\ + \begin{tikzcd} +A \arrow[d] \arrow[r] & B \arrow[d] \\ +A' \arrow[d] \arrow[r] & B' \arrow[d] \\ +A'' \arrow[r] & B'' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + With for every horizontal arrow, a vertical identity: + \begin{tikzcd} +A \arrow[d, "id"] \arrow[r, "f"] & B \arrow[d, "id"] \\ +A \arrow[r, "f"] & B +\end{tikzcd} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing as Double Categories} + \begin{itemize} + \item The objects are the types + \item The \emph{vertical arrows} are the type dynamism judgments $A + \ltdyn B$ (so a thin category) + \item The \emph{horizontal arrows} are the terms $x :A \vdash t : B$ + \item The \emph{squares} are the term dynamism: + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + x:A \arrow[dd] \arrow[rr, "t"] & & B \arrow[dd] \\ + & \sqsubseteq & \\ + x':A' \arrow[rr, "t'"] & & B' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \[ + x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' : B \ltdyn B' + \] + \end{column} + \end{columns} + + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing as Double Categories} + Horizontal composition of squares is a \emph{substitution principle} is + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + A \arrow[r,"u"]\arrow[d] & B \arrow[r,"t"]\arrow[d] & C\arrow[d]\\ + A' \arrow[r,"u'"] & B'\arrow[r,"t'"]& C + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + A \arrow[d] \arrow[r, "id"] & A \arrow[d] \\ + B \arrow[r, "id"] & B + \end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule{x \ltdyn x' \vdash t \ltdyn t'\\ u \ltdyn u'} + {t[u/x] \ltdyn t'[u'/x]} + \end{mathpar} + \end{column} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule + {} + {x\ltdyn x' : A \ltdyn A' \vdash x \ltdyn x' : A \ltdyn A'} + \end{mathpar} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing as Double Categories} + Vertical composition of squares is \emph{transitivity and reflexivity}. + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} +\Gamma \arrow[d] \arrow[r,"t"] & A \arrow[d] \\ +\Gamma' \arrow[d] \arrow[r,"t'"] & A' \arrow[d] \\ +\Gamma'' \arrow[r,"t''"] & A'' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \begin{tikzcd} +A \arrow[d, "id"] \arrow[r, "t"] & B \arrow[d, "id"] \\ +A \arrow[r, "t"] & B +\end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule{\Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A'\\ + \Gamma' \ltdyn \Gamma'' \vdash t' \ltdyn t'' : A' \ltdyn A''} + {\Gamma \ltdyn \Gamma'' \vdash t \ltdyn t'' : A \ltdyn A''} + \end{mathpar} + \end{column} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A} + {\Gamma \ltdyn \Gamma \vdash t \ltdyn t : A \ltdyn A} + \end{mathpar} + \end{column} + \end{columns} +\end{frame} + +%\subsection{A Universal Property for Casts} + +\begin{frame}{Type Casts} + To actually get dynamic type checking, we need casts. + + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A \and A \ltdyn B} + {\Gamma \vdash \upcast A B t : B} + + \inferrule + {\Gamma \vdash u : B \and A \ltdyn B} + {\Gamma \vdash \dncast A B u : A} + \end{mathpar} + + \pause By having the term ordering defined \emph{first}, we can + \emph{characterize} them by a universal property. +\end{frame} + +\begin{frame}{Universal Property of Casts} + How should casts behave? + Intuition: the casted term should act as much like the original term + as possible while still satisfying the new type. + + So for a downcast, the casted term should refine the original: + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\dncast A B u \ltdyn u : A \ltdyn B} + \end{mathpar} + + \pause But this would be satisfied even by the type error $\err_A$. + \pause + We want to make sure that the casted term has \emph{as much} + behavior as possible: i.e., it's the \emph{biggest} term below + $u$. + \begin{mathpar} + \inferrule + {t \ltdyn u : A \ltdyn B} + {t \ltdyn \dncast A B u : A} + \end{mathpar} + \pause Order-theoretic summary: $\dncast A B u$ is the greatest + lower bound of $u$ in $A$. +\end{frame} + +\begin{frame}{Universal Property of Casts} + Upcasts are the exact dual + + For an upcast, the casted term should be \emph{more} dynamic than + the original: + \begin{mathpar} + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \upcast A B t : A \ltdyn B} + \end{mathpar} + + But it has no more behaviors than dictated by the original term, + it's the \emph{least} term above $t$. + + \begin{mathpar} + \inferrule + {t \ltdyn u : A \ltdyn B} + {\upcast A B t \ltdyn u : B} + \end{mathpar} + \pause Order-theoretic summary: $\upcast A B u$ is the least upper + bound of $t$ in $B$. +\end{frame} +% TODO: ugly +\begin{frame}[fragile]{Universal Property for Casts} + The upcast is the universal way to turn a vertical arrow horizontal + \emph{covariantly} and the downcast is the universal way to do it + contravariantly. + + \[\begin{tikzcd} +A' \arrow[d] & A \arrow[r, "f_!"] \arrow[l, "f^*"'] \arrow[d, "f"] & A' \arrow[d] \\ +A' & A' \arrow[r] \arrow[l] & A' + \end{tikzcd} + \] + + \begin{itemize} + \item Can prove that this makes $f_!$ a left adjoint of $f^*$. + \item A double category where every vertical arrow induces these + universal horizontal arrows is called a \textbf{(proarrow) + equipment} or a \textbf{framed bicategory}. + \end{itemize} +\end{frame} + +\begin{frame}{Gradual CCC} + A Gradual CCC is (morally) a cartesian closed category internal to + thin categories with + \begin{itemize} + \item A vertically terminal object ($\dyn$) + \item A least element of every hom set (type error), satisfying $\ldots$ + \item For every $A \ltdyn A'$, an upcast and downcast making the + double category an equipment, such that the downcast is a retract + of the upcast. + \end{itemize} +\end{frame} +\begin{frame}{Gradual CCC} + A Gradual CCC is a representable cartesian multicategory internal to + thin categories with + \begin{itemize} + \item A vertically terminal object ($\dyn$) + \item A least element of every hom set (type error), satisfying $\ldots$ + \item For every $A \ltdyn A'$, an upcast and downcast making the + double category an equipment, such that the downcast is a retract + of the upcast. + \end{itemize} +\end{frame} + +\begin{frame}{Gradual Type Theory} + \begin{itemize} + \item Made type theory with types, terms, type and term ordering. + \item Proved soundness and completeness theorems for our Gradual + Type Theory. + \end{itemize} +\end{frame} + +\section{Applications} +%\subsection{Theorems about Gradual Languages} +\begin{frame}{New Results in Gradual Typing} + Can prove results that hold in any ``reasonable'' gradually typed + language.\pause + \begin{itemize} + \item Validate optimizations: elimination of redundant casts. + \item Uniqueness Theorems + \end{itemize} +\end{frame} + +\begin{frame}{Uniqueness Theorem} + \begin{theorem}{Uniqueness of Function Casts} + In Gradual Type Theory if $A \ltdyn A'$ and $B \ltdyn B'$, the + following equivalences hold: + \[ + \dncast {A \to B} {A' \to B'} t \equiv \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + and + \[ + \upcast {A \to B} {A' \to B'} t \equiv \lambda x':A'. \upcast{B}{B'}(t (\dncast A {A'} x')) + \] + \end{theorem} + + Proof uses specification for casts, and $\beta, \eta$ axioms for + function types. + + \begin{corollary} + Any \emph{observably different} implementation violates $\beta$, + $\eta$ or graduality! + \end{corollary} +\end{frame} + +\begin{frame}{Uniqueness Theorems} + Uniqueness theorems say ``there is only one way to do it'': + \begin{itemize} + \item Can prove them for casts between the \emph{same} connective + using $\beta, \eta$ + \item Only casts that don't have uniqueness theorems are the ``tag types'': + \begin{mathpar} + (\dyn \to \dyn) \ltdyn \dyn \and + (\dyn \times \dyn) \ltdyn \dyn \and + \end{mathpar} + The actual design flexibility is here. + \end{itemize} +\end{frame} + +%\subsection{Model Construction} + +%% \begin{frame}{Semantic Contract Translation} +%% \begin{itemize} +%% \item Start with a locally thin cartesian closed 2-category $C$ +%% local bottoms and with a $D \in C$ which has specified ep pairs: +%% \begin{enumerate} +%% \item $(D \to D) \triangleleft D$ +%% \item $(D \times D) \triangleleft D$ +%% \end{enumerate} +%% \item Make $C$ into a vertically thin double category whose vertical +%% arrows are ep pairs: $EP(C)$ +%% \item Take a \emph{vertical slice} of $EP(C)$ over $D$ $EP(C)/D$ +%% \end{itemize} +%% \end{frame} + +%% \begin{frame}{Semantic Contract Translation} +%% \begin{itemize} +%% \item A gradual type $A$ is an ep pair $A \triangleleft D$ +%% \item If $A \ltdyn B$, then there exists a (unique) ep pair $A +%% \triangleleft B$ that factorizes: +%% \[ A \triangleleft D \equiv A \triangleleft B \triangleleft D \] +%% \item A term $\Gamma \vdash t : A$ is a morphism $\times \Gamma \to A$ +%% \item A term ordering +%% \[ \Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A' \] +%% is an ordering +%% \[ \upcast{A}{A'} \circ t \circ \dncast{\Gamma}{\Gamma'} \ltdyn t' \] +%% (or one of 3 other equivalent placements) +%% \end{itemize} +%% \end{frame} + +\begin{frame}[fragile]{A Domain Model} + Need domains with 2 notions of error (divergence, crash) and 2 + notions of order. + \begin{itemize} + \item A gradual pre-domain is a set $X$ with an $\omega$CPO + structure $(X,\leq)$ and a preorder $(X, \ltdyn)$ with $\ltdyn$ + closed under $\omega\leq$-chains. + \item A gradual domain is a gradual pre-domain with a $\leq$-least + element $\bot$ and a $\ltdyn$-least element $\err$ + \item A morphism $f : X \to Y$ is continuous in $\leq$ and monotone + in $\ltdyn$. + \end{itemize} + $1_{\bot,\err}$: + \begin{columns} + $\leq$: + \begin{column}{0.45\textwidth} + \begin{tikzcd} +* \arrow[d, no head] & \mho \arrow[ld, no head] \\ +\bot & +\end{tikzcd} + \end{column} + $\ltdyn$: + \begin{column}{0.45\textwidth} + \begin{tikzcd} +* \arrow[d, no head] & \bot \arrow[ld, no head] \\ +\err & +\end{tikzcd} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{A Domain Model} + \begin{itemize} + \item Can construct a suitable reflexive object using the domain + structure and standard techniques (using $\leq$ ep pairs). + \item Combine with the earlier construction to get a + domain-theoretic model of gradual typing (using $\sqsubseteq$ ep + pairs). + \end{itemize} +\end{frame} + +\begin{frame}{Domain Contract Translation} + \begin{itemize} + \item A gradual type $A$ is an ep pair $A \triangleleft D$ with + respect to the \emph{error ordering} + \item If $A \ltdyn B$, then there exists a (unique) ep pair $A + \triangleleft B$ that factorizes: + \[ A \triangleleft D \equiv A \triangleleft B \triangleleft D \] + \item A term $\Gamma \vdash t : A$ is a morphism $\times \Gamma \to A$ + \item A term ordering + \[ \Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A' \] + is an $\ltdyn$ ordering + \[ \upcast{A}{A'} \circ t \circ \dncast{\Gamma}{\Gamma'} \ltdyn t' \] + (or one of 3 other equivalent placements) + \end{itemize} +\end{frame} + +\section{Conclusion} + +%% \begin{frame}{Questions} +%% \begin{itemize} +%% \item Double Categories for Domain Theory? +%% \item Strong analogy to parametricity +%% \item Intuition for Gradual Typing has a refinement typing flavor, +%% but GTT is Church-style, so doesn't show up yet. +%% \end{itemize} +%% \end{frame} + +\begin{frame}{Conclusion} + \begin{itemize} + \item Using double categories, type casts get a + specification/universal property. + \item Type-theoretic presentation makes it easy to combine with + other features. + \begin{itemize} + \item In progress: Gradual Call-by-push-value with positives and + negatives. + \end{itemize} + % Present + \item More inside! + \begin{itemize} + \item More theorems in GTT + \item Formulation using Cartesian Multicategories internal to + Preorders. + \item Category-theoretic construction generalizing the domain + model as a \emph{vertical} slice category. + \end{itemize} + \end{itemize} +\end{frame} + +\end{document} + +%% Local Variables: +%% compile-command: "latexmk -C talk.tex; latexmk talk.tex" +%% End: diff --git a/talk/cmu-fall-2018/mfps-talk.tex b/talk/cmu-fall-2018/mfps-talk.tex new file mode 100644 index 0000000..54b671d --- /dev/null +++ b/talk/cmu-fall-2018/mfps-talk.tex @@ -0,0 +1,779 @@ +\documentclass{beamer} +\usetheme{boxes} + +\newcommand{\notepr}[1]{} + +\usepackage{rotating} +\usepackage{listings} +\usepackage{xcolor} +%% \lstset{ +%% basicstyle=\ttfamily, +%% escapeinside=~~ +%% } + +\usepackage{microtype} +\usepackage{latexsym,amsmath,amssymb,amsthm} +\usepackage{mathpartir} +\usepackage{tikz} +\usepackage{tikz-cd} +\input{talk-defs} +\newcommand{\ltdyn}{\dynr} +\newcommand{\uarrow}{\mathrel{\rotatebox[origin=c]{-30}{$\leftarrowtail$}}} +\newcommand{\darrow}{\mathrel{\rotatebox[origin=c]{30}{$\twoheadleftarrow$}}} +\renewcommand{\upcast}[2]{\langle{#2}\uarrow{#1}\rangle} +\renewcommand{\dncast}[2]{\langle{#1}\darrow{#2}\rangle} +\newcommand{\gtdyn}{\sqsupseteq} + +\newcommand{\converge}{\Downarrow} +\newcommand{\diverge}{\Uparrow} + +\title{Semantic Foundations for Gradual Typing} +\author{Max S. New} +\institute{Northeastern University} +\date{\today} + +\begin{document} + +\begin{frame} + \titlepage + \begin{small} + Joint work with\\ + Amal Ahmed (Northeastern Uni.),\\ + Dan Licata (Wesleyan Uni.) + \end{small} +\end{frame} + +\begin{frame} + \frametitle{Outline} + \tableofcontents +\end{frame} + +\section{Why Semantic Gradual Typing} + +\begin{frame}{My Dream of Gradual Typing} + % Don't dwell too long, 3rd talk in the session. + Why should people that like \emph{static typing} like \emph{gradual + typing}? + \begin{itemize} + \item We need to interact with unverified inputs + %% \begin{enumerate} + %% \item External environment/user interface + %% \item Legacy code + %% \item Untrusted, possibly adversarial code + %% \end{enumerate} + \item Want to be able to add \emph{new} typing features to + languages, but still easily interact with old code. + %% \begin{enumerate} + %% %% \item Types were unclear before, but are clear now + %% \item Maybe the type features didn't exist before! + %% %% \item Didn't know how/didn't have time to convince the type + %% %% checker. + %% \end{enumerate} + \end{itemize} + Want language-level support for these use cases. +\end{frame} + +\begin{frame}{The Problem with Gradual Typing} + \begin{itemize} + \item Useful type systems have strong type-based reasoning + principles, and categorical semantics! + \begin{enumerate} + \item Extensionality principles ($\eta$ laws) + \item Representation independence + \item Parametricity + \end{enumerate} + \item \pause But the state of the art$\ldots$ + \begin{enumerate} + \item Many incompatible semantics of casts have been proposed: + Lazy, Eager, UD, Transient, Monotonic state, Proxied state + \begin{itemize} + \item Which ones give which reasoning principles? + \end{itemize} + \item Blame Safety Theorem: + \begin{enumerate} + \item Gradually typed programs \emph{do} go wrong, but + \item \pause If a type error occurs, the well-typed + part of the program is never \emph{blamed}. + \end{enumerate} + %% \item Other work: Parametricity for Gradual Typing, but ``ad + %% hoc'': what was the principle behind it. + \end{enumerate} + \end{itemize} +\end{frame} + +\begin{frame}{A Guide: Domain Theory} + \begin{itemize} + \item Gradual Typing is very similar to domain theory with universal + types + \begin{enumerate} + \item Dynamic type $\sim$ Universal object + \item Type casts $\sim$ Embedding-projection pairs + \end{enumerate} + \item But the usual notion of ep pair is not an \emph{adequate} + model of gradual typing: + \begin{enumerate} + \item Models type errors as $\bot$, same as diverging computation. + \item Come back to this later + \end{enumerate} + \item Analogy exploited in work on semantics of contracts in the + 2000s by Findler, Blume and Felleisen (some unpublished). + \end{itemize} +\end{frame} + +\section{Gradual CCCs and GTT} +\begin{frame}{First Step: Gradual CCCs and Gradual Type Theory} + \begin{footnotesize} + \begin{center} + \begin{tabular}{|c|c|} + \hline + \textbf{Type Theory} & \textbf{Category} \\ + \hline + \hline + Negative Typed Lambda Calculus & Cartesian Closed Categories\\ + \hline + Negative \textbf{Gradual} Lambda Calculus? & Gradual CCC?\\ + \hline + \end{tabular} + \end{center} + \end{footnotesize} + \begin{itemize} + \item Model the ``cast calculus'' of gradual typing, not the surface + syntax. + \item Develop concrete models based on domains and logical + relations. + \end{itemize} +\end{frame} + +\begin{frame}{Desiderata} + What should GCCC/GTT be? + \begin{itemize} + \item Have a fully dynamically typed sub-language that is + computationally complete for the language. + \begin{itemize} + \item \pause There should be a \emph{universal type} $\dyn$ such that + every type is a \emph{retract} of it. + \end{itemize} + \item \pause Have static types, that \emph{mean the right thing}. + \begin{itemize} + \item \pause The category should be a CCC (satsifies $\beta, \eta$) + \end{itemize} + \item \pause Have a smooth transition from dynamic to static typing. + \begin{itemize} + \item \pause ??? + \end{itemize} + \end{itemize} +\end{frame} + +%\subsection{Graduality} +\begin{frame}{Graduality} + How do we formalize the idea of a ``smooth transition''? + \begin{itemize} + \item Siek-Vitousek-Cimini-Boyland, SNAPL 2015, propose the + \emph{gradual guarantee}. + \item \pause Intuition: changing the precision of the types of the + program should only add or remove checking and not otherwise + change behavior. + \item \pause More Formally: refining the \emph{types} of a gradual + program should result in a refinement of \emph{behavior}. + \item \pause Analogous to parametricity: changing the type can only result + in certain changes to behavior. + \end{itemize} +\end{frame} + +% Refining a type means... +\begin{frame}{Type Dynamism Ordering} + $A \ltdyn B$ means $A$ is ``less dynamic''/ ``more precise'' than $B$. + \pause + \begin{mathpar} + \begin{array}{rrl} + \text{types} & A,B,C \bnfdef & \dyn \bnfalt A \to B \bnfalt A \times B \bnfalt 1 + \end{array}\\ + \pause + \inferrule + {} + {A \ltdyn \dyn} + + \inferrule + {} + {A \ltdyn A} + + \inferrule + {A \ltdyn B \ltdyn C} + {A \ltdyn C}\\ + + \inferrule + {A \ltdyn A' \and B \ltdyn B'} + {A \to B \ltdyn A' \to B'} + + \inferrule + {A \ltdyn A' \and B \ltdyn B'} + {A \times B \ltdyn A' \times B'} + \end{mathpar} + \pause + Why? Consider + \[ N \to N \sqsubseteq \dyn \to \dyn \] +\end{frame} + +% Now what does it mean to refine the behavior of a program... +\begin{frame}{Operational Definition of Graduality} + Extend type ordering to terms by congruence: + \begin{itemize} + \item Define $t \ltdyn u$ to hold if $t$ is formed by replacing all + type annotaitons in $u$ with a less dynamic type. + \end{itemize} + \begin{theorem}[Gradual Guarantee] + If $t \ltdyn u$ and $\cdot \vdash t : A$ and $\cdot \vdash u : B$, + then either + \begin{enumerate} + \item $t \converge \err$ + \item $t \diverge, u \diverge$ + \item $t \converge v, u \converge v'$ and $v \ltdyn v'$ + \end{enumerate} + \end{theorem} + \pause Defines an ordering on effects, but where error is the least + element, not divergence. +\end{frame} + +\begin{frame}{Graduality Violations} + Rules out: + \begin{itemize} + \item Making program more static turning a type error into a + success(!) + \item Or changing the successful result + \item Or causing a terminating program to diverge + \end{itemize} + + Nota Bene: + \begin{itemize} + \item Rules out unrestricted \emph{catching} of type errors + \emph{within} the gradual language. + \item So if the language is sequential, type errors must crash. + \end{itemize} +\end{frame} + +\begin{frame}{Extensional Term Dynamism} + To model this theorem, we want an \emph{axiomatic ,extensional} + version of the term ordering. I.e., we include the rules: + + \begin{mathpar} + \inferrule* + {~} + {\err \ltdyn t} + + \inferrule*[Right=$\beta\to$] + {~} + {(\lambda x.t) u \equidyn t[u/x]} + + \inferrule*[Right=$\eta\to$] + {t : A \to B} + {t \equidyn \lambda x. t x} + + \inferrule*[Right=$\beta\times$] + {~} + {\pi_i (t_1, t_2) \equidyn t_i} + + \inferrule*[Right=$\eta\times$] + {t : A \times B} + {t \equidyn (\pi_1 t, \pi_2 t)} + + \inferrule*[Right=$\eta 1$] + {t : 1} + {t \equidyn ()} + \end{mathpar} +\end{frame} + +\begin{frame}{Term Dynamism Ordering} + \alt<2>{\emph{Heterogeneously typed congruence rules}}{Also add \emph{congruence rules}} + \begin{mathpar} + \alt<2>{\inferrule {t \ltdyn t' : A \to B \ltdyn A' \to B' \and u + \ltdyn u' : A \ltdyn A'} {t\, u \ltdyn t' \, u' : B \ltdyn + B'}} {\inferrule {t \ltdyn t' \and u \ltdyn u'} {t\, u \ltdyn + u \ltdyn u'}} + + \alt<2>{\inferrule + {\Gamma,x:A\ltdyn \Gamma',x':B \vdash t \ltdyn u : B \to B'} + {\Gamma \ltdyn \Gamma' \vdash \lambda x : A. t \ltdyn \lambda y : B . u : A \to B \ltdyn B \to B'}} + {\inferrule + {x:A\ltdyn y:B \vdash t \ltdyn u \and A \ltdyn B} + {\lambda x : A. t \ltdyn \lambda y : B . u}} + + \alt<2>{ + {\inferrule{t_1 \ltdyn u_1 : A_1 \ltdyn B_1 \and t_2\ltdyn u_2 : A_2 \ltdyn B_2}{(t_1,t_2)\ltdyn (u_1,u_2) : A_1 \times A_2 \ltdyn B_1 \times B_2}} + } + {\inferrule{t_1 \ltdyn u_1 \and t_2\ltdyn u_2}{(t_1,t_2)\ltdyn (u_1,u_2)}} + + \alt<2> + {\inferrule{t \ltdyn u : A_1 \times A_2 \ltdyn B_1 \times B_2}{\pi_i t \ltdyn \pi_i u : A_i \ltdyn B_i}} + {\inferrule{t \ltdyn u}{\pi_i t \ltdyn \pi_i u}\\} + + \alt<2>{ + \inferrule + {\Gamma \vdash t : A} + {\Gamma \ltdyn \Gamma \vdash t \ltdyn t}} + {\inferrule{~}{t \ltdyn t}} + + \alt<2>{ + \inferrule{\Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A'\\ + \Gamma' \ltdyn \Gamma'' \vdash t' \ltdyn t'' : A' \ltdyn A''} + {\Gamma \ltdyn \Gamma'' \vdash t \ltdyn t'' : A \ltdyn A''} + }{\inferrule{t \ltdyn t' \ltdyn t''}{t \ltdyn t''}} + \end{mathpar} +\end{frame} + +\begin{frame}{Formalizing Graduality} + How to formalize this in our categories? + \begin{enumerate} + \item Types and terms are ordered, type constructors and term + constructors are monotone\ldots\pause Use internal categories! + \item \pause Want categories internal to preorders, i.e., thin categories. + \item \pause Categories internal to categories are well-studied and called + \emph{double categories}. + \end{enumerate} +\end{frame} + +%\subsection{Double Categories} +\begin{frame}{Double Categories} + A double category is a kind of 2-category with two different kinds + of 1-arrows between the same objects.\\ + Examples: + \begin{itemize} + \item Functions and Relations + \item EP Pairs and Continuous functions + \item Functors and Profunctors + \item Adjunctions and Functors + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{Double Categories} + A double category consists of + \begin{itemize} + \item A set of objects $O$ + \item A set of \emph{vertical arrows} $p,q,r$, with category structure. + \begin{tikzcd} + A \arrow[d, "p"]\\ B + \end{tikzcd} + \item A set of \emph{horizontal arrows}, with category structure. + \[ + \begin{tikzcd} + A \arrow[r, "f"] & B + \end{tikzcd} + \] + \item A set of \emph{squares} which are ``framed'' by horizontal, vertical arrows: + \[\begin{tikzcd} + A \arrow[dd,"p"] \arrow[rr, "f"] & & B \arrow[dd,"r"] \\ + & \alpha & \\ + C \arrow[rr, "g"] & & D + \end{tikzcd}\] + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{Double Categories} + Squares can be composed in 2 ways\\ + \begin{columns} + \begin{column}{0.45\textwidth} + First, \emph{horizontally}\\ + \begin{tikzcd} + A \arrow[r]\arrow[d] & B \arrow[r]\arrow[d] & C\arrow[d]\\ + A' \arrow[r] & B'\arrow[r]& C + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + With for every vertical arrow, a horizontal identity: + \begin{tikzcd} +A \arrow[d, "p"] \arrow[r, "id"] & A \arrow[d, "p"] \\ +B \arrow[r, "id"] & B +\end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + Second, \emph{vertically}\\ + \begin{tikzcd} +A \arrow[d] \arrow[r] & B \arrow[d] \\ +A' \arrow[d] \arrow[r] & B' \arrow[d] \\ +A'' \arrow[r] & B'' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + With for every horizontal arrow, a vertical identity: + \begin{tikzcd} +A \arrow[d, "id"] \arrow[r, "f"] & B \arrow[d, "id"] \\ +A \arrow[r, "f"] & B +\end{tikzcd} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing as Double Categories} + \begin{itemize} + \item The objects are the types + \item The \emph{vertical arrows} are the type dynamism judgments $A + \ltdyn B$ (so a thin category) + \item The \emph{horizontal arrows} are the terms $x :A \vdash t : B$ + \item The \emph{squares} are the term dynamism: + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + x:A \arrow[dd] \arrow[rr, "t"] & & B \arrow[dd] \\ + & \sqsubseteq & \\ + x':A' \arrow[rr, "t'"] & & B' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \[ + x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' : B \ltdyn B' + \] + \end{column} + \end{columns} + + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing as Double Categories} + Horizontal composition of squares is a \emph{substitution principle} is + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + A \arrow[r,"u"]\arrow[d] & B \arrow[r,"t"]\arrow[d] & C\arrow[d]\\ + A' \arrow[r,"u'"] & B'\arrow[r,"t'"]& C + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \begin{tikzcd} + A \arrow[d] \arrow[r, "id"] & A \arrow[d] \\ + B \arrow[r, "id"] & B + \end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule{x \ltdyn x' \vdash t \ltdyn t'\\ u \ltdyn u'} + {t[u/x] \ltdyn t'[u'/x]} + \end{mathpar} + \end{column} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule + {} + {x\ltdyn x' : A \ltdyn A' \vdash x \ltdyn x' : A \ltdyn A'} + \end{mathpar} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{Gradual Typing as Double Categories} + Vertical composition of squares is \emph{transitivity and reflexivity}. + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{tikzcd} +\Gamma \arrow[d] \arrow[r,"t"] & A \arrow[d] \\ +\Gamma' \arrow[d] \arrow[r,"t'"] & A' \arrow[d] \\ +\Gamma'' \arrow[r,"t''"] & A'' + \end{tikzcd} + \end{column} + \begin{column}{0.45\textwidth} + \begin{tikzcd} +A \arrow[d, "id"] \arrow[r, "t"] & B \arrow[d, "id"] \\ +A \arrow[r, "t"] & B +\end{tikzcd} + \end{column} + \end{columns} + \begin{columns} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule{\Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A'\\ + \Gamma' \ltdyn \Gamma'' \vdash t' \ltdyn t'' : A' \ltdyn A''} + {\Gamma \ltdyn \Gamma'' \vdash t \ltdyn t'' : A \ltdyn A''} + \end{mathpar} + \end{column} + \begin{column}{0.45\textwidth} + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A} + {\Gamma \ltdyn \Gamma \vdash t \ltdyn t : A \ltdyn A} + \end{mathpar} + \end{column} + \end{columns} +\end{frame} + +%\subsection{A Universal Property for Casts} + +\begin{frame}{Type Casts} + To actually get dynamic type checking, we need casts. + + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A \and A \ltdyn B} + {\Gamma \vdash \upcast A B t : B} + + \inferrule + {\Gamma \vdash u : B \and A \ltdyn B} + {\Gamma \vdash \dncast A B u : A} + \end{mathpar} + + \pause By having the term ordering defined \emph{first}, we can + \emph{characterize} them. +\end{frame} + +\begin{frame}{Casts bend the Rules} + How should casts behave? + Intuition: the casted term should act as much like the original term + as possible while still satisfying the new type. + + So for a downcast, the casted term should refine the original: + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\dncast A B u \ltdyn u : A \ltdyn B} + \end{mathpar} + + \pause But this would be satisfied even by the type error $\err_A$. + \pause + We want to make sure that the casted term has \emph{as much} + behavior as possible: i.e., it's the \emph{biggest} term below + $u$. + \begin{mathpar} + \inferrule + {t \ltdyn u : A \ltdyn B} + {t \ltdyn \dncast A B u : A} + \end{mathpar} + \pause Order-theoretic summary: $\dncast A B u$ is the greatest + lower bound of $u$ in $A$. +\end{frame} + +\begin{frame}{Casts bend the Rules} + Upcasts are the exact dual + + For an upcast, the casted term should be \emph{more} dynamic than + the original: + \begin{mathpar} + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \upcast A B t : A \ltdyn B} + \end{mathpar} + + But it has no more behaviors than dictated by the original term, + it's the \emph{least} term above $t$. + + \begin{mathpar} + \inferrule + {t \ltdyn u : A \ltdyn B} + {\upcast A B t \ltdyn u : B} + \end{mathpar} + \pause Order-theoretic summary: $\upcast A B u$ is the least upper + bound of $t$ in $B$. +\end{frame} +% TODO: ugly +\begin{frame}[fragile]{A Universal Property for Casts} + The upcast is the universal way to turn a vertical arrow horizontal + \emph{covariantly} and the downcast is the universal way to do it + contravariantly. + + \[\begin{tikzcd} +A' \arrow[d] & A \arrow[r, "upcast"] \arrow[l, "dncast"'] \arrow[d] & A' \arrow[d] \\ +A' & A' \arrow[r] \arrow[l] & A' + \end{tikzcd} + \] + + A double category where every vertical arrow induces these universal + horizontal arrows is called an \textbf{equipment}. +\end{frame} + +\begin{frame}{Gradual CCC} + A Gradual CCC is a cartesian closed category internal to thin + categories with + \begin{itemize} + \item A vertically terminal object ($\dyn$) + \item A least element of every hom set (type error), satisfying $\ldots$ + \item For every $A \ltdyn A'$, an upcast and downcast making the + double category an equipment, such that the downcast is a retract + of the upcast. + \end{itemize} +\end{frame} + +\begin{frame}{Gradual Type Theory} + \begin{itemize} + \item Made type theory with types, terms, type and term ordering. + \item Proved soundness and completeness theorems for our Gradual + Type Theory. + \item Slightly simplifying here: we used multicategories, rather + than CCCs directly, but should be essentially the same. + \end{itemize} +\end{frame} + +\section{Applications} +%\subsection{Theorems about Gradual Languages} +\begin{frame}{New Results in Gradual Typing} + Can prove results that hold in any ``reasonable'' gradually typed + language.\pause + \begin{itemize} + \item Validate optimizations: elimination of redundant casts. + \item Uniqueness Theorems + \end{itemize} +\end{frame} + +\begin{frame}{Uniqueness Theorem} + \begin{theorem}{Uniqueness of Function Casts} + In Gradual Type Theory if $A \ltdyn A'$ and $B \ltdyn B'$, the + following equivalences hold: + \[ + \dncast {A \to B} {A' \to B'} t \equiv \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + and + \[ + \upcast {A \to B} {A' \to B'} t \equiv \lambda x':A'. \upcast{B}{B'}(t (\dncast A {A'} x')) + \] + \end{theorem} + + Proof uses specification for casts, and $\beta, \eta$ axioms for + function types. + + \begin{corollary} + Any \emph{observably different} implementation violates $\beta$, + $\eta$ or graduality! + \end{corollary} +\end{frame} + +\begin{frame}{Uniqueness Theorems} + Uniqueness theorems say ``there is only one way to do it'': + \begin{itemize} + \item Can prove them for casts between the \emph{same} connective + using $\beta, \eta$ + \item Only casts that don't have uniqueness theorems are the ``tag types'': + \begin{mathpar} + (\dyn \to \dyn) \ltdyn \dyn \and + (\dyn \times \dyn) \ltdyn \dyn \and + \end{mathpar} + The actual design flexibility is here. + \end{itemize} +\end{frame} + +%\subsection{Model Construction} +\begin{frame}{Semantic Contract Translation} + \begin{itemize} + \item Start with a locally thin cartesian closed 2-category $C$ + local bottoms and with a $D \in C$ which has specified ep pairs: + \begin{enumerate} + \item $(D \to D) \triangleleft D$ + \item $(D \times D) \triangleleft D$ + \end{enumerate} + \item Make $C$ into a vertically thin double category whose vertical + arrows are ep pairs: $EP(C)$ + \item Take a \emph{vertical slice} of $EP(C)$ over $D$ $EP(C)/D$ + \end{itemize} +\end{frame} + +\begin{frame}{Semantic Contract Translation} + \begin{itemize} + \item A gradual type $A$ is an ep pair $A \triangleleft D$ + \item If $A \ltdyn B$, then there exists a (unique) ep pair $A + \triangleleft B$ that factorizes: + \[ A \triangleleft D \equiv A \triangleleft B \triangleleft D \] + \item A term $\Gamma \vdash t : A$ is a morphism $\times \Gamma \to A$ + \item A term ordering + \[ \Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A' \] + is an ordering + \[ \upcast{A}{A'} \circ t \circ \dncast{\Gamma}{\Gamma'} \ltdyn t' \] + (or one of 3 other equivalent placements) + \end{itemize} +\end{frame} + +\begin{frame}[fragile]{A Domain Model} + Need domains with 2 notions of error (divergence, crash) and 2 + notions of order. + \begin{itemize} + \item A gradual pre-domain is a set $X$ with an $\omega$CPO + structure $(X,\leq)$ and a preorder $(X, \ltdyn)$ with $\ltdyn$ + closed under $\omega\leq$-chains. + \item A gradual domain is a gradual pre-domain with a $\leq$-least + element $\bot$ and a $\ltdyn$-least element $\err$ + \item A continuous function $f : X \to Y$ must be continuous in + $\leq$ and monotone in $\ltdyn$. + \end{itemize} + $1_{\bot,\err}$: + \begin{columns} + $\leq$: + \begin{column}{0.45\textwidth} + \begin{tikzcd} +* \arrow[d, no head] & \mho \arrow[ld, no head] \\ +\bot & +\end{tikzcd} + \end{column} + $\ltdyn$: + \begin{column}{0.45\textwidth} + \begin{tikzcd} +* \arrow[d, no head] & \bot \arrow[ld, no head] \\ +\err & +\end{tikzcd} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{A Domain Model} + \begin{itemize} + \item Can construct a suitable reflexive object using the domain + structure and standard techniques (using $\leq$ ep pairs). + \item Combine with the earlier construction to get a + domain-theoretic model of gradual typing (using $\sqsubseteq$ ep + pairs). + \end{itemize} +\end{frame} + +%% \begin{frame}{Domain Contract Translation} +%% \begin{itemize} +%% \item A gradual type $A$ is an ep pair $A \triangleleft D$ with +%% respect to the \emph{error ordering} +%% \item If $A \ltdyn B$, then there exists a (unique) ep pair $A +%% \triangleleft B$ that factorizes: +%% \[ A \triangleleft D \equiv A \triangleleft B \triangleleft D \] +%% \item A term $\Gamma \vdash t : A$ is a morphism $\times \Gamma \to A$ +%% \item A term ordering +%% \[ \Gamma \ltdyn \Gamma' \vdash t \ltdyn t' : A \ltdyn A' \] +%% is an $\ltdyn$ ordering +%% \[ \upcast{A}{A'} \circ t \circ \dncast{\Gamma}{\Gamma'} \ltdyn t' \] +%% (or one of 3 other equivalent placements) +%% \end{itemize} +%% \end{frame} + +\section{Conclusion} + +%% \begin{frame}{Questions} +%% \begin{itemize} +%% \item Double Categories for Domain Theory? +%% \item Strong analogy to parametricity +%% \item Intuition for Gradual Typing has a refinement typing flavor, +%% but GTT is Church-style, so doesn't show up yet. +%% \end{itemize} +%% \end{frame} + +\begin{frame}{Conclusion} + \begin{itemize} + % Past + \item Using double categories, type casts get a + specification/universal property. + \begin{itemize} + \item Can prove theorems about \emph{many} gradual languages: only + one way to do function, product casts with $\eta$. + \end{itemize} + % Future + \item Internal category semantics means it's easy to combine with + other type theoretic features. + \begin{itemize} + \item In progress: Gradual Call-by-push-value with positives and + negatives. + \end{itemize} + % Present + \item Two papers + \begin{itemize} + \item ``Call-by-Name Gradual Type Theory'', New and Licata, FSCD 2018 + \item ``Graduality from Embedding-projection Pairs'', New and Ahmed, ICFP 2018 + \end{itemize} + \end{itemize} +\end{frame} + +\end{document} + +%% Local Variables: +%% compile-command: "latexmk -C talk.tex; latexmk talk.tex" +%% End: diff --git a/talk/cmu-fall-2018/neu-talk.tex b/talk/cmu-fall-2018/neu-talk.tex new file mode 100644 index 0000000..ade92ee --- /dev/null +++ b/talk/cmu-fall-2018/neu-talk.tex @@ -0,0 +1,1134 @@ +\documentclass{beamer} +\usetheme{boxes} + +\newcommand{\notepr}[1]{} + +%% \newif\ifshort +%% \newif\iflong + +%% \shorttrue +%% %% \longtrue + +\usepackage{rotating} +\usepackage{listings} +\usepackage{xcolor} +\lstset{ + basicstyle=\ttfamily, + escapeinside=~~ +} + +\usepackage{microtype} +\usepackage{latexsym,amsmath,amssymb,amsthm} +\usepackage{mathpartir} +\usepackage{tikz} +\usepackage{tikz-cd} +\input{talk-defs} +\newcommand{\ltdyn}{\dynr} +\newcommand{\uarrow}{\mathrel{\rotatebox[origin=c]{-30}{$\leftarrowtail$}}} +\newcommand{\darrow}{\mathrel{\rotatebox[origin=c]{30}{$\twoheadleftarrow$}}} +\renewcommand{\upcast}[2]{\langle{#2}\uarrow{#1}\rangle} +\renewcommand{\dncast}[2]{\langle{#1}\darrow{#2}\rangle} +\newcommand{\gtdyn}{\sqsupseteq} + +\title{Call-by-name Gradual Type Theeory} +\author{Max S. New} +\institute{Northeastern University} +\date{\today} + +\begin{document} + +\begin{frame} + \titlepage + Joint work with Dan Licata, Amal Ahmed + + \notepr{I know the mere mention of call-by-name probably puts you to + sleep so before I show you my type theory I want to explain at the + bigger picture level what I'm trying do to} +\end{frame} + +\begin{frame}{Call-by-name Gradual Type Theory} + \begin{enumerate} + \item A specific type theory for call-by-name gradually typing. + \item A \emph{method} for developing new gradually typed languages. + \end{enumerate} +\end{frame} + +\begin{frame} + \frametitle{Outline} + \tableofcontents +\end{frame} + +\section{Intro to Gradual Typing} + +\begin{frame}[fragile]{Gradually Typing is about Program Evolution} +\begin{columns} +\begin{column}{0.5\textwidth} +\begin{onlyenv}<1> +\begin{footnotesize} +\begin{lstlisting} +view(st) -> { ... } + +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { + st = initial_state; + while true { + evt = input(); + state = update(evt,st); + display(view(st)); + } +} +\end{lstlisting} +\end{footnotesize} +\end{onlyenv} +\begin{onlyenv}<2-> +\begin{footnotesize} +\begin{lstlisting} +view(st) -> { ... } + +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { .. } +\end{lstlisting} +\end{footnotesize} +\end{onlyenv} +\end{column} +\begin{column}{0.1\textwidth} + \begin{onlyenv}<3-> + $\Longrightarrow$ +\end{onlyenv} +\end{column} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<3-> +\begin{lstlisting} +view(st) -> { ... } + +~\colorbox{green}{type State = (Boolean, ?)}~ +~\colorbox{green}{update : (?, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} +\end{footnotesize}} \end{column} +\end{columns} +\end{frame} + +\begin{frame}[fragile]{Gradually Typing is about Program Evolution} +\begin{columns} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<1-> +\begin{lstlisting} +view(st) -> { ... } + +~\colorbox{green}{type State = (Boolean, ?)}~ +~\colorbox{green}{update : (?, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} +\end{footnotesize}} +\end{column} +\begin{column}{0.1\textwidth} +\begin{onlyenv}<2-> + $\Longrightarrow$ +\end{onlyenv} +\end{column} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<2-> +\begin{lstlisting} +view(st) -> { ... } +~\colorbox{green}{type State = (Boolean, Num)}~ +~\colorbox{green}{type Evt = NewData Num }~ + ~\colorbox{green}{| Cancel}~ +~\colorbox{green}{update : (Evt, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Up to date +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} +\end{footnotesize}} +\end{column} +\end{columns} +\begin{onlyenv}<3-> + Code runs at every point, even though only \texttt{update} is typed. +\end{onlyenv} +\end{frame} + +\begin{frame}{Gradual Typing Goals} + Gradually Typed languages let dynamically typed and statically typed + code interoperate within a single language. + \begin{enumerate} + \item Can write scripts, experimental code in the dynamically typed + fragment. + \item Can later add types for documentation, refactoring, bug + squashing. + \item But don't have to add types to the \emph{entire} program to + run it! + \end{enumerate} +\end{frame} + +\begin{frame}{What do the Types Mean?} + How do the types affect the semantics of the program? + Two answers. + \begin{enumerate} + \item Optional Typing: they don't, just run the type erased + programs. Dynamic typing is already safe anyway. + \item Gradual Typing: a type system should give us a type + \emph{soundness} theorem, need to check types at runtime in order + to satisfy that. + \end{enumerate} +\end{frame} + +\begin{frame}{Evaluation in Gradual Typing} + Say we have a function $\lambda x. t : \texttt{bool} \to \texttt{string}$ and + we apply it to a dynamically typed value $v : \dyn$. + \pause + If $v$ is a boolean, fine + \[ (\lambda x. t)(\texttt{true}) \mapsto t[\texttt{true}/x] + \] + \pause + Otherwise, we raise an error + \[ (\lambda x. t)(5) \mapsto \texttt{type error: expected ...} + \] +\end{frame} + +\begin{frame}{Evaluation in Gradual Typing} + What if it's is a higher order function $\lambda x.t : + (\texttt{number} \to \texttt{string})\to \texttt{num}$? + \pause + If $v$ is not a function, we error + \[ (\lambda x.t)(\texttt{true}) \mapsto \texttt{type error:...}\] + But if $v$ is a function? How can we verify that it takes numbers + to strings? + \begin{enumerate} + \item\pause Run the type checker at runtime :( + \item\pause Findler-Felleisen wrapping implementation: + \[ (\lambda x.t)(\lambda y. u) \mapsto t[\lambda y. \texttt{string?}(u)/x]\] + \item \pause Some combination: Annotate every value with best-known type + information, use wrapping if the type information is not precise + enough to tell. + \end{enumerate} +\end{frame} + +\begin{frame}{Evaluation in Gradual Typing} + \begin{enumerate} + \item Which implementation should we choose? + \item How do we implement checking for \texttt{\$NEXT\_BIG\_TYPE\_FEATURE}? + \end{enumerate} + \pause + The type soundness theorem can help! +\end{frame} + +\begin{frame}{Type Soundness} + \begin{enumerate} + \item For simple types, usually defined as a safety property: if $t + : A$ then $t$ diverges or $t$ runs to one of the allowed errors or + $t$ runs to a value that has type $A$. + \item \pause But there are also stronger, \emph{relational} soundness + theorems + \begin{enumerate} + \item Function extensionality: $f_1 = f_2 : A \to B$ if for every + $x : A$, $f_1(x)$ has the same behavior as $f_2(x)$. + \item Universal Polymorphism: $v : \forall \alpha. A$, then + $v[B_1]$ and $v[B_2]$ ``do the same thing''. + \item Local State/Existential Polymorphism: If I don't expose my + representation, I can change it without breaking the program. + \end{enumerate} + \end{enumerate} + \pause + The stronger the type soundness theorem, the more constrained we are + in design. +\end{frame} + +\begin{frame}{Designing Gradual Languages} + \begin{enumerate} + \item Previous work (Gradualizer, Abstracting Gradual Typing) can + ``automatically'' define gradual languages with a \emph{safety} + theorem. + \item But both fail on relational soundness theorems: they produce + violations of \emph{all} our examples: extensionality, + parametricity, representation independence. + \end{enumerate} + \pause + Conclusion: need to think beyond operational semantics +\end{frame} + +\section{CBN Gradual Type Theory} + +\begin{frame}{Gradual Type Theory} + Idea: using \emph{axiomatic semantics}, flip language design on its + head.\\ + \pause + Operational view: + \begin{enumerate} + \item Design operational semantics + \item Try to prove soundness + \end{enumerate} + \pause + Axiomatic view: + \begin{enumerate} + \item Declare by fiat that the relational type soundness theorem and + the gradual guarantee hold. + \item See if we can \emph{derive} operational rules as program + equivalences. + \end{enumerate} + \pause Allows us to explore the limits of the design space: many + languages as models, any theorems we can prove will apply to all of + them. +\end{frame} + +\begin{frame}{Gradual Type Theory: Main Theorem} + \begin{block}{Theorem: Function Casts Uniqueness Principle} + For a call-by-name gradual language, to satisfy the $\eta$ law + and the gradual guarantee, casts between function types + \emph{must be} equivalent to the Findler-Felleisen ``wrapping'' + implementation. + \end{block} + \pause + Two components: + \begin{itemize} + \item Axiomatic Semantics + \item Gradual Guarantee + \end{itemize} +\end{frame} + +\begin{frame}{Axiomatic Semantics of $\lambda$-calculus} + In call-by-name/need: + \begin{mathpar} + \inferrule*[right=$\beta$] + {~} + {(\lambda x. t) t' \equiv t[t'/x]} + + \inferrule*[right=$\eta$] + {t : A \to B} + {t \equiv (\lambda x. t \,x)} + \end{mathpar} + + \pause + $\beta$ says how to run the program.\\\pause + $\eta$ is equivalent to the following + \begin{mathpar} + \inferrule*[right=extensionality] + {\Gamma, x : A \vdash t\,x \equiv t'\,x : B} + {\Gamma \vdash t \equiv t' : A \to B} + \end{mathpar} +\end{frame} + +\begin{frame}{Axiomatic Semantics of $\lambda$-calculus} + In call-by-value, we need to modify the equations: + \begin{mathpar} + \inferrule*[right=$cbv-\beta$] + {~} + {(\lambda x. t) v \equiv t[v/x]} + + \inferrule*[right=$cbv-\eta$] + {v : A \to B} + {v \equiv (\lambda x. v \,x)} + + \end{mathpar} + \pause + We did call-by-name first because we don't need to introdue this + distinction to do functions.\\ + Note that we don't do sums or booleans. +\end{frame} + +\begin{frame}{Gradual Guarantee} + Formal property for saying that there is a ``smooth'' transition + from dynamic to static typing. predictable behavior. +\end{frame} + +\begin{frame}[fragile]{Gradual Guarantee} +\begin{columns} +\begin{column}{0.5\textwidth} +\begin{onlyenv}<1-> +\begin{footnotesize} +\begin{lstlisting} +view(st) -> { ... } + +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { .. } +\end{lstlisting} +\end{footnotesize} +\end{onlyenv} +\end{column} +\begin{column}{0.1\textwidth} + \begin{onlyenv}<1> + $\Longrightarrow$ + \end{onlyenv} + \begin{onlyenv}<2-> + $\gtdyn$ +\end{onlyenv} +\end{column} +\begin{column}{0.5\textwidth}{% +\begin{footnotesize} +\begin{onlyenv}<1-> +\begin{lstlisting} +view(st) -> { ... } + +~\colorbox{green}{type State = (Boolean, ?)}~ +~\colorbox{green}{update : (?, State) -> State}~ +update(event, state) -> { + if first(state) + then ... ; Loading + else ... ; Valid Data +} + +main() -> { ... } +\end{lstlisting} +\end{onlyenv} + \end{footnotesize}} +\end{column} +\end{columns} + \begin{onlyenv}<2-> + Adding types \emph{refines} program behavior + + \end{onlyenv} + +\end{frame} + +\begin{frame}{Gradual Guarantee} + Syntactically, + \[ t \ltdyn t' \] + if $t$ and $t'$ have the same type erasure, and every type in $t$ is + ``less dynamic'' than the corresponding type in $t'$ + + \pause Semantically, if $t, t' : \texttt{Bool}$, then + \begin{enumerate} + \item $t$ runs to a type error (and $t'$ can do anything) + \item $t$ and $t'$ diverge + \item $t \mapsto \texttt{true}$ and $t' \mapsto \texttt{true}$ + \item $t \mapsto \texttt{false}$ and $t' \mapsto \texttt{false}$ + \end{enumerate} +\end{frame} + +\subsection{Syntax of GTT} + +\begin{frame}{Syntax of Gradual Type Theory} + 4 sorts in Gradual Type Theory + \begin{enumerate} + \item Types $A,B,C$ + \item Terms $x_1 : A_1,\ldots, x_n:A_n \vdash t : A$ + \item Type Dynamism: $A \ltdyn B$ means $A$ is less dynamic than $B$ + \item Term Dynamism: $x_1 \ltdyn y_1 : A_1 \ltdyn B_1,\ldots \vdash t \ltdyn u : A \ltdyn B$ + \end{enumerate} + Type and term dynamism axiomatize gradual guarantee. +\end{frame} + +\begin{frame}{Types and Terms} + Add dynamic type, casts, error to typed lambda calculus. + \begin{mathpar} + \begin{array}{rrl} + \text{types} & A,B,C \bnfdef & \dyn \bnfalt A \to B \bnfalt A \times B \bnfalt 1\\ + \text{terms} & s,t,u \bnfdef& x \bnfalt \lambda (x : A). t \bnfalt t u\\&& + \bnfalt (t, u) \bnfalt \pi_i t \bnfalt ()\\&& + \bnfalt \err_{A} \bnfalt \upcast A B t \bnfalt \dncast A B t + \end{array} + \end{mathpar} +\end{frame} +%% \begin{frame}{Terms} +%% Terms are all intrinsically typed, we have variables and +%% substitution as usual. +%% %% and we can also add function symbols as +%% %% primitives. + +%% \begin{mathpar} +%% \inferrule +%% {x:A \in \Gamma} +%% {\Gamma \vdash x : A} + +%% \inferrule +%% {\Gamma \vdash u : B\and +%% \Gamma, x: B \vdash t : A} +%% {\Gamma \vdash t[u/x] : A} + +%% \inferrule +%% {\Gamma \vdash t_1 : A_1,\ldots, \Gamma \vdash t_n : A_n \and f : (A_1,\ldots,A_n) \to B} +%% {\Gamma \vdash f(t_1,\ldots,t_n) : B} +%% \end{mathpar} +%% \end{frame} + +\begin{frame}{Type Dynamism} + \[ A \ltdyn B \] + read: $A$ is ``less dynamic''/''more precise'' than $B$. \\ + \begin{itemize} + \item Syntactic intuition: $B$ has more $\dyn$s in it. + \item Semantic intuition: the type given by $A$ has a more + restrictive interface than $B$. + \end{itemize} + +\end{frame} + +\begin{frame}{Type Dynamism Axioms} + \begin{mathpar} + \inferrule{~}{A \dynr A}\and + \inferrule{A \dynr B \and B \dynr C}{A \dynr C}\and + \inferrule{~}{A \ltdyn \dyn}\\ + \inferrule{A \ltdyn A' \and B \ltdyn B'}{A \to B \ltdyn A' \to B'}\and + \inferrule{A \ltdyn A' \and B \ltdyn B'}{A \times B \ltdyn A' \times B'} + \end{mathpar} + +\end{frame} + +%% \begin{frame}{Term Dynamism INTUITION} +%% Models the refinement relation of gradual guarantee. Here's an example +%% \emph{model} for intuition.\\ +%% \pause +%% For every $A \ltdyn A'$, if +%% \begin{mathpar} +%% \cdot \vdash t \ltdyn t' : A \ltdyn A' +%% \end{mathpar} +%% Then either +%% \begin{enumerate} +%% \item $t$ runs to a type error. +%% \item $t,t'$ both diverge +%% \item $t \Downarrow v$ and $t' \Downarrow v'$ and $v \ltdyn v'$. +%% \end{enumerate} + +%% \notepr{ +%% The $A \ltdyn A'$ restriction ensures we have a sensible +%% way to interpret that. +%% } +%% \end{frame} + +\begin{frame}{Term Dynamism} + \begin{mathpar} + x_1 \ltdyn x_1' : A_1 \ltdyn A_1' ,\ldots \vdash t \ltdyn t' : B \ltdyn B' + \end{mathpar} + read: ``if all the $x_i$s are less than $x_i'$s then $t$ is less than $t'$''.\\ + \pause + Note that it is only well formed if all of the $A_i \ltdyn A_i'$ and + $B \ltdyn B'$. and + \begin{mathpar} + x_1:A_1,\ldots\vdash t : B\\ + x_1':A_1',\ldots\vdash t' : B'\\ + \end{mathpar} + %% We sometimes abbreviate the context ordering as $\Phi \vdash t + %% \ltdyn t' : B \ltdyn B'$. + \pause + This very explicit notation gets very big, so I'll simplify in the + talk. Abbreviate the above as: + \[ + \Phi \vdash t \ltdyn t' : B \ltdyn B' + \] +\end{frame} + +\begin{frame}{Term Dynamism Axioms} + Identity and substitution are monotone. + %% For open terms, identity and substitution properties that should + %% evoke a logical relation interpretation. Substitution says that + %% every term is monotone in its free variables. + \begin{mathpar} + \inferrule + {(x \ltdyn x' : A \ltdyn A') \in \Phi} + {\Phi \vdash x \ltdyn x' : A \ltdyn A'} + + \inferrule + {x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' \and + u \ltdyn u' : A \ltdyn A'} + {t[u/x] \ltdyn t'[u'/x']} + \end{mathpar} + %% \pause Note that these are the same as ordinary identity and + %% substitution but ``monotonically''. +\end{frame} + +\begin{frame}{Term Dynamism} + Like type dynamism, term dynamism has reflexivity and transitivity + rules: + %% note + %% how well-formedness relies on the reflexivity and transitivity of + %% type dynamism + + \begin{mathpar} + \inferrule + {\vdash t : B} + {t \ltdyn t : B \ltdyn B} + + \inferrule + {t \ltdyn t' : B \ltdyn B'\\ + t' \ltdyn t'' : B' \ltdyn B''} + {t \ltdyn t'' : B \ltdyn B''} + \end{mathpar} +\end{frame} + +\begin{frame}[fragile]{Term Dynamism Squares} + These can be nicely visualized as squares: + For $\Phi : \Gamma \ltdyn \Gamma'$, we can think of + \[ \Phi \vdash t \ltdyn t' : A \ltdyn A' \] + as + \[ + \begin{tikzcd} + \Gamma \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "t" ] + & + \Gamma' \arrow[d , "t'"] + \\ + A \arrow[ dashed , r, "\preciser" ] + & + A' + \end{tikzcd} + \] + +\end{frame} + +\begin{frame}[fragile]{Term Dynamism Squares} + Then substitution and transitivity are stacking and sequencing of squares + with variables and reflexivity as identities: + \[ +\begin{tikzcd} +\Gamma \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "t" ] + & +\Gamma \arrow[d , "t"] + \\ +A \arrow[ dashed , r, "\preciser" ] +& +A +\end{tikzcd} +\qquad +\begin{tikzcd} +\Gamma \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "t" ] + & +\Gamma' \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "t'" ] +& +\Gamma'' \arrow[d , "t''"] + \\ +A \arrow[ dashed , r, "\preciser" ] +& +A' \arrow[ dashed , r, "\preciser" ] +& +A'' +\end{tikzcd} +\]\[ +\begin{tikzcd} +\Gamma,x:A \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "x" ] + & +\Gamma',x:A' \arrow[d , "x"] + \\ +A \arrow[ dashed , r, "\preciser" ] +& +A' +\end{tikzcd} +\qquad +\begin{tikzcd} +\Gamma \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "u" ] + & +\Gamma' \arrow[ d, swap, "u'" ] +\\ +\Gamma,x:A \arrow[ dashed , r, "\preciser" ] \arrow[ d, swap, "t" ] + & +\Gamma',x:A' \arrow[ d, swap, "t'" ] +\\ +B \arrow[ dashed , r, "\preciser" ] +& +B' +\end{tikzcd} +\] +\end{frame} + +\begin{frame}{Dynamic Type and Type Error} + The dynamic type is the most dynamic type. + + \[ A \ltdyn \dyn \] + + The type error is the \emph{least} dynamic term + \notepr{recall the intuition I gave before} + + \[ \Phi \vdash \err_A \ltdyn t : A \ltdyn A' \] +\end{frame} +\begin{frame}{Function Type} + First, we give have the usual term structure for the function type + \begin{mathpar} + \inferrule + {\Gamma, x : A \vdash t : B} + {\Gamma \vdash \lambda x:A. t : A \to B} + + \inferrule + {\Gamma \vdash t : A \to B\and \Gamma \vdash u : A} + {\Gamma \vdash t u : B} + \end{mathpar} + \pause + And we add the $\beta, \eta$ laws as ``equi-dynamism'' axioms: + + \begin{mathpar} + \inferrule + {} + {(\lambda x:A. t) u \equidyn t[u/x]} + + \inferrule + {t : A \to B} + {t \equidyn (\lambda x:A. t x)} + \end{mathpar} +\end{frame} + +\begin{frame}{Function Type} + Next to model the gradual guarantee, we assert that everything is monotone. + \begin{mathpar} + \inferrule + {A \ltdyn A' \and B \ltdyn B'} + {A \to B \ltdyn A' \to B'}\\ + \inferrule + {x \ltdyn x' : A \ltdyn A' \vdash t \ltdyn t' : B \ltdyn B'} + {\lambda x:A. t \ltdyn \lambda x':A'.t' } + + \inferrule + {t \ltdyn t' : A \to B \ltdyn A' \to B'\and + u \ltdyn u' : A \ltdyn A' %% \and B \ltdyn B' + } + {t\, u \ltdyn t'\, u' : B \ltdyn B'} + \end{mathpar} +\end{frame} + +\begin{frame}{Product Type} + Product and (unit types) get a similar treatment. + \begin{mathpar} + \inferrule + {\Gamma \vdash t_1 : A_1\\ \Gamma \vdash t_2 : A_2} + {\Gamma \vdash (t_1,t_2) : A_1 \times A_2} + + \inferrule + {\Gamma \vdash t : A_1 \times A_2} + {\Gamma \vdash \pi_i t : A_i}\\ + + \inferrule + {} + {\pi_i(t_1,t_2) \equidyn t_i} + + \inferrule + {t : A_1 \times A_2} + {t \equidyn (\pi_1 t, \pi_2 t)}\\ + + \inferrule{t_1 \ltdyn t_1' : A_1 \ltdyn A_1' \and t_2 \ltdyn t_2' : A_2 \ltdyn A_2'}{(t_1,t_1') \ltdyn (t_2,t_2') : A_1 \times A_2 \ltdyn A_1' \times A_2'} + + \inferrule{t \ltdyn t' : A_1 \times A_2 \ltdyn A_1' \times A_2' \and A_i \ltdyn A_i'}{\pi_i t \ltdyn \pi_i t' : A_i \ltdyn A_i'} + \end{mathpar} +\end{frame} + +\begin{frame}{Casts} + How to define the casts axiomatically? +\end{frame} + +\begin{frame}{Operational View of Casts} + \begin{center} + This was a slide with the wadler findler operational semantics + \includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio]{Wadler-Findler-operational-semantics} + \end{center} +\end{frame} + +\begin{frame}{Casts} + How to define the casts axiomatically? + \begin{itemize} + \item We \emph{characterize} them by a property with respect to $\ltdyn$. + \item We can do this when our casts are \emph{upcasts} or + \emph{downcasts}. + \end{itemize} +\end{frame} + +\begin{frame}{Casts} + When $A \ltdyn B$, we get two functions: the \emph{upcast} $\upcast + A B$ and the \emph{downcast} $\dncast A B$. + + \begin{mathpar} + \inferrule + {\Gamma \vdash t : A \and A \ltdyn B} + {\Gamma \vdash \upcast A B t : B} + + \inferrule + {\Gamma \vdash u : B \and A \ltdyn B} + {\Gamma \vdash \dncast A B u : A} + \end{mathpar} +\end{frame} + +\begin{frame}{Oblique Casts} + Not every cast in gradual typing is an upcast or a downcast, for + instance the cast + + \[ \obcast {\texttt{Bool} \times \dyn} {\dyn \times \texttt{Bool}} \] + + But we can use a trick by Henglein using the dynamic type: + + \[ \obcast{A}{B}t = \dncast B \dyn \upcast A \dyn t\] +\end{frame} + +\begin{frame}{Casts} + How should casts behave? + Intuition: the casted term should act as much like the original term + as possible while still satisfying the new type. + \pause + + So for a downcast, the casted term should refine the original: + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\dncast A B u \ltdyn u : A \ltdyn B} + \end{mathpar} + + \pause But this would be satisfied even by the type error $\err_A$. + \pause + We want to make sure that the casted term has \emph{as much} + behavior as possible: i.e., it's the \emph{biggest} term below + $u$. + \begin{mathpar} + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash t \ltdyn \dncast A B u : A} + \end{mathpar} + \pause Order-theoretic summary: $\dncast A B u$ is the greatest + lower bound of $u$ in $A$. +\end{frame} + +\begin{frame}{Casts} + Upcasts are the exact dual + + For an upcast, the casted term should be \emph{more} dynamic than + the original: + \begin{mathpar} + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \upcast A B t : A \ltdyn B} + \end{mathpar} + + But it has no more behaviors than dictated by the original term, + it's the \emph{least} term above $t$. + + \begin{mathpar} + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash \upcast A B t \ltdyn u : B} + \end{mathpar} + \pause Order-theoretic summary: $\upcast A B u$ is the least upper + bound of $t$ in $B$. +\end{frame} + +\begin{frame}{Casts} + \begin{enumerate} + \item Note that this is a \emph{precise} specification because lubs + and glbs are unique (up to $\equidyn$). + \item Since they are exact \emph{duals} most theorems about one + dualize for the other. + \end{enumerate} +\end{frame} + +\subsection{Theorems in GTT} + +\begin{frame}{Consequences of GTT} + \begin{itemize} + \item Now we have a somewhat rich logic of term dynamism, what can + we \emph{prove} in the theory? + \item Theorems in the theory should be true in + any model, i.e. a simple lazy gradually typed language. + \end{itemize} +\end{frame} + +\begin{frame}{Wrapping Implementation} + We can prove that the wrapping + implementation of contracts is the unique implementation of casts + between function types. I.e. if $A \ltdyn A'$ and $B \ltdyn B'$ and $t : A' \to B'$ then + + \[ + \dncast {A \to B} {A' \to B'} t \equidyn \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + and + \[ + \upcast {A \to B} {A' \to B'} t \equidyn \lambda x':A'. \upcast{B}{B'}(t (\dncast A {A'} x')) + \] +\end{frame} + +\begin{frame}{Wrapping Implementation} + One direction: + To prove + \[ \dncast {A \to B} {A' \to B'} t \ltdyn \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + By extensionality, it is sufficient to show + \[ + x : A \vdash (\dncast {A \to B} {A' \to B'} t) x \ltdyn \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + Then it follows by congruence because: + \begin{enumerate} + \item $(\dncast {A \to B} {A' \to B'} t)\ltdyn t$ + \item $x \ltdyn \upcast A {A'} x$ + \item if $u \ltdyn u'$ then $u \ltdyn \dncast {B} {B'} u'$ + \end{enumerate} + \pause + Other 3 cases are similar/dual. +\end{frame} +\begin{frame}{Wrapping Implementation} + If we weaken eta + \begin{itemize} + \item If $t \ltdyn (\lambda x. t\,x)$: + \[ + \dncast {A \to B} {A' \to B'} t \ltdyn \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + \[ + \upcast {A \to B} {A' \to B'} t \ltdyn \lambda x':A'. \upcast{B}{B'}(t (\dncast A {A'} x')) + \] + \item If $t \gtdyn (\lambda x. t\,x)$: + \[ + \dncast {A \to B} {A' \to B'} t \gtdyn \lambda x:A. \dncast{B}{B'}(t (\upcast A {A'} x)) + \] + \[ + \upcast {A \to B} {A' \to B'} t \gtdyn \lambda x':A'. \upcast{B}{B'}(t (\dncast A {A'} x')) + \] + \end{itemize} +\end{frame} + +\begin{frame}{Casts are Galois Connections} + \notepr{ First, we can show that every pair of upcast and downcast forms a + Galois connection, which} + + Two ``round-trip'' properties: + \pause + Casting down and back up has ``more errors'' + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\upcast A B \dncast A B u \ltdyn u : B} + \end{mathpar} + \pause + And casting up and back down has ``fewer errors'' + \begin{mathpar} + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \dncast A B \upcast A B t : A} + \end{mathpar} + \pause In practice, we probably want to strengthen this to an + equality, so we can add the ``retract axiom'' as well to the system + (though none of the theorems I'll show depend on it): + \begin{mathpar} + \inferrule*[right=RETRACT-AXIOM] + {t : A \and A \ltdyn B} + {\dncast A B \upcast A B t \ltdyn t : A} + \end{mathpar} +\end{frame} + +\begin{frame}{Identity and Composition} + Reflexivity gives the identity cast and transitivity gives the + composition of casts + \begin{enumerate} + \item $\dncast A A t \equidyn t$ + \item If $A \ltdyn B \ltdyn C$, then $\dncast A C t \equidyn \dncast A B \dncast B C t$ + \end{enumerate} + \pause + These are part of the operational semantics for instance: + \begin{align*} + \dncast{\dyn}{\dyn} v &\mapsto v\\ + \dncast{A \to B}{\dyn} v &\mapsto \dncast {A \to B}{\dyn \to \dyn}\dncast {\dyn \to \dyn} \dyn v + \end{align*} +\end{frame} + +%% \begin{frame}{Identity Cast} +%% The upcast/downcast from $A$ to $A$ is the identity function, +%% i.e. $\upcast A A t \equidyn t$. Each direction is an instance of +%% the specification of upcasts: + +%% \begin{mathpar} +%% \inferrule*[right=UR] +%% {~} +%% {t \ltdyn \upcast{A}{A}{t} : A}\and + +%% \inferrule*[right=UL] +%% {t \ltdyn t : A \ltdyn A} +%% {\upcast{A}{A}{t} \ltdyn t : A} +%% \end{mathpar} + +%% Downcast proof is precise dual. +%% \end{frame} + +%% \begin{frame}{Casts (de)compose} +%% If $A \ltdyn B \ltdyn C$ then a cast between $A$ and $B$ factors +%% through $C$: + +%% \begin{mathpar} +%% \inferrule +%% {A \ltdyn B \ltdyn C} +%% {\dncast A B \dncast B C t \equidyn \dncast A C t} +%% \end{mathpar} + +%% Example: casts between dynamic type and function types factor +%% through ``most dynamic'' function type: +%% \begin{mathpar} +%% \dncast {A \to B}{\dyn \to \dyn}\dncast \dyn {\dyn \to \dyn} t \equidyn \dncast{A \to B}{\dyn} t +%% \end{mathpar} +%% This is often a rule of the operational semantics. +%% \end{frame} + +\begin{frame}{Non-theorem: Disjointness of Constructors} + The only standard operational rules we cannot prove are the ones like + \[ + \obcast {A \times B} {A' \to B'} t \mapsto \err + \] + that say that the type constructors $\times, \to, 1$ are all + disjoint. + + We also have a model where it is false. We can always add axioms to + make it true though. + + %% In fact, written as equidynamism instead of stepping, these are + %% \emph{independent} of the other axioms: we can construct adequate + %% models where it's true and ones where it's false(!) + + %% The reason is that our definition of function types, product types + %% et cetera only defines them \emph{up to isomorphism}, not up to cast + %% equivalence. And we know very little about the dynamic type, just + %% that everything embeds in it. +\end{frame} + +%% \begin{frame}{Term Dynamism Reduces to Globular Dynamism} +%% The presence of the casts means that we can reduce all type dynamism +%% to ordering \emph{at the same type}, which is more typical for +%% logical relations. +%% \begin{mathpar} +%% {x \ltdyn y : A \ltdyn B \vdash t \ltdyn u : C \ltdyn D}\\ +%% \iff\\ +%% {x \ltdyn x : A \ltdyn A \vdash t \ltdyn \dncast C D u[\upcast A B x/y] : C \ltdyn C}\\ +%% \iff\\ +%% {x \ltdyn x : A \ltdyn A \vdash \upcast C D t \ltdyn u[\upcast A B x/y] : D \ltdyn D}\\ +%% \iff\\ +%% {y \ltdyn y : B \ltdyn B \vdash \upcast C D t[\dncast A B y/x] \ltdyn u : D \ltdyn D}\\ +%% \iff\\ +%% {y \ltdyn y : B \ltdyn B \vdash t[\dncast A B y/x] \ltdyn \dncast C D u : C \ltdyn C}\\ +%% \end{mathpar} +%% \end{frame} + +\subsection{Models of GTT} + +\begin{frame}{Models of GTT} + Just a theory, need models to show it's any good. + \begin{enumerate} + \item In the paper we have a denotational model that is similar to + Type Racket and Scott's model of untyped lambda calculus. + \item We've made a logical relation model for a call-by-value + language, but don't have the axiomatic call-by-value theory yet. + \end{enumerate} +\end{frame} + +%% \begin{frame}{Models of GTT} +%% Just a theory... +%% \begin{enumerate} +%% \item We give models in which $0 \not\ltdyn 1$ and even better +%% $\Omega \not\ltdyn \err$. Shows the theory is \emph{consistent}. +%% \item We give a domain theoretic model that is an analogue of +%% Scott's semantics of untyped lambda calculus, but where \emph{type +%% error} is not modeled as divergence. +%% \end{enumerate} +%% \end{frame} + +%% \begin{frame}{Categorical Model} +%% We define an abstract notion of model using categories enriched in +%% preorders: encodes the idea that ``everything is monotone'' using +%% enrichment. +%% \end{frame} + +%% \begin{frame}{Types as EP Pairs} +%% Using that, we define an abstract model construction theorem broken +%% up into 3 steps. +%% We summarize the model here. + +%% TODO: table +%% input: language/category with notion of ordering *within each type*, a specified dynamic type and the upcsat/downcast ep pairs +%% output: model of gradual type theory +%% Types: ep pairs into a specific dynamic type +%% Type Dynamism: ep pair factorization +%% Terms: terms +%% Term Dynamism: insert ep pairs, then ordering +%% \end{frame} + +\section{Conclusion} + +\begin{frame}{Summary} + \begin{enumerate} + \item By axiomatizing term precision/dynamism, we can give casts a + specification. + \item Gradual guarantee + extensionality = wrapping, so not wrapping + implies either no gradual guarantee or no extensionality. + \end{enumerate} +\end{frame} + +\begin{frame}{Objections} + \begin{enumerate} + \item Should the language have $\eta$ for arbitrary terms or just + those that never error? + \begin{enumerate} + \item Violated by ``eager'' cast semantics. + \item Does it make sense for eta to be an expansion? Then maybe + wrapping is the ``maximal'' implementation instead of the unique + one. + \end{enumerate} + \item Gradual guarantee is, like parametricity, brittle: can't catch + type errors! But maybe it is still helpful to design cast calculi + even if we throw it away later. + \end{enumerate} +\end{frame} + +\begin{frame}{Future Work} + \begin{enumerate} + \item Call-by-value! Already have a logical relation, but want the + nice axiomatic presentation. + \item Relation to Abstracting Gradual Typing: is AGT an abstract + interpretation of GTT? + \item Polymorphism: can we use the techniques here to ``explain'' + the complex cast calculi for dynamic typing? + \item State: what are the \emph{program equivalence} tradeoffs + between monotonic and wrapping references? + \end{enumerate} +\end{frame} + +\begin{frame}{Conclusion} + \begin{itemize} + \item Gradualize Axiomatic Semantics + \item\pause Paper accepted to FSCD '18, extended version available on + arxiv: ``Call-by-Name Gradual Type Theory'' by New and Licata + \end{itemize} + + \pause + \begin{mathpar} + \inferrule + {u : B \and A \ltdyn B} + {\dncast A B u \ltdyn u : A \ltdyn B} + + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash t \ltdyn \dncast A B u : A}\\ + + \inferrule + {t : A \and A \ltdyn B} + {t \ltdyn \upcast A B t : A \ltdyn B} + + \inferrule + {\Phi \vdash t \ltdyn u : A \ltdyn B} + {\Phi \vdash \upcast A B t \ltdyn u : B} + \end{mathpar} + +\end{frame} +\end{document} + +%% Local Variables: +%% compile-command: "pdflatex talk.tex" +%% End: diff --git a/talk/cmu-fall-2018/outline.org b/talk/cmu-fall-2018/outline.org new file mode 100644 index 0000000..b299c0f --- /dev/null +++ b/talk/cmu-fall-2018/outline.org @@ -0,0 +1,66 @@ +* Outline + 1) The problem: gradual typing is good, but the state of the art is + questionable. Cast semantics is in flux. Re-use old slides + - Introduce graduality as a key + 2) CBN GTT: simple axiomatics, show the proof for function types, + mention category theoretic connections. Re-use old slides + - Mention double category semantics briefly, which are closely + related to parametricity + 3) CBPV GTT: captures CBV and CBN, refines analysis of casts: + upcasts are pure values, downcasts are stacks. Make new slides! +* TODO Today + - [ ] (Before the talk). Re-org for parts 1 and 2 of the talk. Merge + NEU and FSCD talks + - [ ] (This afternoon) flesh out intro + - [ ] (This afternoon) Add some CBPV slides at the end + - [ ] (Sundary) Finish slides, practice. +* Expanded Outline +** Intro + - Gradual typing is about migration from dynamic to static typing + styles + - the "dynamics" of GT + - Very basic account of mechanics + - very simple overview of the "mechanics" of GT, no details + - But the state of the art is *nasty* + - many different cast semantics have been proposed + - just present a pile of rules + - Our approach: type theoretic, axiomatic. Take the properties of + types and casts that we want, decompose complex structures, + *derive* the complex definitions from a small type-theoretic core + - Key ingredient: *graduality* + - Key obstacle: dynamic typing inherently involves *effects*, so + we need equational theories that are sound even in the presence + of effects +** CBN GTT + - Motivate: + - Based on FSCD 2018 + - start with a simple, CBN lambda calculus, only do the negative + types. This works even with effects, but not + - Syntax + - types, type precision + - terms, term precision + - Casts + - Theorems + - Casts form galois connections, in practice strengthen this to + ep pair + - Uniqueness theorems: casts must be their functorial actions + - Semantics + - Category internal to preorders: well-studied under the name + "double categories" + - A double category that has all upcasts/downcasts is called a + "proarrow equipment" or a "framed bicategory", some nice work + by Mike Shulman on this +** CBPV GTT (conditionally accepted to POPL 2019) + - Motivate: + - most gradual typing is CBV + - want to prove program equivalences in the presence of effects + - Intro to CBPV + - Casts as values/stacks + - same definition(!) + - recover + - Used non-gradual CBPV as a metalanguage for defining models + - Dynamic type for Scheme in CBPV + - Experience + - CBPV rocks: keep the nice equational theory, but handle effects + - The syntax guided us naturally to the right solution + - type theoretic approach plays nicely with others diff --git a/talk/cmu-fall-2018/talk-defs.tex b/talk/cmu-fall-2018/talk-defs.tex new file mode 100644 index 0000000..28094d9 --- /dev/null +++ b/talk/cmu-fall-2018/talk-defs.tex @@ -0,0 +1,55 @@ +\usepackage{stmaryrd} +\usepackage{graphicx} + +\newcommand{\preciser}{\sqsubseteq} +\newcommand{\dynr}{\sqsubseteq} +\newcommand{\equiprecise}{\mathrel{\sqsupseteq\sqsubseteq}} +\newcommand{\equidyn}{\mathrel{\sqsupseteq\sqsubseteq}} +\newcommand{\vpreciser}{\rotatebox[origin=c]{-90}{$\preciser$}} + +\newcommand{\dyn}{{?}} +\newcommand{\obcast}[2]{\langle{#2}\Leftarrow{#1}\rangle} +\newcommand{\upcast}[2]{\langle{#2}\leftarrowtail{#1}\rangle} +\newcommand{\upcastpr}[2]{\langle{#2}\hookleftarrow{#1}\rangle} +\newcommand{\dncast}[2]{\langle{#1}\twoheadleftarrow{#2}\rangle} + +\newcommand{\type}{\,\text{type}} +\newcommand{\ctx}{\,\text{ctx}} +\newcommand{\dom}{\text{dom}} + +\newcommand{\cat}{\mathbb} +\newcommand{\Ctx}{\text{Ctx}} +\newcommand{\Multi}{\text{Multi}} +\newcommand{\PTT}{\text{PTT}} +\newcommand{\PTTS}{PttS } +\newcommand{\CPM}{CPM } +\newcommand{\GTT}{\text{GTT}} +\newcommand{\pocat}{\text{PoCat}} +\newcommand{\sem}[1]{\llbracket{#1}\rrbracket} +\newcommand{\interp}[1]{\llparenthesis{#1}\rrparenthesis} +\newcommand{\floor}[1]{\lfloor{#1}\rfloor} + +\newcommand{\err}{\mho} +\newcommand{\id}{\text{id}} +\newcommand{\app}{\text{app}} +\newcommand{\pair}{\text{pair}} +\newcommand{\unit}{\text{unit}} +\newcommand{\tl}{\triangleleft} + +\newcommand{\coreflection}{\text{CoReflect}} + +% mathpartir hacks +\newcommand{\axiom}[2]{\inferrule*[Right=#1]{~}{#2}} +\newcommand{\invinferrule}{{\mprset}{fraction={===}}\inferrule} + +\newcommand{\funupcast}[6]{\lambda #6 : #3. \upcast{#2}{#4}{(#5 (\dncast{#1}{#3}{#6}))}} +\newcommand{\fundncast}[6]{\lambda #6 : #3. \dncast{#2}{#4}{(#5 (\upcast{#1}{#3}{#6}))}} + +\newcommand{\produpcast}[5]{(\upcast{#1}{#3}\pi_0{#5}, \upcast{#2}{#4}\pi_0{#5})} + +\newcommand{\bnfalt}{\mathrel{\bf \,\mid\,}} +\newcommand{\bnfdef}{\mathrel{\bf ::=}} + +%% Local Variables: +%% compile-command: "latexmk -c main.tex; latexmk -pdf main.tex" +%% End: diff --git a/talk/popl-winter-2019/outline.org b/talk/popl-winter-2019/outline.org new file mode 100644 index 0000000..c3e9824 --- /dev/null +++ b/talk/popl-winter-2019/outline.org @@ -0,0 +1,88 @@ +* Technical Bits + - How much CBPV do we discuss? (little) + - Do we show the specification for casts (yes) + - Do we show the dynamic computation type? (maybe) +* Outline +** Intro + - Goals of GT (copy the ICFP slides here) + - Issue: what should the dynamic semantics be? We want the types to + remain meaningful + - This paper: Axiomatize our desires: typed reasoning/optimization + and gradual evolution. Explore the consequences + - Main result: Strong typed based reasoning *uniquely determines* + most of the cast semantics. + - Prove for CBV, CBN and mix of these semantics for simple types. +** Background +*** Graduality + - Graduality says that changing the types of a program to be more + *precise* means that the resulting program should have more + refined behavior. +*** CBPV + - Gradual Typing inherently involves + - Higher-order dynamic types ? -> ? <| ? so we have recursion + - Runtime errors + - Need an axiomatic framework that accommodates effects + - Our savior: CBPV + - subsumes CBV and CBN in a strong sense + - What is CBPV? + - An effect calculus into which call-by-value and call-by-name + languages can be embedded + - Based on beta, eta axioms: lambda calculus for effects + - Generalizes Moggi's metalanguage from (strong) monads to + adjunctions +** Axioms (Theory) +** Theorems (Consequences) +** Models? +** Conclusion +* Real Outline +** Intro +*** What is Gradual Typing for? +*** The problem: many different semantics, trade off soundness +*** Design Principles have emerged +*** Our Contribution: Axiomatize these design principles, show that some of them have *unique solutions* +** What are the Principles? +*** Graduality +*** Type Based Reasoning +**** The Humble Eta Principle(s) + - Naive Eta Principle + - CBV Eta Principle for functions + - Show Sum or Bool Eta principle + - fails in CBN +** Axiomatizing the Principles +*** Extend CBPV + - 3 ingredients: ordering, dynamic and casts + - For each type add congruence, beta, eta + - Emphasize *modularity* +** Consequences of the Axioms +*** Laundry List of Theorems + - Purity of upcasts/dual purity of downcsats (related to + Wadler-Findler/TH-Felleisen's blame soundness property) + - applications to optimization? + - Compositionality + - identity, composition + - EP Pairs + - Uniqueness Principles + - End on a bang: this works for call-by-value, call-by-name, etc +** More Inside + - More theorems + - Soundness of the axioms + - We construct (many) models by translation to CBPV. Implement + the dynamic types using recursive value/computation types + - Dynamic value type as a sum, Dynamic comp type as a product + - Logical Relation to prove soundness of the ordering wrt an + operational semantics +** Related/Future Work + - AGT: doesn't satisfy eta principles, can it be modified to do so? + - Greenman-Felleisen: show weakened principles but not blah + - Degen-Thiemmann? +** Future Work + - Inductive, Coinductive, Recursive types should all have a similar + theorem. + - One of our models is quite similar to Scheme, maybe CBPV should + be taken more seriously as an intermediate lang? +** Conclusions + - Axiomatic Approach shows us the strength of the graduality concept + - but also that it is a reimagining of EP pairs + - CBPV is a convenient metalanguage to prove results for many + different evaluation orders + - Our models suggest new diff --git a/talk/popl-winter-2019/talk-from-scratch.key b/talk/popl-winter-2019/talk-from-scratch.key new file mode 100644 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