From 8329fdff26f1808fb097aee87e19160fbc24596f Mon Sep 17 00:00:00 2001 From: "ajeffrey@roblox.com" Date: Fri, 16 Jul 2021 18:28:05 -0500 Subject: [PATCH] Yet more tidying of the HATRA paper --- papers/hatra21/hatra21.pdf | Bin 3014622 -> 3012758 bytes papers/hatra21/hatra21.tex | 85 ++++++++++++++++++++----------------- 2 files changed, 46 insertions(+), 39 deletions(-) diff --git a/papers/hatra21/hatra21.pdf b/papers/hatra21/hatra21.pdf index 4f3bc0c1fc9c30ced048d87c0b0ddb5e3986d58b..c5127d9811598d943272548d727e36d16eb23128 100644 GIT binary patch delta 85532 zcmX__V|OMDv}9x3wr$(CZQFig+h)hMZ5thRY}=W4=B_*E+x`QmR_)sTZ3j-jZwC$x z1r7xX1q}rY1rLP?g$#uXg${)Yg$;!Zg%3ptMGQp>MGi#?MGZv@MGwUY#SFy?#SX;@ z#SO&^#SbM2B@87BB@QJCB@HDDB@d-&p*&at=px1P6HlqOwkzH zVw~=uM++Pne(tiz7fFn@Q65HTEs99equr1tgQKMA9oAw9L)zVSJ&4I@93Q)n z)ltx9*;`nV0X@}-;PHO?r=J*tm+(Ixyf~k2^S)nQ+EsL31N;rAJEB|Y5y$A5e5r$+ zj2Vq^4t;>AkbcZBl>Rn|mrz7vv@VXYpn6 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+\newcommand{\squnder}[1]{\color{red}\underline{{\color{black}#1}}\color{black}} \newcommand{\infer}[2]{\frac{\textstyle#1}{\textstyle#2}} \newcommand{\erase}{\mathrm{erase}} \newcommand{\evCtx}{\mathcal{E}} \newcommand{\NIL}{\mathsf{nil}} \newcommand{\TRUE}{\mathsf{true}} \newcommand{\FALSE}{\mathsf{false}} +\newcommand{\NUMBER}{\mathsf{number}} \newcommand{\ERROR}{\mathsf{error}} \newcommand{\IF}{\mathsf{if}\,} \newcommand{\THEN}{\,\mathsf{then}\,} @@ -153,7 +154,7 @@ even to creators who are not explicitly providing types. \section{Types} \subsection{Infallible types} -Goal: \emph{support type-driven tools in all programs}. +Goal: \emph{support type-driven tools for all programs}. Programs spend much of their time under development in an incomplete state, even if the final arifact is well-typed. Type-driven tools should support this, by providing type information for all programs. @@ -161,10 +162,10 @@ An analogy is infallible parsers, which perform error recovery and provide an AS Program analysis can still flag type errors, for example with red squiggly underlining. Formalizing this, rather than a judgement -$\Gamma\vdash M:T$, for an input terms $M$, there is a judgement -$\Gamma \vdash M \Rightarrow M' : T$ where $M'$ is an output term -where some subterms are flagged $\squnder{M}$. Write $\erase(M)$ -for the result of erasing flagged type errors: $\erase(\squnder{M}) = \erase(M)$. +$\Gamma\vdash M:T$, for an input term $M$, there is a judgement +$\Gamma \vdash M \Rightarrow N : T$ where $N$ is an output term +where some subterms are \emph{flagged} $\squnder{N}$. Write $\erase(N)$ +for the result of erasing flaggings: $\erase(\squnder{N}) = \erase(N)$. %% For example the usual %% type rules for field access becomes: @@ -194,9 +195,9 @@ for the result of erasing flagged type errors: $\erase(\squnder{M}) = \erase(M)$ The goal of infallible types is that every term can be typed: \begin{itemize} \item \emph{Typability}: for every $M$ and $\Gamma$, - there are $M'$ and $T$ such that $\Gamma \vdash M \Rightarrow M' : T$. -\item \emph{Erasure}: if $\Gamma \vdash M \Rightarrow M' : T$ - then $\erase(M) = \erase(M')$ + there are $N$ and $T$ such that $\Gamma \vdash M \Rightarrow N : T$. +\item \emph{Erasure}: if $\Gamma \vdash M \Rightarrow N : T$ + then $\erase(M) = \erase(N)$ \end{itemize} Some issues raised by infallible types: \begin{itemize} @@ -209,7 +210,9 @@ Some issues raised by infallible types: \end{itemize} Related work: lots on type error reporting~\cite{???}, and on heuristics for program repair~\cite{???}, but not type repair, or on -the semantics of programs with type errors. +the semantics of programs with type errors. Many compilers perform +error recovery during typechecking, but do not provide a semantiocs +for programs with type errors. \subsection{Strict types} @@ -225,19 +228,26 @@ progress. This requires: \item \emph{Operational semantics}: a reduction judgement $M \rightarrow N$ on terms. \item \emph{Values}: a subset of terms representing a successfully completed evaluation. \end{itemize} -We then represent error states as stuck states (terms that are not +Error states at runtime are represented as stuck states (terms that are not values but cannot reduce), and showing that no well-typed program is stuck. This is not true if typing is infallible, but can fairly -straightforwardly be adapted: +straightforwardly be adapted, by extending the operational semantics to flagged terms: \begin{itemize} -\item \emph{Progress}: if ${} \vdash M \Rightarrow M'$, then either $M \rightarrow N$ or $M$ is a value or $M'$ is flagged. -\item \emph{Preservation}: if ${} \vdash M \Rightarrow M'$ and $M \rightarrow N$ and ${} \vdash N \Rightarrow N'$ and $N'$ is flagged, then $M'$ is flagged. +\item If $M \rightarrow M'$ then $\squnder{M} \rightarrow \squnder{M'}$. +\item If $V$ is a value, then $\squnder{V} \rightarrow V$. +\end{itemize} +and defining: +\begin{itemize} +\item \emph{Progress}: if ${} \vdash M \Rightarrow N$, then either $N \rightarrow N'$ or $N$ is a value or $N$ has a flagged subterm. +\item \emph{Preservation}: if $N \rightarrow N'$ and $N'$ has a flagged subterm, then $N$ has a flagged subterm. \end{itemize} Some issues raised by infallible types: \begin{itemize} \item How should the judgements and their metatheory be set up? -\item How should generic functions be handled? -\item What does type inference or bidirectional typechecking look like in this setting? +\item How should type inference and generic functions be handled? +\item Is the operational semantics of flagged values + ($\squnder{V} \rightarrow V$) the right one? +\item Will higher-order code require wrappers on functions? \end{itemize} Related work: blame analysis~\cite{???}. @@ -246,8 +256,9 @@ Related work: blame analysis~\cite{???}. Goal: \emph{no false positives.} For developers who are not interested in defect detection, type-driven -tools such as autocomplete can still be useful, and type-directed -development can still be useful. For such developers, Luau provides a +tools and techniques such as autocomplete, API documentation +and support for refactoring can still be useful. +For such developers, Luau provides a \emph{nonstict mode}, which we hope will eventually be useful for all developers. This does \emph{not} aim for soundness, but instead has the goal of ``no false positives``, in the sense that any flagged code @@ -255,21 +266,18 @@ is guaranteed to produce a runtime error when executed. On the face of it, this is undecidable, since a program such as $(\IF f() \THEN \ERROR \END)$ will produce a runtime error when $f()$ is -$\TRUE$, but we can aim for a weaker property, thjat all flagged code +$\TRUE$, but we can aim for a weaker property, that all flagged code is either dead code or will produce an error. Either of these is a defect, so deserves flagging, even if the tool does not know -which. For example, using this definition -$(\IF f() \THEN \squnder{\ERROR} \END)$ is not a false positive, but -$(\squnder{\IF f() \THEN \ERROR \END})$ might be. +which reason applies. For example, using this definition +$(\IF f() \THEN \squnder{\ERROR} \END)$ is not a false positive. We can formalize this by defining an \emph{evaluation context} -$\evCtx[\bullet]$, defining a \emph{redex} of $M$ to be a subterm $N$ -such that $M = \evCtx[N]$, and defining $M{\Uparrow}$ when there is no -value $V$ such that $M \rightarrow^* V$. We can then define: - +$\evCtx[\bullet]$, and saying $M$ is \emph{incorrectly flagged} +if it is of the form $\evCtx[\squnder{V}]$. We can then define: \begin{itemize} -\item \emph{Snappy name}: if ${} \vdash M \Rightarrow M'$ and $M'$ has a flagged redex - then $M{\Uparrow}$. +\item \emph{Correct flagging}: if ${} \vdash M \Rightarrow M'$ and + $M' \rightarrow^* N'$ then $N'$ is correctly flagged. \end{itemize} Some issues raised by nonstrict types: \begin{itemize} @@ -278,21 +286,20 @@ Some issues raised by nonstrict types: flagging will often move from function definitions to call sites. \item This definition will not allow an unchecked use of an optional value - to be flagged, for example if we define $\FUNCTION g()\,\IF f() \THEN \RETURN 5 \ELSE \RETURN \NIL \END$ - then a strict type system can flag $1 + g()$ but a nonstrict one cannot. + to be flagged, for example if $f() : \NUMBER?$ (meaning $f$ may optionally return a number) + then a strict type system can flag $1 + f()$ but a nonstrict one cannot. \item Property update of tables in languages like Luau always succeeds - (the property is inserted if it did not exist), which interacts - badly with compositional analysis. For example, if module $A$ - exports a table $t=\{p=5\}$, module $B$ assumes $t.p$ is a number, - and module $C$ sets $t.p=\FALSE$, then we cannot flag $C$ since the - assignment will succeed, and we cannot flag $A$ or $B$ without some - whole-program analysis. + (the property is inserted if it did not exist), and so functions which + update properties cannot be flagged. + +\item Does nonstrict typing require whole program analysis, + to find all the possible types a property might be updated with? \item The natural formulation of function types in a nonstrict setting - is that of~\cite{???}: if $f: T \rightarrow U$ and $f(v) \rightarrow^* w$ - then $v:T$ and $w:U$. This formulation is \emph{covariant} in $T$, - not \emph{contavariant}. + is that of~\cite{???}: if $f: T \rightarrow U$ and $f(V) \rightarrow^* W$ + then $V:T$ and $W:U$. This formulation is \emph{covariant} in $T$, + not \emph{contavariant}; what impact does this have? \end{itemize} Related work: success types~\cite{???} and incorrectness logic~\cite{???}.